Finite subgroups of $PSU(3)$ I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
 A: As others have said, this is well documented in the literature, though recent references are fairly sparse.
Here are some sketch arguments (none original). The interesting case is that of finite subgroups.
Note that finite subgroups of ${\rm GL}(3,\mathbb{C})$ are conjugate to subgroups of the corresponding unitary group.
Note also that since $\mathbb{C}$ is divisible, given a finite irreducible subgroup $X$ of ${\rm GL}(3,\mathbb{C}),$ we may replace generators of $X$ by scalar multiples of determinant $1$ to produce an irreducible group $Y$ with $Y/Z(Y) \cong X/Z(X).$
Note that $Y$ is a subgroup of ${\rm SL}(3,\mathbb{C}).$
Hence you really want to know the finite irreducible subgroups of ${\rm GL}(3,\mathbb{C})$ (modulo their centers).
These have long been classified- it would take too long to go over that classification in detail here, so I invoke that classification. Let $G$ be such a group, which, as remarked above may be taken to lie inside ${\rm SL}(3,\mathbb{C})$ with no loss of generality: there is a further subdivision.
If (up to equivalence), the given representation of $G$ can be induced from (necessarily $1$-dimensional representation of) a proper subgroup, then $G$ has an Abelian normal subgroup $A$ with $G/A$ a subgroup (of order divisible by $3$) of the symmetric group $S_{3}.$
If the given representation can not be so induced, then all Abelian normal subgroups of $G$ are central (and of order dividing $3$ by unimodularity). This is the so-called primitive case.
In this case (from Blichfeldt), when $G$ has no central-normal solvable subgroup, we have $G/Z(G) \cong A_{5}, {\rm PSL}(2,7)$ or $A_{6}.$ If (still in the primitive case) $G$ has a non-central solvable normal subgroup, then $G$ has an extra-special normal subgroup $U$ of order $27$ such that $G/U$ is isomorphic to a subgroup (of even order) of ${\rm SL}(2,3).$
