  We study the phase diagram of a system of $2\times2\times2$ hard cubes on a
three dimensional cubic lattice. Using Monte Carlo simulations, we show that
the system exhibits four different phases as the density of cubes is increased:
disordered, layered, sublattice ordered, and columnar ordered. In the layered
phase, the system spontaneously breaks up into parallel slabs of size $2\times
L \times L$ where only a very small fraction cubes do not lie wholly within a
slab. Within each slab, the cubes are disordered; translation symmetry is thus
broken along exactly one principal axis. In the solid-like sublattice ordered
phase, the hard cubes preferentially occupy one of eight sublattices of the
cubic lattice, breaking translational symmetry along all three principal
directions. In the columnar phase, the system spontaneously breaks up into
weakly interacting parallel columns of size $2\times 2\times L$ where only a
very small fraction cubes do not lie wholly within a column. Within each
column, the system is disordered, and thus translational symmetry is broken
only along two principal directions. Using finite size scaling, we show that
the disordered-layered phase transition is continuous, while the
layered-sublattice and sublattice-columnar transitions are discontinuous. We
construct a Landau theory written in terms of the layering and columnar order
parameters, which is able to describe the different phases that are observed in
the simulations and the order of the transitions. Additionally, our results
near the disordered-layered transition are consistent with the $O(3)$
universality class perturbed by cubic anisotropy as predicted by the Landau
theory.
