It is well-known that finite subgroups of $PGL\_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are classical.


If $K$ is an arbitrary field then theres are Beauville's notes about $PGL\_2(K)$:
<http://arxiv.org/abs/0909.3942>


In dimension $n=3$ there also classical results about subgroups of $PGL(3,\mathbb{C})$. One can also show the connection between subgroups of $SU(3)$ and $PGL(3,\mathbb{C})$. See this discussion for details:
<https://math.stackexchange.com/questions/42904/finite-subgroups-of-pgl3-c>


So my questions is what can we say about finite subgroups of $PGL(3,K)$ where $K$ is not necessarily algebraicly closed? More concretely, knowing, for example, finite subgroups of $PGL(3,\mathbb{C})$ can one classify finite subgroups in $PGL(3,K)$ for any subfield $K\subset\mathbb{C}$ (up to conjugacy)?



Finite subgroups of $PGL(3,K)$ 
*Edited in view of Derek Holt's comment on Schur indices*: These things are well studied in the literature. You probably want to restrict to irreducible subgroups, and it's probably just as well to work with ${\rm GL}(3,K).$ In such a low dimensions, the Schur index usually will not play much of a role. The Schur index usually arises because it can happen that a complex irreducible character $\chi$ may take values in a field $K,$ but might not be afforded by a representation over $K.$ There is, however, a smallest integer $m\_{K}(\chi)$ such that the character $m\_{K}(\chi) \chi$ is afforded by a representation over $K$ and $m\_{K}(\chi)$ divides $\chi(1).$ If $m\_{K}(\chi) =3,$ then representation affording $\chi$ can only be realised over a degree $3$ extension of $K$.Except in degenerate cases, we won't have irreducible representations over $K$ which are not absolutely irreducible, since such a a representation would break up over some extension field of $K$ as a sum of Galois conjugate representations of the same degree.
 The finite irreducible subgroups of ${\rm GL}(3,\mathbb{C})$ have been known for a century or so. Such an imprimitive group has an Abelian normal subgroup $A$ such that $G/A$ is isomorphic to a subgroup of $S\_{3}.$ The primitive ones may be rescaled so that all elements are unimodular, and once this is done, we obtain $G/Z(G)$ isomorphic to $A\_{5}, A\_{6},{\rm PSL}(2,7)$ or else $G$ is a solvable group with $G/O\_{3}(G)$ isomorphic to ${\rm SL}(2,3)$ and $[G:Z(G)] = 216.$


