  On a Lie group $G$, we investigate the discreteness of the spectrum of
Schr\"odinger operators of the form $\mathcal{L} +V$, where $\mathcal{L}$ is a
subelliptic sub-Laplacian on $G$ and the potential $V$ is a locally integrable
function which is bounded from below. We prove general necessary and sufficient
conditions for arbitrary potentials, and we obtain explicit characterizations
when $V$ is a polynomial on $G$ or belongs to a local Muckenhoupt class. We
finally discuss how to transfer our results to weighted sub-Laplacians on $G$.
