My question is:


How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)?


With more discribe:


If $S^n\backslash \Gamma$ is a manifold, 


I just want to know that how many subgroups of SO(n) can be realized as the $\Gamma$ above?


If for generic n, it may be difficult, can we just figure out the case when n=3,4, I only care about $S^3$ and $S^4$



A Question about SO(n)
There are a pair of double covers $\text{SU}(2) \times \text{SU}(2) \to \text{SO}(4) \to \text{SO}(3) \times \text{SO}(3)$, and the first resp. the second more or less reduces the classification of finite subgroups of $\text{SO}(4)$ to the classification of finite subgroups of $\text{SU}(2)$ resp. $\text{SO}(3)$ (these classifications are in turn more or less equivalent and well-known) using [Goursat's lemma](http://en.wikipedia.org/wiki/Goursat%27s_lemma). This classification can be found more explicitly in Conway and Smith's [*On Quaternions and Octonions*](http://books.google.com/books/about/On_Quaternions_and_Octonions.html?id=E_HCwwxMbfMC) (Section 4.3). 


There is also a double cover $\text{Sp}(2) \to \text{SO}(5)$ which reduces the classification of finite subgroups of $\text{SO}(5)$ to the classification of finite subgroups of $\text{Sp}(2)$. This is probably easier but I don't know if it's well-known (in any case I don't know it). 


