  The volume operator plays a central role in both the kinematics and dynamics
of canonical approaches to quantum gravity which are based on algebras of
generalized Wilson loops. We introduce a method for simplifying its spectral
analysis, for quantum states that can be realized on a cubic three-dimensional
lattice. This involves a decomposition of Hilbert space into sectors
transforming according to the irreducible representations of a subgroup of the
cubic group. As an application, we determine the complete spectrum for a class
of states with six-valent intersections.
