The holosymmetric group Q of an n-dimensional crystal lattice determined by a given lattice basis B is considered. This group is contained in the n-dimensional orthogonal group 0(n) so its elements preserve the orthog- onality of basis vectors and their lengths. These conditions yield the decom- position of lattice basis into orthogonal sublattices and next the factoriza- tion of the holosymmetric group, which can be written as a direct product of complete monomial groups of k-dimensional (k < n) holosymmetric groups. Simple, decomposable and primitive holosymmetric groups are discussed. The results for n < 4 are presented.