What are the finite subgroups of $SU_2(C)$? Is there any reference which classifies the finite subgroups of $SU_2(C)$ up to conjugacy ?
What I know is that lifting the finite subgroups of $SO_3(R)$ by the map $SU_2(C) \rightarrow PSU_2(C)$ gives rise to the groups : cyclic groups of even order, dicyclic groups, binary tetrahedral/octahedral/icosahedral groups. But I dont know if they are the only one.
 A: The image of a finite subgroup of $\text{SU}(2)$ in $\text{SO}(3)$ is a finite subgroup of $\text{SO}(3)$; moreover, the kernel is either trivial or $\{ \pm 1 \}$. But $-1$ is the unique element of order $2$ in $\text{SU}(2)$, so any group of even order contains it. 
I claim all the finite subgroups of odd order are cyclic. This follows because the inclusion $G \to \text{SU}(2)$ cannot define an irreducible representation of $G$ (since otherwise $2 | |G|$), hence it must break up into a direct sum of dual $1$-dimensional representations. 
So once you know the finite subgroups of $\text{SO}(3)$, you already know the finite subgroups of $\text{SU}(2)$. 
A: Here is a full list of (closed) subgroups of $ SU_2 $.
There are three maximal subgroups (up to conjugacy). The first maximal subgroup, $ N $, has dimension $ 1 $ and has two connected components
$$
N=\left \{\begin{pmatrix}e^{i\theta}&0 \\ 0 & e^{-i\theta}
\end{pmatrix} \mid 0\leq\theta<2\pi  \right \} 
\cup
\left \{\begin{pmatrix} 0 & -e^{-i\theta}\\ e^{i\theta}&0\end{pmatrix} \mid 0\leq\theta<2\pi  \right \} \\
=<U_1,\begin{pmatrix} 0 & -1\\ 1&0\end{pmatrix}>
$$
and contains subgroups
$$
U_1=\left \{\begin{pmatrix}e^{i\theta}&0 \\ 0 & e^{-i\theta}
\end{pmatrix} \mid 0\leq\theta<2\pi  \right \} 
$$
and all cyclic groups of order $ n $
$$
 C_n=\left \{\begin{pmatrix}e^{2 \pi k i/n}&0 \\ 0 & e^{-2 \pi k i/n}
\end{pmatrix} \mid 0 \leq k < n  \right \} 
$$
and all dicyclic groups of order $ 4n $ (note that a dicylic group has only one element of order $ 2 $ and so is not isomorphic to dihedral groups like $ D_n $ which always have $ n $ elements of order $ 2 $)
\begin{align*}
 {Dic}_{n}=&\left \{\begin{pmatrix}e^{2 \pi k i/2n}&0 \\ 0 & e^{-2 \pi k i/2n}
\end{pmatrix} \mid 0 \leq k < 2n  \right \} \\
& \cup
\left \{\begin{pmatrix} 0 & -e^{-2 \pi k i/2n}\\ e^{2 \pi k i/2n}&0\end{pmatrix} \mid 0 \leq k < 2n  \right \}\\
=&<C_{2n},\begin{pmatrix} 0 & -1\\ 1&0\end{pmatrix}>
\end{align*}
Note that we have SES
$$
 U_1 \to N \to C_2 
$$
which look like the SES for
$$
U_1 \to O_2(\mathbb{R}) \to C_2
$$
But these groups are distinct. In particular the only element of $ SU_2 $ with order $ 2 $ is the matrix with eigenvalues $ -1,-1 $
$$
\begin{pmatrix} -1 & 0\\ 0&-1\end{pmatrix}
$$
by contrast $ O_2(\mathbb{R}) $ has infinitely many elements of order $ 2 $. It is also interesting to note that $ U_1 $ is the maximal torus of $ SU_2 $ and that $ N $ is $ N(T) $ the normalizer of the maximal torus. So the short exact sequence above is really
$$
1 \to T \to N(T) \to W \to 1 
$$
where $ T $ is the maximal torus, $ N(T) $ is the normalizer and $ W $ is the Weyl group (the Weyl group for $ SU_2 $ is symmetric group on two letters, equivalently two element cyclic).
The two other maximal subgroups are finite: the binary icosahedral group $ 2.A_5 \cong SL(2,5) $ of order $ 120 $ and the binary octahedral group $ 2.S_4 \cong SL(2,3):2 $ of order $ 48 $,which has GAP ID $SmallGroup(48,28)$ . The intersection of these two groups is exactly $ 2.A_4 \cong SL(2,3) $ the binary tetrahedral group of order 24. These three binary polyhedral groups are the three exceptional subgroups of $ SU_2 $. They are the only primitive subgroups. They are also all Lie primitive (not contained in any proper positive dimensional closed subgroup).
To reiterate, the closed subgroups of $ SU_2 $ are classified exactly as follows:
The finite subgroups are given by the ADE classification with $ A_n $ corresponding to cyclic order $ n $, $ D_n $ corresponding to dicyclic order $ 4n $, $ E_6 $ corresponding to binary tetrahedral, $ E_7 $ to binary octahedral and $ E_8 $ to binary icosahedral.
There are only three closed subgroups of positive dimension: $ SU_2 $ itself, the maximal torus $ U_1 $, and $ N=N(T) $ the normalizer of the maximal torus.
A: Yes, the ADE classification enumerates all possible finite subgroups of $SU(2)$. Just a trivial correction, when it comes to the Abelian cyclic groups, A, the order may be both even and odd because $U(1)$ inside $SU(2)$ has all these $Z_N$ subgroups. The odd ones aren't linked to subgroups of $SO(3)$ in the same way.
The dicyclic groups, D, and the three exceptions, E, are as you wrote.
