TITLE: Discrete subgroups of products of SU(2)
QUESTION [6 upvotes]: It is known that there are -up to conjugation- 5 classes of discrete subgroups of SU(2). One way to show this is by means of the McKay correspondence. My question is more regarding products of $SU(2)$, say with $n$ factors. Well, since I am definitely not familiar with the correspondence above, I would like to ask whether I have any hope to classify all finite subgroups of the product of 2 or 3 copies of $SU(2)$ by following the same line of thought. Or in case someone knows a more economical alternative, I would be very happy to listen to it.

REPLY [3 votes]: Incidentally, $SO(4)=SU(2)\times SU(2)/\{\pm (I\times I)\}$, so the classification of discrete subgroups of $SO(4)$ is (almost) equivalent to the classification of discrete subgroups of $SU(2)\times SU(2)$ (every discrete subgroup of $SU(2)\times SU(2)$ will correspond to a subgroup of $SO(4)$ containing $\pm I$). The discrete subgroups of $SO(4)$ correspond to orientable 3-dimensional spherical orbifolds, which have been explicitly worked out by Dunbar (see also here).