I would like to have an overview of how a subgroup of a vector space over $\mathbb R$ of dimension $n$ can look like.
Is there a complete classification available? I know that there are for examples the linear subspaces, subgroups which are ismorphic to $\mathbb Z^k$ or $\mathbb Q^k$ (with $k\leq n$) and probably many more.
What if I impose some extra condition? For example, I know that the only discrete subgroups are those isomorphic to $\mathbb Z^k$.
So what happens if I ask for local compactness? (I know this rules out the subgroups isomorphic to $\mathbb Q^k$). Or for the Lie subgroups? Are in this case the linear subspaces and the discrete subgroups the only candidates?