A system for categorizing symmetries of molecules. C_n groups contain only an n-fold rotation axis. C_nv groups, in addition to the n-fold rotation axis, have a mirror plane that contains the axis of rotation (and mirror planes associated with the existence of the n-fold axis). C_nh groups, in addition to the n-fold rotation axis, have a mirror plane perpendicular to the axis. S_n groups have an n-fold rotation–reflection axis. D_n groups have an n-fold rotation axis and a two-fold axis perpendicular to the n-fold axis (and two-fold axes associated with the existence of the n-fold axis). D_nh groups have all the symmetry operations of D_n and also a mirror plane perpendicular to the n-fold axis. D_nd groups contain all the symmetry operations of D_n and also mirror planes that contain the n-fold axis and bisect the angles between the two-fold axes. In the Schoenflies notation C stands for ‘cyclic’, S stands for ‘spiegel’ (mirror), and D stands for ‘dihedral’. The subscripts h, v, and d stand for horizontal, vertical, and diagonal respectively, where these words refer to the position of the mirror planes with respect to the n-fold axis (considered to be vertical). In addition to the noncubic groups referred to so far, there are cubic groups, which have several rotation axes with the same value of n. These are the tetrahedral groups T, T_h, and T_d, the octahedral groups O and O_h, and the icosahedral group I. The Schoenflies system is commonly used for isolated molecules, while the Hermann–Mauguin system is commonly used in crystallography. The Schoenflies notation was named after Arthur Moritz Schoenflies (1853–1928), who devised it in 1891.