  We develop a symmetry classification scheme to find ground states of pseudo
spin-1/2, spin-1, and spin-2 spin-orbit coupled spinor Bose-Einstein
condensates, and show that as the SO(2) symmetry of simultaneous spin and space
rotations is broken into discrete cyclic groups, various types of lattice
structures emerge in the absence of a lattice potential, examples include two
different kagaome lattices for pseudo spin-1/2 condensates and a nematic vortex
lattice in which uniaxial and biaxial spin textures align alternatively for
spin-2 condensates. For the pseudo spin-1/2 system, although mean-field states
always break time-reversal symmetry, there exists a time-reversal invariant
many-body ground state, which is fragmented and expected to be observed in a
micro-condensate.
