I define a finite subgroup $H$ of a group $G$, finite-maximal if for any $g\in G\setminus H$, $\langle G,g\rangle$ is infinite. My question is now to find the finite-maximal subgroups of $\mathrm{SO}_n(\mathbb R)$ and of $\mathrm{O}_n(\mathbb R)$. It is clear that if $n=2$, there are no finite-maximal subgroups. If $n=3$, we get the (full) octahedral and (full) icosahedral symmetry groups. If $n=4$ I think we get $[3,3,5]$ and $[[3,4,3]]$, but probably also what Conway calls $\pm[O\times I]$. Are these first cases correct? What about higher values of $n$?