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 | P2 | 2 | 5 | 3 | 4 prototypes |
| 4 | P2₁ | 2 | 1 | 3 | 11 prototypes |
| 5 | C2 | 4 | 3 | 9 | 11 prototypes |
| 6 | Pm | 2 | 3 | 3 | 3 prototypes |
| 7 | Pc | 2 | 1 | 9 | 11 prototypes |
| 8 | Cm | 4 | 2 | 9 | 8 prototypes |
| 9 | Cc | 4 | 1 | 18 | 11 prototypes |
| 10 | P2/m | 4 | 15 | 3 | 7 prototypes |
| 11 | P2₁/m | 4 | 6 | 3 | 19 prototypes |
| 12 | C2/m | 8 | 10 | 9 | YCl₃ · 79 prototypes |
| 13 | P2/c | 4 | 7 | 9 | 13 prototypes |
| 14 | P2₁/c | 4 | 5 | 9 | SiO₂ (MFI), Na₃AlH₆ · 98 prototypes |
| 15 | C2/c | 8 | 6 | 18 | 75 prototypes |
Orthorhombic (59)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 16 | P222 | 4 | 21 | 1 | 2 prototypes |
| 17 | P222₁ | 4 | 5 | 3 | 3 prototypes |
| 18 | P2₁2₁2 | 4 | 3 | 3 | 4 prototypes |
| 19 | P2₁2₁2₁ | 4 | 1 | 1 | 19 prototypes |
| 20 | C222₁ | 8 | 3 | 3 | 7 prototypes |
| 21 | C222 | 8 | 12 | 3 | 3 prototypes |
| 22 | F222 | 16 | 11 | 1 | 2 prototypes |
| 23 | I222 | 8 | 11 | 1 | 4 prototypes |
| 24 | I2₁2₁2₁ | 8 | 4 | 1 | 2 prototypes |
| 25 | Pmm2 | 4 | 9 | 3 | 2 prototypes |
| 26 | Pmc2₁ | 4 | 3 | 6 | 6 prototypes |
| 27 | Pcc2 | 4 | 5 | 3 | 2 prototypes |
| 28 | Pma2 | 4 | 4 | 6 | 2 prototypes |
| 29 | Pca2₁ | 4 | 1 | 6 | 8 prototypes |
| 30 | Pnc2 | 4 | 3 | 6 | 3 prototypes |
| 31 | Pmn2₁ | 4 | 2 | 6 | 12 prototypes |
| 32 | Pba2 | 4 | 3 | 3 | 3 prototypes |
| 33 | Pna2₁ | 4 | 1 | 6 | 13 prototypes |
| 34 | Pnn2 | 4 | 3 | 3 | 4 prototypes |
| 35 | Cmm2 | 8 | 6 | 3 | 1 prototype |
| 36 | Cmc2₁ | 8 | 2 | 6 | 17 prototypes |
| 37 | Ccc2 | 8 | 4 | 3 | 2 prototypes |
| 38 | Amm2 | 8 | 6 | 6 | 8 prototypes |
| 39 | Aem2 | 8 | 4 | 6 | 4 prototypes |
| 40 | Ama2 | 8 | 3 | 6 | 5 prototypes |
| 41 | Aea2 | 8 | 2 | 6 | 4 prototypes |
| 42 | Fmm2 | 16 | 5 | 3 | 3 prototypes |
| 43 | Fdd2 | 16 | 2 | 3 | 8 prototypes |
| 44 | Imm2 | 8 | 5 | 3 | 5 prototypes |
| 45 | Iba2 | 8 | 3 | 3 | 3 prototypes |
| 46 | Ima2 | 8 | 3 | 6 | 7 prototypes |
| 47 | Pmmm | 8 | 27 | 1 | YBa₂Cu₃O₇ · 3 prototypes |
| 48 | Pnnn, origin 2 | 8 | 13 | 2 | 2 prototypes |
| 49 | Pccm | 8 | 18 | 3 | 3 prototypes |
| 50 | Pban, origin 2 | 8 | 13 | 6 | 2 prototypes |
| 51 | Pmma | 8 | 12 | 6 | 9 prototypes |
| 52 | Pnna | 8 | 5 | 6 | 4 prototypes |
| 53 | Pmna | 8 | 9 | 6 | 4 prototypes |
| 54 | Pcca | 8 | 6 | 6 | 3 prototypes |
| 55 | Pbam | 8 | 9 | 3 | 17 prototypes |
| 56 | Pccn | 8 | 5 | 3 | 4 prototypes |
| 57 | Pbcm | 8 | 5 | 6 | 12 prototypes |
| 58 | Pnnm | 8 | 8 | 3 | FeS₂ · 16 prototypes |
| 59 | Pmmn, origin 2 | 8 | 7 | 6 | 11 prototypes |
| 60 | Pbcn | 8 | 4 | 6 | 11 prototypes |
| 61 | Pbca | 8 | 3 | 2 | 27 prototypes |
| 62 | Pnma | 8 | 4 | 6 | LiFePO₄, Mg₂SiO₄, Li₁₄Zn(GeO₄)₄, Al(OH)(BDC) · 113 prototypes |
| 63 | Cmcm | 16 | 8 | 6 | 60 prototypes |
| 64 | Cmce | 16 | 7 | 6 | 23 prototypes |
| 65 | Cmmm | 16 | 18 | 3 | 19 prototypes |
| 66 | Cccm | 16 | 13 | 3 | 6 prototypes |
| 67 | Cmme | 16 | 15 | 6 | 7 prototypes |
| 68 | Ccce, origin 2 | 16 | 9 | 12 | 1 prototype |
| 69 | Fmmm | 32 | 16 | 1 | 10 prototypes |
| 70 | Fddd, origin 2 | 32 | 8 | 2 | 9 prototypes |
| 71 | Immm | 16 | 15 | 1 | 13 prototypes |
| 72 | Ibam | 16 | 11 | 3 | 3 prototypes |
| 73 | Ibca | 16 | 6 | 2 | 2 prototypes |
| 74 | Imma | 16 | 10 | 6 | 13 prototypes |
Tetragonal (68)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 75 | P 4 | 4 | 4 | 2 | 2 prototypes |
| 76 | P4₁ | 4 | 1 | 2 | 3 prototypes |
| 77 | P4₂ | 4 | 4 | 2 | 3 prototypes |
| 78 | P4₃ | 4 | 1 | 2 | 2 prototypes |
| 79 | I 4 | 8 | 3 | 2 | 2 prototypes |
| 80 | I4₁ | 8 | 2 | 2 | 1 prototype |
| 81 | P4̅ | 4 | 8 | 2 | 1 prototype |
| 82 | I4̅ | 8 | 7 | 2 | Ag₂HgI₄ · 6 prototypes |
| 83 | P 4/m | 8 | 12 | 2 | 2 prototypes |
| 84 | P4₂/m | 8 | 11 | 2 | 2 prototypes |
| 85 | P 4/n :2 | 8 | 7 | 4 | 7 prototypes |
| 86 | P4₂/n, origin 2 | 8 | 7 | 4 | 6 prototypes |
| 87 | I 4/m | 16 | 9 | 2 | 9 prototypes |
| 88 | I4₁/a, origin 2 | 16 | 6 | 4 | CaWO₄, NaAlH₄ · 5 prototypes |
| 89 | P422 | 8 | 16 | 2 | 2 prototypes |
| 90 | P42₁2 | 8 | 7 | 2 | 4 prototypes |
| 91 | P4₁22 | 8 | 4 | 2 | 1 prototype |
| 92 | P4₁2₁2 | 8 | 2 | 2 | SiO₂ · 9 prototypes |
| 93 | P4₂22 | 8 | 16 | 2 | 1 prototype |
| 94 | P4₂2₁2 | 8 | 7 | 2 | 2 prototypes |
| 95 | P4₃22 | 8 | 4 | 2 | 1 prototype |
| 96 | P4₃2₁2 | 8 | 2 | 2 | 3 prototypes |
| 97 | I422 | 16 | 11 | 2 | 2 prototypes |
| 98 | I4₁22 | 16 | 7 | 2 | 2 prototypes |
| 99 | P4mm | 8 | 7 | 2 | BaTiO₃ · 2 prototypes |
| 100 | P4bm | 8 | 4 | 2 | 4 prototypes |
| 101 | P4₂cm | 8 | 5 | 2 | 2 prototypes |
| 102 | P4₂nm | 8 | 4 | 2 | 2 prototypes |
| 103 | P4cc | 8 | 4 | 2 | 2 prototypes |
| 104 | P4nc | 8 | 3 | 2 | 2 prototypes |
| 105 | P4₂mc | 8 | 6 | 2 | 1 prototype |
| 106 | P4₂bc | 8 | 3 | 2 | 1 prototype |
| 107 | I4mm | 16 | 5 | 2 | 4 prototypes |
| 108 | I4cm | 16 | 4 | 2 | 2 prototypes |
| 109 | I4₁md | 16 | 3 | 2 | 2 prototypes |
| 110 | I4₁cd | 16 | 2 | 2 | 1 prototype |
| 111 | P4̅2m | 8 | 15 | 2 | 3 prototypes |
| 112 | P4̅2c | 8 | 14 | 2 | 1 prototype |
| 113 | P4̅2₁m | 8 | 6 | 2 | 6 prototypes |
| 114 | P4̅2₁c | 8 | 5 | 2 | 5 prototypes |
| 115 | P4̅m2 | 8 | 12 | 2 | 5 prototypes |
| 116 | P4̅c2 | 8 | 10 | 2 | 3 prototypes |
| 117 | P4̅b2 | 8 | 9 | 2 | 1 prototype |
| 118 | P4̅n2 | 8 | 9 | 2 | 2 prototypes |
| 119 | I4̅m2 | 16 | 10 | 2 | 6 prototypes |
| 120 | I4̅c2 | 16 | 9 | 2 | 3 prototypes |
| 121 | I4̅2m | 16 | 10 | 2 | 7 prototypes |
| 122 | I4̅2d | 16 | 5 | 2 | CuFeS₂ · 7 prototypes |
| 123 | P4/mmm | 16 | 21 | 2 | Sr₃Fe₂O₇ · 31 prototypes |
| 124 | P4/mcc | 16 | 14 | 2 | 3 prototypes |
| 125 | P4/nbm, origin 2 | 16 | 14 | 4 | 5 prototypes |
| 126 | P4/nnc, origin 2 | 16 | 11 | 4 | 3 prototypes |
| 127 | P4/mbm | 16 | 12 | 2 | 10 prototypes |
| 128 | P4/mnc | 16 | 9 | 2 | 5 prototypes |
| 129 | P4/nmm, origin 2 | 16 | 11 | 4 | LaFeAsO · 26 prototypes |
| 130 | P4/ncc, origin 2 | 16 | 7 | 4 | CaCuSi₄O₁₀ · 6 prototypes |
| 131 | P4₂/mmc | 16 | 18 | 2 | 3 prototypes |
| 132 | P4₂/mcm | 16 | 16 | 2 | 2 prototypes |
| 133 | P4₂/nbc, origin 2 | 16 | 11 | 4 | 2 prototypes |
| 134 | P4₂/nnm, origin 2 | 16 | 14 | 4 | 1 prototype |
| 135 | P4₂/mbc | 16 | 9 | 2 | 4 prototypes |
| 136 | P4₂/mnm | 16 | 11 | 2 | TiO₂, MgH₂ · 18 prototypes |
| 137 | P4₂/nmc, origin 2 | 16 | 8 | 4 | 7 prototypes |
| 138 | P4₂/ncm, origin 2 | 16 | 10 | 4 | 3 prototypes |
| 139 | I4/mmm | 32 | 15 | 2 | K₂NiF₄, Tl₂Ba₂Ca₂Cu₃O₁₀ · 51 prototypes |
| 140 | I4/mcm | 32 | 13 | 2 | 18 prototypes |
| 141 | I4₁/amd, origin 2 | 32 | 9 | 4 | TiO₂ · 21 prototypes |
| 142 | I4₁/acd, origin 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 | P3₁ | 3 | 1 | 1 | 3 prototypes |
| 145 | P3₂ | 3 | 1 | 1 | 1 prototype |
| 146 | R 3 :h | 9 | 2 | 2 | 6 prototypes |
| 147 | P3̅ | 6 | 7 | 1 | 7 prototypes |
| 148 | R3̅, hex. axes | 18 | 6 | 2 | BiI₃, PbMo₆S₈ · 27 prototypes |
| 149 | P312 | 6 | 12 | 1 | 2 prototypes |
| 150 | P321 | 6 | 7 | 1 | 8 prototypes |
| 151 | P3₁12 | 6 | 3 | 1 | 2 prototypes |
| 152 | P3₁21 | 6 | 3 | 1 | SiO₂ · 5 prototypes |
| 153 | P3₂12 | 6 | 3 | 1 | 2 prototypes |
| 154 | P3₂21 | 6 | 3 | 1 | 4 prototypes |
| 155 | R32, hex. axes | 18 | 6 | 2 | 6 prototypes |
| 156 | P3m1 | 6 | 5 | 1 | 5 prototypes |
| 157 | P31m | 6 | 4 | 1 | 1 prototype |
| 158 | P3c1 | 6 | 4 | 1 | 2 prototypes |
| 159 | P31c | 6 | 3 | 1 | Si₃N₄ · 3 prototypes |
| 160 | R3m, hex. axes | 18 | 3 | 2 | 21 prototypes |
| 161 | R3c, hex. axes | 18 | 2 | 2 | 5 prototypes |
| 162 | P3̅1m | 12 | 12 | 1 | 5 prototypes |
| 163 | P3̅1c | 12 | 9 | 1 | 7 prototypes |
| 164 | P3̅m1 | 12 | 10 | 1 | CdI₂ · 34 prototypes |
| 165 | P3̅c1 | 12 | 7 | 1 | 4 prototypes |
| 166 | R3̅m, hex. axes | 36 | 9 | 2 | CdCl₂, LiCoO₂, Bi₂Se₃, NbS₂ · 75 prototypes |
| 167 | R3̅c, hex. axes | 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 | P6₁ | 6 | 1 | 1 | 3 prototypes |
| 170 | P6₅ | 6 | 1 | 1 | 2 prototypes |
| 171 | P6₂ | 6 | 3 | 1 | 2 prototypes |
| 172 | P6₄ | 6 | 3 | 1 | 1 prototype |
| 173 | P6₃ | 6 | 3 | 1 | SiC · 10 prototypes |
| 174 | P6̅ | 6 | 12 | 1 | BN · 5 prototypes |
| 175 | P 6/m | 12 | 12 | 1 | 4 prototypes |
| 176 | P6₃/m | 12 | 9 | 1 | Ca₅(PO₄)₃F, Ca₅(PO₄)₃(OH) · 19 prototypes |
| 177 | P622 | 12 | 14 | 1 | 3 prototypes |
| 178 | P6₁22 | 12 | 3 | 1 | 4 prototypes |
| 179 | P6₅22 | 12 | 3 | 1 | 2 prototypes |
| 180 | P6₂22 | 12 | 11 | 1 | 6 prototypes |
| 181 | P6₄22 | 12 | 11 | 1 | 4 prototypes |
| 182 | P6₃22 | 12 | 9 | 1 | 7 prototypes |
| 183 | P6mm | 12 | 6 | 1 | 3 prototypes |
| 184 | P6cc | 12 | 4 | 1 | 3 prototypes |
| 185 | P6₃cm | 12 | 4 | 1 | 7 prototypes |
| 186 | P6₃mc | 12 | 4 | 1 | ZnS · 26 prototypes |
| 187 | P6̅m2 | 12 | 15 | 1 | 13 prototypes |
| 188 | P6̅c2 | 12 | 12 | 1 | 4 prototypes |
| 189 | P6̅2m | 12 | 12 | 1 | 12 prototypes |
| 190 | P6̅2c | 12 | 9 | 1 | 6 prototypes |
| 191 | P6/mmm | 24 | 18 | 1 | Li₃N · 17 prototypes |
| 192 | P6/mcc | 24 | 13 | 1 | Be₃Al₂Si₆O₁₈ · 3 prototypes |
| 193 | P6₃/mcm | 24 | 12 | 1 | 7 prototypes |
| 194 | P6₃/mmc | 24 | 12 | 1 | Mg, C, NiAs, MoS₂, H₂O, TaS₂ · 74 prototypes |
Cubic (36)
| No. | Symbol | Operations | Wyckoff positions | Settings | Structures here |
|---|---|---|---|---|---|
| 195 | P23 | 12 | 10 | 1 | 2 prototypes |
| 196 | F23 | 48 | 8 | 1 | 3 prototypes |
| 197 | I23 | 24 | 6 | 1 | 4 prototypes |
| 198 | P2₁3 | 12 | 2 | 1 | 14 prototypes |
| 199 | I2₁3 | 24 | 3 | 1 | 3 prototypes |
| 200 | Pm3̅ | 24 | 12 | 1 | 4 prototypes |
| 201 | Pn3̅, origin 2 | 24 | 8 | 2 | 5 prototypes |
| 202 | Fm3̅ | 96 | 9 | 1 | K₃C₆₀ · 4 prototypes |
| 203 | Fd3̅, origin 2 | 96 | 7 | 2 | 3 prototypes |
| 204 | Im3̅ | 48 | 8 | 1 | 11 prototypes |
| 205 | Pa3̅ | 24 | 4 | 1 | FeS₂ · 21 prototypes |
| 206 | Ia3̅ | 48 | 5 | 1 | 4 prototypes |
| 207 | P432 | 24 | 11 | 1 | 2 prototypes |
| 208 | P4₂32 | 24 | 13 | 1 | 3 prototypes |
| 209 | F432 | 96 | 10 | 1 | 2 prototypes |
| 210 | F4₁32 | 96 | 8 | 1 | 2 prototypes |
| 211 | I432 | 48 | 10 | 1 | 1 prototype |
| 212 | P4₃32 | 24 | 5 | 1 | 6 prototypes |
| 213 | P4₁32 | 24 | 5 | 1 | 5 prototypes |
| 214 | I4₁32 | 48 | 9 | 1 | 3 prototypes |
| 215 | P4̅3m | 24 | 10 | 1 | 7 prototypes |
| 216 | F4̅3m | 96 | 9 | 1 | ZnS · 22 prototypes |
| 217 | I4̅3m | 48 | 8 | 1 | Zn(mim)₂ · 8 prototypes |
| 218 | P4̅3n | 24 | 9 | 1 | Na₈(Al₆Si₆O₂₄)Cl₂, Na₈(Al₆Si₆O₂₄)S₂ · 6 prototypes |
| 219 | F4̅3c | 96 | 8 | 1 | 3 prototypes |
| 220 | I4̅3d | 48 | 5 | 1 | 9 prototypes |
| 221 | Pm3̅m | 48 | 14 | 1 | Po, CsCl, SrTiO₃, ReO₃ · 19 prototypes |
| 222 | Pn3̅n, origin 2 | 48 | 9 | 2 | 4 prototypes |
| 223 | Pm3̅n | 48 | 12 | 1 | 10 prototypes |
| 224 | Pn3̅m, origin 2 | 48 | 12 | 2 | Cu₂O · 6 prototypes |
| 225 | Fm3̅m | 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 | Fm3̅c | 192 | 10 | 1 | 5 prototypes |
| 227 | Fd3̅m, origin 2 | 192 | 9 | 2 | C, MgAl₂O₄, Fe₃O₄, LiMn₂O₄, Na₅₆(Al,Si)₁₉₂O₃₈₄, Y₂Ti₂O₇, Li₄Ti₅O₁₂ · 27 prototypes |
| 228 | Fd3̅c, origin 2 | 192 | 8 | 2 | 3 prototypes |
| 229 | Im3̅m | 96 | 12 | 1 | W, AgI · 14 prototypes |
| 230 | Ia3̅d | 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.