  We study $\Gamma_3$ quadrupole orders in a face-centered cubic lattice. The
$\Gamma_3$ quadrupole moments under cubic symmetry possess a unique cubic
invariant in their free energy in the uniform ($q=0$) sector and the triple-q
sector for the X points $q=(2\pi,0,0),(0,2\pi,0)$, and $(0,0,2\pi)$.
Competition between this cubic anisotropy and anisotropic quadrupole-quadrupole
interactions causes a drastic impact on the phase diagram both in the ground
state and at finite temperatures. We show details about the model construction
and its properties, the phase diagram, and the mechanism of the various
triple-$q$ quadrupole orders reported in our preceding letter [J. Phys. Soc.
Jpn. 90, 43701 (2021), arXiv:2102.06346]. By using a mean-field approach, we
analyze a quadrupole exchange model that consists of a crystalline-electric
field scheme with the ground-state $\Gamma_3$ non-Kramers doublet and the
excited singlet $\Gamma_1$ state. We find various triple-$q$ orders in the
four-sublattice mean-field approximation. A few partial orders of quadrupoles
are stabilized in a wide range of parameter space at a higher transition
temperature than single-$q$ orders. With lowering the temperature, these
partial orders undergo phase transitions into further symmetry broken phases in
which nonvanishing quadrupole moments emerge at previously disordered sites.
The obtained phases in the mean-field approximation are investigated by a
phenomenological Landau theory, which clearly shows that the cubic invariant
plays an important role for stabilizing the triple-$q$ states. We also discuss
its implications for recent experiments in a few f- and d-electron compounds.
