Triclinic
a ≠ b ≠ c
α ≠ β ≠ γ

1

C₁

1: P1

-

-

-

-

-

n

1

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

1

S₂ = C_i

2: P1

-

-

-

-

-

y

2

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

Monoclinic
a ≠ b ≠ c
α ≠ 90°, β = γ = 90°

2

C₂

3: P2, 4: P2₁, 5: C2

1

-

-

-

-

n

4: sucrose

2

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

m

C_1h = C_s

6: Pm, 7: Pc, 8: Cm, 9: Cc

-

-

-

-

1

n

2

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

$\frac{2}{m}$

C_2h

10: P2/m, 11: P2₁/m, 12: C2/m, 13: P2/c, 14: P2₁/c, 15: C2/c

1

-

-

-

1

y

4

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

Orthorhombic
a ≠ b ≠ c
α = β = γ = 90°

222

V = D₂

16: P222, 17: P222₁, 18: P2₁2₁2, 19: P2₁2₁2₁, 20: C222₁, 21: C222, 22: F222, 23: I222, 24: I2₁2₁2₁

3

-

-

-

-

n

4

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

mm2

C_2v

25: Pmm2, 26: Pmc2₁, 27: Pcc2, 28: Pma2, 29: Pca2₁, 30: Pnc2, 31: Pmn2₁, 32: Pba2, 33: Pna2₁, 34: Pnn2 35: Cmm2, 36: Cmc2₁, 37: Ccc2, 38: Amm2, 39: Aem2, 40: Ama2, 41: Aea2, 42: Fmm2, 43: Fdd2, 44: Imm2, 45: Iba2, 46: Ima2

1

-

-

-

2

n

4

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

mmm

V_h = D_2h

47: Pmmm, 48: Pnnn, 49: Pccm, 50: Pban, 51: Pmma, 52: Pnna, 53: Pmna, 54: Pcca, 55: Pbam, 56: Pccn, 57: Pbcm, 58: Pnnm, 59: Pmmn, 60: Pbcn, 61: Pbca, 62: Pnma 63: Cmcm, 64: Cmce, 65: Cmmm, 66: Cccm, 67: Cmme, 68: Ccce, 69: Fmmm, 70: Fddd, 71: Immm, 72: Ibam, 73: Ibca, 74: Imma

3

-

-

-

3

y

47: YBa₂Cu₃O_7-x
62: Fe₃C
64: αGa
70: αS

8

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

Tetragonal

4

C₄

75: P4, 76: P4₁, 77: P4₂, 78: P4₃, 79: I4, 80: I4₁

-

-

1

-

-

n

4

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

4

S₄

81: P4, 82: I4

1

-

-

-

-

n

4

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

$\frac{4}{m}$

C_4h

83: P4/m, 84: P4₂/m, 85: P4/n, 86: P4₂/n, 87: I4/m, 88: I4₁/a

-

-

1

-

1

y

8

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

422

D₄

89: P422, 90: P42₁2, 91: P4₁22, 92: P4₁2₁2, 93: P4₂22, 94: P4₂2₁2, 95: P4₃22, 96: P4₃2₁2, 97: I422, 98: I4₁22

4

-

1

-

-

n

8

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

4mm

C_4v

99: P4mm, 100: P4bm, 101: P4₂cm, 102: P4₂nm, 103: P4cc, 104: P4nc, 105: P4₂mc, 106: P4₂bc, 107: I4mm, 108: I4cm, 109: I4₁md, 110: I4₁cd

-

-

-

-

4

n

99: PZT PbZr_xTi_1-xO₃ x<0.52

8

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

42m

V_d = D_2d

111: P42m, 112: P42c, 113: P42₁m, 114: P42₁c, 115: P4m2, 116: P4c2, 117: P4b2, 118: P4n2, 119: I4m2, 120: I4c2, 121: I42m, 122: I42d

3

-

-

-

2

n

8

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

$\frac{4}{m}mm$

D_4h

123: P4/mmm, 124: P4/mcc, 125: P4/nbm, 126: P4/nnc, 127: P4/mbm, 128: P4/mnc, 129: P4/nmm, 130: P4/ncc, 131: P4₂/mmc, 132: P4₂/mcm, 133: P4₂/nbc, 134: P4₂/nnm, 135: P4₂/mbc, 136: P4₂/mnm, 137: P4₂/nmc, 138: P4₂/ncm, 139: I4/mmm, 140: I4/mcm, 141: I4₁/amd, 142: I4₁/acd

4

-

1

-

5

y

123: Tetrataenite, 139: In
141: βSn

16

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

Trigonal

3

C₃

143: P3, 144: P3₁, 145: P3₂, 146: R3

-

1

-

-

-

n

3

symmetric, asymmetric

symmetric, asymmetric

3

S₆ = C_6i

147: P3, 148: R3

-

1

-

-

-

y

6

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

32

D₃

149: P312, 150: P321, 151: P3₁12, 152: P3₁21, 153: P3₂12, 154: P3₂21, 155: R32

3

1

-

-

-

n

154: α-Quartz

6

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

3m

C_3v

156: P3m1, 157: P31m, 158: P3c1, 159: P31c, 160: R3m, 161: R3c

-

1

-

-

3

n

161: ferroelectric LiNbO₃

6

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

3m

D₃d

162: P31m, 163: P31c, 164: P3m1, 165: P3c1, 166: R3m, 167: R3c

3

1

-

-

3

y

167: calcite, paraelectric LiNbO₃, sapphire (α-Al₂O₃)

12

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

Hexagonal

6

C₆

168: P6, 169: P6₁, 170: P6₅, 171: P6₂, 172: P6₄, 173: P6₃

-

-

-

1

-

n

6

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

6

C_3h = S₃

174: P6

-

1

-

-

1

n

6

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

$\frac{6}{m}$

C_6h

175: P6/m, 176: P6₃/m

-

-

-

1

1

y

12

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

622

D₆

177: P622, 178: P6₁22, 179: P6₅22, 180: P6₂22, 181: P6₄22, 182: P6₃22

6

-

-

1

-

n

180: β-Quartz

12

reduced, symmetric, asymmetric

reducedsymmetric, asymmetric

6mm

C_6v

183: P6mm, 184: P6cc, 185: P6₃cm, 186: P6₃mc

-

-

-

1

6

n

186: Wurtzite,

12

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

6m2

D_3h

187: P6m2, 188: P6c2, 189: P62m, 190: P62c

3

1

-s

-

4

n

12

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

$\frac{6}{m}mm$

D_6h

191: P6/mmm, 192: P6/mcc, 193: P6₃/mcm, 194: P6₃/mmc

6

-

-

1

7

y

194: hcp, Mg, Be, Sc, α-Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Gd, Tb, Dy, Ho, Er, Tm, Lu, Hf, Re, Os, Tl, graphite, MoS₂, ice Ih

24

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

Cubic

23

T

195: P23, 196: F23, 197: I23, 198: P2₁3, 199: I2₁3

3

4

-

-

-

n

12

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

m3

T_h

200: Pm3, 201: Pn3, 202: Fm3, 203: Fd3, 204: Im3, 205: Pa3, 206: Ia3

3

4

-

-

3

y

24

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

432

O

207: P432, 208: P4₂32, 209: F432, 210: F4₁32, 211: I432, 212: P4₃32, 213: P4₁32, 214: I4₁32

6

4

3

-

-

n

24

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

43m

T_d

215: P43m, 216: F43m, 217: I43m, 218: P43n, 219: F43c, 220: I43d

3

4

-

-

6

n

216: Zincblende, GaAs, GaP, InAs, SiC

24

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric

m3m

O_h

221: Pm3m, 222: Pn3n, 223: Pm3n, 224: Pn3m 225: Fm3m, 226: Fm3c, 227: Fd3m, 228: Fd3c 229: Im3m, 230: Ia3d

6

4

3

-

9

y

221: CsCl, cubic perovskite
225: fcc, Al, Cu, Ni, Ag, Pt, Au, Pb, γ-Fe, NaCl
227: diamond, C, Si, Ge, α-Sn, spinel
229: bcc, Na, K, Cr, α-Fe, β-Ti, Nb, Mo, Ta

48

reduced, symmetric, asymmetric

reduced, symmetric, asymmetric