The enumeration of finite subgroups of $\operatorname{PGL}_2(\mathbb{C})$ is one of the classic classification problems: mathematicians in many fields know well that the answer is cyclic groups, dihedral groups, $A_4$, $S_4$ and $A_5$. Moreover each of these groups occurs exactly once, up to conjugacy.
I hadn't thought about this classification very much for many years (if at all), but recently I have had some impetus to do so. In particular I am currently reading this wonderful note of A. Beauville, which treats the classification of finite subgroups of $\operatorname{PGL}_2(K)$ for any field $K$ of characteristic $0$. Here the new work is not so much figuring out exactly which groups can be realized over a given $K$, but rather working out the classification up to $K$-rational conjugacy, which turns out to be an interesting problem in Galois cohomology.
In some recent ruminations I have been thinking about Galois cohomology of finite subgroups of $\operatorname{PGL}_N(K)$, but it occurs to me that I don't know anything concrete past $N = 2$. Well, I can see that every finite group occurs as a subgroup of $\operatorname{PGL}_N(K)$ for any field $K$ and sufficiently large $N$, so obviously one can only ask for so much.
So...what about $\operatorname{PGL}_3(\mathbb{C})$? In particular:
1) What are the finite subgroups of $\operatorname{PGL}_3(\mathbb{C})$.
2) Is it still the case that if a finite group can be embedded in $\operatorname{PGL}_3(\mathbb{C})$, the embedding is unique up to conjugacy? If so, is there a general principle at work here?
Note that Beauville gives proofs of everything in the $\operatorname{PGL}_2$ case. But much of his classification arguments turn on the "accidental isomorphism" $\operatorname{PGL}_2 \cong SO(q)$, where $q = x^2 + yz$. Perhaps if these arguments generalized to $\operatorname{PGL}_N$ in some well-known way, he would have given proofs which generalize as well...