$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SU{SU}\newcommand{\C}{\mathbb{C}}$My question is about large order finite non-abelian subgroups of $\GL_n\C$ without an abelian normal subgroup. I could look at $\SU_n$ as well since all finite subgroups of $\GL_n\C$ can be conjugated to an $\SU_n$ subgroup. For low $n$ it seems there isn't a regular pattern, for $\SU_2$ we have the icosahedral group, for $\SU_3$ there is the Valentiner group, for $\SU_4$ we have the result from Hanany and He (arXiv link) which gives a full classification. But Jordan's and Collins' well-known results say there might be something happening at $n=70$. I also understand that a full classification of all finite subgroups is hopeless for general $n$.

So let's limit ourselves to $n>70$, maybe we can find a pattern for these. Is it possible to give a finite non-abelian subgroup, without an abelian normal subgroup, parametrized by $n$, such that for any $n$ it's a “large” subgroup? What I mean is that it doesn't have to have the largest order (if finding the largest order finite non-abelian subgroup is impossible), but reasonably large number of elements. Or if having no abelian normal subgroup is difficult, let's have as small an abelian normal subgroup as possible.

I know this is rather vague. But the reason I'm hoping something like this is possible is that groups $\PSL_kq$ do show up for various $n$ and maybe, perhaps for only even or only odd $n$, one can have explicitly $k$ and $q$ as a function of $n$ such that the order of $\PSL_kq$ is “large” and is a subgroup of $\GL_n\C$.

In other words, are there specific embeddings of $\PSL_kq$ into $\GL_n\C$ which work for all sufficiently large $n$?

I went through the following, which partly inspired the question:

[1] The finite subgroups of SU(n)

[2] Finite simple groups and $ \operatorname{SU}_n $

[3] https://math.stackexchange.com/questions/497853/closed-lie-subgroups-of-su3

[4] https://math.stackexchange.com/questions/4272017/finite-maximal-closed-subgroups-of-lie-groups