The finite subgroups of $SU(2)$ consist of the symmetry groups of the Platonic solids plus the finite subgroups of $O(2)$. I would like to know if there are any results concerning $SU(3)$. In general, are there classification results as to the finite subgroups of compact simple Lie groups over $\mathbb{C}$?

**Update:** Thanks everyone for the references.