  We investigate the existence of ground states for the focusing Nonlinear
Schr\"odinger Equation on the infinite three-dimensional cubic grid. We extend
the result found for the analogous two-dimensional grid by proving an
appropriate Sobolev inequality giving rise to a family of critical
Gagliardo-Nirenberg inequalities that hold for every nonlinearity power from
$10/3$ and $6$, namely, from the $L^2$-critical power for the same problem in
$\mathbb{R}^3$ to the critical power for the same problem in $\mathbb{R}$.
Given the Gagliardo-Nirenberg inequality, the problem of the existence of
ground state can be treated as already done for the two-dimensional grid.
