Crystal Lab › The 230 space groups
The 230 space groups
3D crystal patterns. Open a group to see its symmetry elements drawn in the cell, its operations, and its Wyckoff positions with a model atom you can move.
The 230 space groups · The 17 plane groups · The 7 frieze groups · The 75 rod groups · The 80 layer groups
Triclinic (2)
Monoclinic (13)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 3 | P 1 2 1 | 2 | 5 | 3 | 4 prototypes |
| 4 | P 1 21 1 | 2 | 1 | 3 | 11 prototypes |
| 5 | C 1 2 1 | 4 | 3 | 9 | 11 prototypes |
| 6 | P 1 m 1 | 2 | 3 | 3 | 3 prototypes |
| 7 | P 1 c 1 | 2 | 1 | 9 | 11 prototypes |
| 8 | C 1 m 1 | 4 | 2 | 9 | 8 prototypes |
| 9 | C 1 c 1 | 4 | 1 | 18 | 11 prototypes |
| 10 | P 1 2/m 1 | 4 | 15 | 3 | 7 prototypes |
| 11 | P 1 21/m 1 | 4 | 6 | 3 | 19 prototypes |
| 12 | C 1 2/m 1 | 8 | 10 | 9 | YCl₃ · 79 prototypes |
| 13 | P 1 2/c 1 | 4 | 7 | 9 | 13 prototypes |
| 14 | P 1 21/c 1 | 4 | 5 | 9 | SiO₂ (MFI), Na₃AlH₆ · 98 prototypes |
| 15 | C 1 2/c 1 | 8 | 6 | 18 | 75 prototypes |
Orthorhombic (59)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 16 | P 2 2 2 | 4 | 21 | 1 | 2 prototypes |
| 17 | P 2 2 21 | 4 | 5 | 3 | 3 prototypes |
| 18 | P 21 21 2 | 4 | 3 | 3 | 4 prototypes |
| 19 | P 21 21 21 | 4 | 1 | 1 | 19 prototypes |
| 20 | C 2 2 21 | 8 | 3 | 3 | 7 prototypes |
| 21 | C 2 2 2 | 8 | 12 | 3 | 3 prototypes |
| 22 | F 2 2 2 | 16 | 11 | 1 | 2 prototypes |
| 23 | I 2 2 2 | 8 | 11 | 1 | 4 prototypes |
| 24 | I 21 21 21 | 8 | 4 | 1 | 2 prototypes |
| 25 | P m m 2 | 4 | 9 | 3 | 2 prototypes |
| 26 | P m c 21 | 4 | 3 | 6 | 6 prototypes |
| 27 | P c c 2 | 4 | 5 | 3 | 2 prototypes |
| 28 | P m a 2 | 4 | 4 | 6 | 2 prototypes |
| 29 | P c a 21 | 4 | 1 | 6 | 8 prototypes |
| 30 | P n c 2 | 4 | 3 | 6 | 3 prototypes |
| 31 | P m n 21 | 4 | 2 | 6 | 12 prototypes |
| 32 | P b a 2 | 4 | 3 | 3 | 3 prototypes |
| 33 | P n a 21 | 4 | 1 | 6 | 13 prototypes |
| 34 | P n n 2 | 4 | 3 | 3 | 4 prototypes |
| 35 | C m m 2 | 8 | 6 | 3 | 1 prototype |
| 36 | C m c 21 | 8 | 2 | 6 | 17 prototypes |
| 37 | C c c 2 | 8 | 4 | 3 | 2 prototypes |
| 38 | A m m 2 | 8 | 6 | 6 | 8 prototypes |
| 39 | A e m 2 | 8 | 4 | 6 | 4 prototypes |
| 40 | A m a 2 | 8 | 3 | 6 | 5 prototypes |
| 41 | A e a 2 | 8 | 2 | 6 | 4 prototypes |
| 42 | F m m 2 | 16 | 5 | 3 | 3 prototypes |
| 43 | F d d 2 | 16 | 2 | 3 | 8 prototypes |
| 44 | I m m 2 | 8 | 5 | 3 | 5 prototypes |
| 45 | I b a 2 | 8 | 3 | 3 | 3 prototypes |
| 46 | I m a 2 | 8 | 3 | 6 | 7 prototypes |
| 47 | P m m m | 8 | 27 | 1 | YBa₂Cu₃O₇ · 3 prototypes |
| 48 | P n n n :2 | 8 | 13 | 2 | 2 prototypes |
| 49 | P c c m | 8 | 18 | 3 | 3 prototypes |
| 50 | P b a n :2 | 8 | 13 | 6 | 2 prototypes |
| 51 | P m m a | 8 | 12 | 6 | 9 prototypes |
| 52 | P n n a | 8 | 5 | 6 | 4 prototypes |
| 53 | P m n a | 8 | 9 | 6 | 4 prototypes |
| 54 | P c c a | 8 | 6 | 6 | 3 prototypes |
| 55 | P b a m | 8 | 9 | 3 | 17 prototypes |
| 56 | P c c n | 8 | 5 | 3 | 4 prototypes |
| 57 | P b c m | 8 | 5 | 6 | 12 prototypes |
| 58 | P n n m | 8 | 8 | 3 | FeS₂ · 16 prototypes |
| 59 | P m m n :2 | 8 | 7 | 6 | 11 prototypes |
| 60 | P b c n | 8 | 4 | 6 | 11 prototypes |
| 61 | P b c a | 8 | 3 | 2 | 27 prototypes |
| 62 | P n m a | 8 | 4 | 6 | LiFePO₄, Mg₂SiO₄, Li₁₄Zn(GeO₄)₄, Al(OH)(BDC) · 113 prototypes |
| 63 | C m c m | 16 | 8 | 6 | 60 prototypes |
| 64 | C m c e | 16 | 7 | 6 | 23 prototypes |
| 65 | C m m m | 16 | 18 | 3 | 19 prototypes |
| 66 | C c c m | 16 | 13 | 3 | 6 prototypes |
| 67 | C m m e | 16 | 15 | 6 | 7 prototypes |
| 68 | C c c e :2 | 16 | 9 | 12 | 1 prototype |
| 69 | F m m m | 32 | 16 | 1 | 10 prototypes |
| 70 | F d d d :2 | 32 | 8 | 2 | 9 prototypes |
| 71 | I m m m | 16 | 15 | 1 | 13 prototypes |
| 72 | I b a m | 16 | 11 | 3 | 3 prototypes |
| 73 | I b c a | 16 | 6 | 2 | 2 prototypes |
| 74 | I m m a | 16 | 10 | 6 | 13 prototypes |
Tetragonal (68)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 75 | P 4 | 4 | 4 | 2 | 2 prototypes |
| 76 | P 41 | 4 | 1 | 2 | 3 prototypes |
| 77 | P 42 | 4 | 4 | 2 | 3 prototypes |
| 78 | P 43 | 4 | 1 | 2 | 2 prototypes |
| 79 | I 4 | 8 | 3 | 2 | 2 prototypes |
| 80 | I 41 | 8 | 2 | 2 | 1 prototype |
| 81 | P -4 | 4 | 8 | 2 | 1 prototype |
| 82 | I -4 | 8 | 7 | 2 | Ag₂HgI₄ · 6 prototypes |
| 83 | P 4/m | 8 | 12 | 2 | 2 prototypes |
| 84 | P 42/m | 8 | 11 | 2 | 2 prototypes |
| 85 | P 4/n :2 | 8 | 7 | 4 | 7 prototypes |
| 86 | P 42/n :2 | 8 | 7 | 4 | 6 prototypes |
| 87 | I 4/m | 16 | 9 | 2 | 9 prototypes |
| 88 | I 41/a :2 | 16 | 6 | 4 | CaWO₄, NaAlH₄ · 5 prototypes |
| 89 | P 4 2 2 | 8 | 16 | 2 | 2 prototypes |
| 90 | P 4 21 2 | 8 | 7 | 2 | 4 prototypes |
| 91 | P 41 2 2 | 8 | 4 | 2 | 1 prototype |
| 92 | P 41 21 2 | 8 | 2 | 2 | SiO₂ · 9 prototypes |
| 93 | P 42 2 2 | 8 | 16 | 2 | 1 prototype |
| 94 | P 42 21 2 | 8 | 7 | 2 | 2 prototypes |
| 95 | P 43 2 2 | 8 | 4 | 2 | 1 prototype |
| 96 | P 43 21 2 | 8 | 2 | 2 | 3 prototypes |
| 97 | I 4 2 2 | 16 | 11 | 2 | 2 prototypes |
| 98 | I 41 2 2 | 16 | 7 | 2 | 2 prototypes |
| 99 | P 4 m m | 8 | 7 | 2 | BaTiO₃ · 2 prototypes |
| 100 | P 4 b m | 8 | 4 | 2 | 4 prototypes |
| 101 | P 42 c m | 8 | 5 | 2 | 2 prototypes |
| 102 | P 42 n m | 8 | 4 | 2 | 2 prototypes |
| 103 | P 4 c c | 8 | 4 | 2 | 2 prototypes |
| 104 | P 4 n c | 8 | 3 | 2 | 2 prototypes |
| 105 | P 42 m c | 8 | 6 | 2 | 1 prototype |
| 106 | P 42 b c | 8 | 3 | 2 | 1 prototype |
| 107 | I 4 m m | 16 | 5 | 2 | 4 prototypes |
| 108 | I 4 c m | 16 | 4 | 2 | 2 prototypes |
| 109 | I 41 m d | 16 | 3 | 2 | 2 prototypes |
| 110 | I 41 c d | 16 | 2 | 2 | 1 prototype |
| 111 | P -4 2 m | 8 | 15 | 2 | 3 prototypes |
| 112 | P -4 2 c | 8 | 14 | 2 | 1 prototype |
| 113 | P -4 21 m | 8 | 6 | 2 | 6 prototypes |
| 114 | P -4 21 c | 8 | 5 | 2 | 5 prototypes |
| 115 | P -4 m 2 | 8 | 12 | 2 | 5 prototypes |
| 116 | P -4 c 2 | 8 | 10 | 2 | 3 prototypes |
| 117 | P -4 b 2 | 8 | 9 | 2 | 1 prototype |
| 118 | P -4 n 2 | 8 | 9 | 2 | 2 prototypes |
| 119 | I -4 m 2 | 16 | 10 | 2 | 6 prototypes |
| 120 | I -4 c 2 | 16 | 9 | 2 | 3 prototypes |
| 121 | I -4 2 m | 16 | 10 | 2 | 7 prototypes |
| 122 | I -4 2 d | 16 | 5 | 2 | CuFeS₂ · 7 prototypes |
| 123 | P 4/m m m | 16 | 21 | 2 | Sr₃Fe₂O₇ · 31 prototypes |
| 124 | P 4/m c c | 16 | 14 | 2 | 3 prototypes |
| 125 | P 4/n b m :2 | 16 | 14 | 4 | 5 prototypes |
| 126 | P 4/n n c :2 | 16 | 11 | 4 | 3 prototypes |
| 127 | P 4/m b m | 16 | 12 | 2 | 10 prototypes |
| 128 | P 4/m n c | 16 | 9 | 2 | 5 prototypes |
| 129 | P 4/n m m :2 | 16 | 11 | 4 | LaFeAsO · 26 prototypes |
| 130 | P 4/n c c :2 | 16 | 7 | 4 | CaCuSi₄O₁₀ · 6 prototypes |
| 131 | P 42/m m c | 16 | 18 | 2 | 3 prototypes |
| 132 | P 42/m c m | 16 | 16 | 2 | 2 prototypes |
| 133 | P 42/n b c :2 | 16 | 11 | 4 | 2 prototypes |
| 134 | P 42/n n m :2 | 16 | 14 | 4 | 1 prototype |
| 135 | P 42/m b c | 16 | 9 | 2 | 4 prototypes |
| 136 | P 42/m n m | 16 | 11 | 2 | TiO₂, MgH₂ · 18 prototypes |
| 137 | P 42/n m c :2 | 16 | 8 | 4 | 7 prototypes |
| 138 | P 42/n c m :2 | 16 | 10 | 4 | 3 prototypes |
| 139 | I 4/m m m | 32 | 15 | 2 | K₂NiF₄, Tl₂Ba₂Ca₂Cu₃O₁₀ · 51 prototypes |
| 140 | I 4/m c m | 32 | 13 | 2 | 18 prototypes |
| 141 | I 41/a m d :2 | 32 | 9 | 4 | TiO₂ · 21 prototypes |
| 142 | I 41/a c d :2 | 32 | 7 | 4 | 11 prototypes |
Trigonal (25)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 143 | P 3 | 3 | 4 | 1 | 5 prototypes |
| 144 | P 31 | 3 | 1 | 1 | 3 prototypes |
| 145 | P 32 | 3 | 1 | 1 | 1 prototype |
| 146 | R 3 :h | 9 | 2 | 2 | 6 prototypes |
| 147 | P -3 | 6 | 7 | 1 | 7 prototypes |
| 148 | R -3 :h | 18 | 6 | 2 | BiI₃, PbMo₆S₈ · 27 prototypes |
| 149 | P 3 1 2 | 6 | 12 | 1 | 2 prototypes |
| 150 | P 3 2 1 | 6 | 7 | 1 | 8 prototypes |
| 151 | P 31 1 2 | 6 | 3 | 1 | 2 prototypes |
| 152 | P 31 2 1 | 6 | 3 | 1 | SiO₂ · 5 prototypes |
| 153 | P 32 1 2 | 6 | 3 | 1 | 2 prototypes |
| 154 | P 32 2 1 | 6 | 3 | 1 | 4 prototypes |
| 155 | R 3 2 :h | 18 | 6 | 2 | 6 prototypes |
| 156 | P 3 m 1 | 6 | 5 | 1 | 5 prototypes |
| 157 | P 3 1 m | 6 | 4 | 1 | 1 prototype |
| 158 | P 3 c 1 | 6 | 4 | 1 | 2 prototypes |
| 159 | P 3 1 c | 6 | 3 | 1 | Si₃N₄ · 3 prototypes |
| 160 | R 3 m :h | 18 | 3 | 2 | 21 prototypes |
| 161 | R 3 c :h | 18 | 2 | 2 | 5 prototypes |
| 162 | P -3 1 m | 12 | 12 | 1 | 5 prototypes |
| 163 | P -3 1 c | 12 | 9 | 1 | 7 prototypes |
| 164 | P -3 m 1 | 12 | 10 | 1 | CdI₂ · 34 prototypes |
| 165 | P -3 c 1 | 12 | 7 | 1 | 4 prototypes |
| 166 | R -3 m :h | 36 | 9 | 2 | CdCl₂, LiCoO₂, Bi₂Se₃, NbS₂ · 75 prototypes |
| 167 | R -3 c :h | 36 | 6 | 2 | Al₂O₃, CaCO₃, Na₃Zr₂Si₂PO₁₂ · 18 prototypes |
Hexagonal (27)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 168 | P 6 | 6 | 4 | 1 | 2 prototypes |
| 169 | P 61 | 6 | 1 | 1 | 3 prototypes |
| 170 | P 65 | 6 | 1 | 1 | 2 prototypes |
| 171 | P 62 | 6 | 3 | 1 | 2 prototypes |
| 172 | P 64 | 6 | 3 | 1 | 1 prototype |
| 173 | P 63 | 6 | 3 | 1 | SiC · 10 prototypes |
| 174 | P -6 | 6 | 12 | 1 | BN · 5 prototypes |
| 175 | P 6/m | 12 | 12 | 1 | 4 prototypes |
| 176 | P 63/m | 12 | 9 | 1 | Ca₅(PO₄)₃F, Ca₅(PO₄)₃(OH) · 19 prototypes |
| 177 | P 6 2 2 | 12 | 14 | 1 | 3 prototypes |
| 178 | P 61 2 2 | 12 | 3 | 1 | 4 prototypes |
| 179 | P 65 2 2 | 12 | 3 | 1 | 2 prototypes |
| 180 | P 62 2 2 | 12 | 11 | 1 | 6 prototypes |
| 181 | P 64 2 2 | 12 | 11 | 1 | 4 prototypes |
| 182 | P 63 2 2 | 12 | 9 | 1 | 7 prototypes |
| 183 | P 6 m m | 12 | 6 | 1 | 3 prototypes |
| 184 | P 6 c c | 12 | 4 | 1 | 3 prototypes |
| 185 | P 63 c m | 12 | 4 | 1 | 7 prototypes |
| 186 | P 63 m c | 12 | 4 | 1 | ZnS · 26 prototypes |
| 187 | P -6 m 2 | 12 | 15 | 1 | 13 prototypes |
| 188 | P -6 c 2 | 12 | 12 | 1 | 4 prototypes |
| 189 | P -6 2 m | 12 | 12 | 1 | 12 prototypes |
| 190 | P -6 2 c | 12 | 9 | 1 | 6 prototypes |
| 191 | P 6/m m m | 24 | 18 | 1 | Li₃N · 17 prototypes |
| 192 | P 6/m c c | 24 | 13 | 1 | Be₃Al₂Si₆O₁₈ · 3 prototypes |
| 193 | P 63/m c m | 24 | 12 | 1 | 7 prototypes |
| 194 | P 63/m m c | 24 | 12 | 1 | Mg, C, NiAs, MoS₂, H₂O, TaS₂ · 74 prototypes |
Cubic (36)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 195 | P 2 3 | 12 | 10 | 1 | 2 prototypes |
| 196 | F 2 3 | 48 | 8 | 1 | 3 prototypes |
| 197 | I 2 3 | 24 | 6 | 1 | 4 prototypes |
| 198 | P 21 3 | 12 | 2 | 1 | 14 prototypes |
| 199 | I 21 3 | 24 | 3 | 1 | 3 prototypes |
| 200 | P m -3 | 24 | 12 | 1 | 4 prototypes |
| 201 | P n -3 :2 | 24 | 8 | 2 | 5 prototypes |
| 202 | F m -3 | 96 | 9 | 1 | K₃C₆₀ · 4 prototypes |
| 203 | F d -3 :2 | 96 | 7 | 2 | 3 prototypes |
| 204 | I m -3 | 48 | 8 | 1 | 11 prototypes |
| 205 | P a -3 | 24 | 4 | 1 | FeS₂ · 21 prototypes |
| 206 | I a -3 | 48 | 5 | 1 | 4 prototypes |
| 207 | P 4 3 2 | 24 | 11 | 1 | 2 prototypes |
| 208 | P 42 3 2 | 24 | 13 | 1 | 3 prototypes |
| 209 | F 4 3 2 | 96 | 10 | 1 | 2 prototypes |
| 210 | F 41 3 2 | 96 | 8 | 1 | 2 prototypes |
| 211 | I 4 3 2 | 48 | 10 | 1 | 1 prototype |
| 212 | P 43 3 2 | 24 | 5 | 1 | 6 prototypes |
| 213 | P 41 3 2 | 24 | 5 | 1 | 5 prototypes |
| 214 | I 41 3 2 | 48 | 9 | 1 | 3 prototypes |
| 215 | P -4 3 m | 24 | 10 | 1 | 7 prototypes |
| 216 | F -4 3 m | 96 | 9 | 1 | ZnS · 22 prototypes |
| 217 | I -4 3 m | 48 | 8 | 1 | Zn(mim)₂ · 8 prototypes |
| 218 | P -4 3 n | 24 | 9 | 1 | Na₈(Al₆Si₆O₂₄)Cl₂, Na₈(Al₆Si₆O₂₄)S₂ · 6 prototypes |
| 219 | F -4 3 c | 96 | 8 | 1 | 3 prototypes |
| 220 | I -4 3 d | 48 | 5 | 1 | 9 prototypes |
| 221 | P m -3 m | 48 | 14 | 1 | Po, CsCl, SrTiO₃, ReO₃ · 19 prototypes |
| 222 | P n -3 n :2 | 48 | 9 | 2 | 4 prototypes |
| 223 | P m -3 n | 48 | 12 | 1 | 10 prototypes |
| 224 | P n -3 m :2 | 48 | 12 | 2 | Cu₂O · 6 prototypes |
| 225 | F m -3 m | 192 | 12 | 1 | Cu, NaCl, CaF₂, Na₂O, AgCl, Zn₄O(BDC)₃, Cu₃(BTC)₂, Mg₂NiH₄, Zr₆O₄(OH)₄(BDC)₆, Li₃Bi, Fe₄[Fe(CN)₆]₃ · 28 prototypes |
| 226 | F m -3 c | 192 | 10 | 1 | 5 prototypes |
| 227 | F d -3 m :2 | 192 | 9 | 2 | C, MgAl₂O₄, Fe₃O₄, LiMn₂O₄, Na₅₆(Al,Si)₁₉₂O₃₈₄, Y₂Ti₂O₇, Li₄Ti₅O₁₂ · 27 prototypes |
| 228 | F d -3 c :2 | 192 | 8 | 2 | 3 prototypes |
| 229 | I m -3 m | 96 | 12 | 1 | W, AgI · 14 prototypes |
| 230 | I a -3 d | 96 | 8 | 1 | Y₃Al₅O₁₂ · 8 prototypes |
Operations are counted per conventional cell (lattice centring included). Symbols follow the International Tables; the first listed setting is the standard one.