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I ask a question about finite subgroup of $GL(2,\mathbb{C})$.

I want to classfy the finite subgroup of $GL(2,\mathbb{C})$

(especially, subgroups of order $p^3$ , $p$ is prime number.)

Is it possible to classfy?

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    The list of finite subgroups of that group is well-known. Google should find you the list and the proofs easily! Try "finite subgroups of gl(2,c)" – 2017-02-26
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    What is the word that I should search using goolge? – 2017-02-26
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    "finite subgroups of gl(2,c)", unsurprisingly (wrap the words between quote marks) – 2017-02-26
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    Is this information from google search only SL(2,C)? Of course, it contains SL(2,C). But, I think it doesn't contain GL(2,C). – 2017-02-26
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    See [this MO-question](http://mathoverflow.net/questions/55995/finite-subgroup-of-gl-2c) for references. – 2017-02-26
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    It follows from the arguments in the answers to [your other question](http://math.stackexchange.com/q/2162430/11619) that a subgroup of $GL_2(\Bbb{C})$ of order $p^3$ is either abelian, or $p=2$. Is that clear to you, or do you want it detailed? – 2017-02-27
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    I want details just in case. – 2017-02-27

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If $G$ is a finite subgroup of $\rm{GL}(2,\mathbb{C})$ then we can state this as there is a homomorphism (representation) $\rho:G\rightarrow \rm{GL}(2,\mathbb{C})$.

It is a standard result that this representation we can look into subgroup $U(2,\mathbb{C})$ of unitary matrices (See, Artin's Algebra).

So, your question reduces to finding finding finite subgroups of $U(2,\mathbb{C})$. This has been well studied.

For example, in Complex Functions by Jones and Singerman, the beginning chapter is devoted to study of finite subgroups of this group. Since the complete proof is very long, it is appropriate here just to mention its source.