The Lie correspondence is well understood. For 'nice enough' Lie groups $G$ (with Lie algebra $\mathfrak{g}$) every sub-group $H < G$ has a Lie algebra $\mathfrak{h} < \mathfrak{g}$ given by $\log(H)$.
Rather than sub algebras I want to know about vector sub spaces of $\mathfrak{g}$. Clearly these don't exponentiate to sub Lie groups of $G$ in general. But what do they exponentiate to? Is there some clear correspondence between sub spaces of $\mathfrak{g}$ and sub something of $G$ and what is this something.
Also, the converse issue. Which subsets $S \subset G$ have that $\log(S)$ is a vector subspace of $\mathfrak{g}$?