Crystallographic Systems

System

Class Name

Hermann-Maguin
Symbol
(Point Group)

Hermann-Maguin Symbols
Space Groups in Point Group

Decoding Space Groups
Bravais Lattice

Triclinic

Triclinic - Pedial

(1) P1 P = Primitive Lattice.

1 = Symmetry Axis (360/1).
1 = Rotoinversion Axis.

Triclinic - Pinacoidal

(1) P1

Monoclinic

Monoclinic - Domatic

(m) Cc, Cm, Pc, and Pm P = Primitive Lattice.
C = 1 Face Centered Lattice.

2 = Symmetry Axis (360/2).
2/ = Mirror Plane perpendicular to Axis.
2₁ = screw symmetric lattice.

Monoclinic - Sphenoidal

(2) C2, P2, and P2₁

Monoclinic - Prismatic

(2/m) C2/c, C2/m, P2/c, P2/m, P2₁/c, and P2₁/m

Orthorhombic

Orthorhombic - Pyramidal

(mm2) Aba2, Abm2, Ama2, Amm2, Ccc2, Cmc2₁, Cmm2, Fdd2, Fmm2, Iba2, Ima2, Imm2, Pba2, Pca2₁, Pcc2, Pma2, Pmc2₁, Pmm2, Pmn2₁, Pna2₁, Pnc2, and Pnn2 P = Primitive Lattice.
C = 1 Face Centered Lattice.
F = 3 Face Centered Lattice.
I = Body Centered Lattice.

2₁ = Screw Axis.
2 = Symmetry Axis (360/2).

a,b,c = Perpendicular Glide Planes.
m,n = Oblique Glide Planes.

Orthorhombic - Disphenoidal

(2 2 2) C222, C222₁, F222, I2₁2₁2₁, I222, P2₁2₁2, P2₁2₁2₁, P222, and P222₁

Orthorhombic - Dipyramidal

(2/m 2/m 2/m) Ccca, Cccm, Cmca, Cmcm, Cmma, Cmmm, Fddd, Fmmm, Ibam, Ibca, Imma, Immm, Pbam, Pban, Pbca, Pbcm, Pbcn, Pcca, Pccm, Pccn, Pmma, Pmmm, Pmmn, Pmna, Pnma, Pnna, Pnnm, and Pnnn

Trigonal

Trigonal - Pyramidal

(3) P3, P3₁, P3₂, and R3 P = Primitive Lattice.
R = Rhombohedral Lattice.

3 = Rotoinversion Axis.
3, 2, 1 = Symmetry Axis (360/n).

c= Perpendicular Glide Planes.
m = Oblique Glide Planes.

Trigonal - Rhombohedral

(3) P3 and R3

Trigonal - Ditrigonal Pyramidal

(3m) P31c, P31m, P3c1, P3m1, R3c, and R3m

Trigonal - Trapezohedral

(3 2) P3₁12, P312, P3₁21, P321, P3₂12, P3₂21, and R3₂

Trigonal - Hexagonal Scalenohedral

(3 2/m) P31c, P31m, P3c1, P3m1, R3c, and R3m

Hexagonal

Hexagonal - Trigonal Dipyramidal

(6) P6 P = Primitive Lattice.

6 = Rotoinversion Axis.
6,2 = Symmetry Axis (360/n).
6_{1, 2, 3, 4, 5} = Screw Axis.

c = Perpendicular Glide Planes.
m = Oblique Glide Planes.

Hexagonal - Pyramidal

(6) P6, P6₁, P6₂, P6₃, P6₄, and P6₅

Hexagonal - Dipyramidal

(6/m) P6/m, P6/m, and P6₃/m

Hexagonal - Ditrigonal Dipyramidal

(6 m2) P62c, P62c, P62m, P6c2, and P6m2

Hexagonal - Dihexagonal Pyramidal

(6mm) P6₃cm, P6₃mc, P6cc, and P6mm

Hexagonal - Trapezohedral

(6 2 2) P622, P6₁22, P6₂22, P6₃22, P6₄22, P6₅22

Hexagonal - Dihexagonal Dipyramidal

(6/m 2/m 2/m) P6/mcc, P6/mmm, P6₃/mcm, and P6₃/mmc

Tetragonal

Tetragonal - Disphenoidal

(4) I4 and P4 P = Primitive Lattice.
I = Body Centered Lattice.

4 = Rotoinversion Axis.
4,2 = Symmetry Axis (360/n).
4_1,2,3 = Screw Axis.

a,b,c = Perpendicular Glide Planes.
m,n = Oblique Glide Planes.

Tetragonal - Pyramidal

(4) I4, I4₁, P4, P4₁, P4₂, and P4₃

Tetragonal - Dipyramidal

(4/m) I4/m, I4₁/a, P4/m, P4/n, P4₂/m, and P4₂/n

Tetragonal - Scalenohedral

(4 2m) I42d, I42m, I4c2, I4m2, P421c, P42₁m, P42c, P42m, P4b2, P4c2, P4m2, and P4n2

Tetragonal - Ditetragonal Pyramidal

(4mm) I4/mcm, I4/mmm, I4₁/acd, I4₁/amd, P4/mbm, P4/mcc, P4/mmm, P4/mnc, P4/nbm, P4/ncc, P4/nmm, P4/nnc, P4₂/mbc, P4₂/mcm, P4₂/mmc, P4₂/mnm, P4₂/nbc, P4₂/ncm, P4₂/nmc, and P4₂/nnm

Tetragonal - Trapezohedral

(4 2 2) I4₁22, I422, P4₁2₁2, P4₁22, P42₁2, P422, P4₂2₁2, P4₂22, P4₃2₁2, and P4₃22

Tetragonal - Ditetragonal Dipyramidal

(4/m 2/m 2/m) I4₁cd, I4₁md, I4cm, I4mm, P4₂bc, P4₂cm, P4₂mc, P4₂nm, P4bm, P4cc, P4mm, and P4nc

Isometric

Isometric - Tetartoidal

(2 3) F23, I2₁3, I23, P2₁3, and P23 P = Primitive Lattice.
F = 3 Face Centered Lattice.
I = Body Centered Lattice.

4 = Rotoinversion Axis.
4,3,2 = Symmetry Axis (360/n).
4_1,2,3 = Screw Axis.

Isometric - Diploidal

(2/m 3) Fd3, Fm3, Ia3, Im3, Pa3, Pm3, and Pn3

Isometric - Hextetrahedral

(4 3m) F43c, F43m, I43d, I43m, P43m, and P43n

Isometric - Gyroidal

(4 3 2) F432, F4₁32, I432, I4₁32, P432, P4₁32, P4₂32, and P4₃32

Isometric - Hexoctahedral

(4/m 3 2/m) F432, F4₁32, I432, I4₁32, P432, P4₁32, P4₂32, and P4₃32