Let $G$ be a locally compact Hausdorff group and $H \le G$ a closed subgroup. Are there properties of $H$ that implies that the quotient $G/H$ is compact?
My guess would be that $H$ needs to be sufficiently spread out, but I don't have a nice way of making this precise.
In the case of locally compact abelian groups, there is something that can be said, although it's not a "classification" but just an equivalent condition that is sometimes helpful: Denote A(H) to be the annihilator of H. If H is a closed subgroup of G, then G/H is compact (aka H is co-compact) iff its dual is discrete. Therefore the isomorphism $$\widehat{G/H} \cong A(H)$$ implies that H is co-compact iff A(H) a discrete subgroup of $\widehat{G}$. Again, this isn't a real classification, but it implies there is a bijective correspondence between closed subgroups with the property you mentioned and discrete subgroups of the Pontryagin dual of G.
