Crystal systems, Laue groups

The 7 crystal systems are obtained when applying basic symmetry operations to a lattice and imposing certain restrictions on the unit cell lengths and interaxial angles. 

Crystal SystemSymmetries of point group (non-centrosymmetric, enantiomorphous point groups) Restrictions
TRICLINICNone (1)    a≠b≠c α≠β≠γ
MONOCLINIC1 twofold axis of rotation or 1 mirror plane (2)a≠b≠c α=γ=90º β≠90º
ORTHOROMBIC3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes (222)a≠b≠c α=β=γ=90º
TETRAGONAL1 fourfold axis of rotation (4, 422)a=b≠c α=β=γ=90º
TRIGONAL1 threefold axis of rotation (3, 32)a=b≠c α=β=90º γ=120º
HEXAGONAL1 sixfold axis of rotation (6, 622)a=b≠c α=β=90º γ=120º
CUBIC4 triads equally inclined at 109.47º (23, 432)a=b=c α=β=γ=90º 

One can only distinguish directly between 11 centrosymmetric groups that are also called 11 Laue groups (X-ray diffraction has the property of adding a center of inversion to the point group of the crystal).