It was shown by Alan Turing that (in a certain precise sense) the only connected Lie groups approximable by finite groups are the compact abelian Lie groups, i.e. $U(1)^n$. See Theorem 2 of [this paper][1].

If you allow infinite discrete subgroups then an approximable connected Lie group must be nilpotent and further any simply-connected nilpotent Lie group is approximable. See [this paper][2].


  [1]: https://www.jstor.org/stable/1968716?seq=5
  [2]: https://www.ias.ac.in/describe/article/pmsc/124/01/0037-0055