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)?