It is well known that the finite subgroups of $SL(2,\mathbb{C})$ up to conjugacy are the binary polyhedral groups (or Klein groups). There are two infinite families (cyclic groups and binary dihedral groups) and three exceptional groups (binary tetrahedral, binary octahedral, binary icosahedral). These groups and the related quotient singularities have a lot of interesting structure and properties. Some of them are listed in the following old and very nice question:


[The finite subgroups of SL(2,C)](https://mathoverflow.net/questions/16026/the-finite-subgroups-of-sl2-c)


I am interested in the following two related questions.


1) What about the finite subgroups of $GL(2,\mathbb{C})$? Every finite subgroup of $GL(2,\mathbb{C})$ can be obtained by extending a finite subgroup of $SL(2,\mathbb{C})$ with a cyclic group of appropriate order. However, if we do not pay attention we might obtain the same subgroup twice. Do you know a reference where this is worked-out carefully and a list of the groups together with generators for each group are given explicitely?


2) What about the finite subgroups of $SL(2,\mathbb{k})$ and $GL(2,\mathbb{k})$ where $\mathbb{k}$ is an algebraically closed field? I am interested in finite groups $G$ such that the characteristic of $\mathbb{k}$ does not divide $|G|$. Can we say that we have exactly the same groups as in the complex situation? Again any reference in this direction is highly appreciated. Since most references I found work only over $\mathbb{C}$, I wonder where problems may arise.



Finite subgroups of $GL(2,K)$ with $K\neq\mathbb{C}$
An answer to your questions is provided by this article of Beauville:


*Beauville, Arnaud*, [**Finite subgroups of (\mathrm{PGL}\_2(K)).**](http://www.arxiv.org/abs/0909.3942), García-Prada, Oscar (ed.) et al., Vector bundles and complex geometry. Conference on vector bundles in honor of S. Ramanan on the occasion of his 70th birthday, Madrid, Spain, June 16–20, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4750-3/pbk). Contemporary Mathematics 522, 23-29 (2010). [ZBL1218.20030](https://zbmath.org/?q=an:1218.20030).


A final version is available [here](https://math.unice.fr/~beauvill/pubs/PGL%282%29.pdf).


