Little groups are enumerated for the irreps and their components in any basis
of O(3) and SO(3) up to rank 9, and for all irreps of C$_{\infty}$, C$_{\infty
h}$, C$_{\infty v}$, D$_{\infty}$ and D$_{\infty h}$. The results are obtained
by a new chain criterion, which distinguishes massive (rotationally
inequivalent) irrep basis functions and allows for multiple branching paths,
and are verified by inspection. These results are relevant to the determination
of the symmetry of a material from its linear and nonlinear optical properties
and to the choices of order parameters for symmetry breaking in liquid
crystals.