Bekka-Mayer in their book below, in II.2 says:
If $\Gamma$ is a discrete and cocompact subgroup of a locally compact group $G$, then $\Gamma$ is a lattice in $G$.
I can't seem to prove it. Of course if $G/\Gamma$ has a finite $G$-invariant measure only then $\Gamma$ is a lattice. I couldn't show it directly.
Surely if one could show $\Delta_G|\Gamma = \Delta_\Gamma$, that would be enough as well. Note that $\Gamma$ being discrete, the modular function of $\Gamma$, $\Delta_\Gamma \equiv 1$.
Bekka, M., & Mayer, M. (2000). Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces (London Mathematical Society Lecture Note Series). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511758898