1. Surprisingly, I found explicit lists of discrete subgroups of the orthogonal group O(n) for up to n=8 dimensions on the [wikipedia page for point groups][1], with rather unspecific references, however. Point groups is another name for discrete subgroups of O(n).
 2. There is an old sequence of two long papers by Threlfall and Seifert, part I [Mathematische Annalen 1931, Volume 104, Issue 1, pp. 1-70][2], part II [1933, Volume 107, Issue 1, pp. 543-586][3], where they apparently do the classification of discrete subgroups of SO(4) by associating to each element of SO(4) a pair of rotations from SO(3). (Although my native language is German, I had a hard time reading (through) this, because I am not used to the terminology that was used at that time.) [Addition: These results are mentioned in the book by Conway and Smith on quaternions and octonions; Conway and Smith say that the list is complete, but contains duplicates.]
 3. I have a rather wild conjecture (true up to three dimensions).
 *Every discrete point group in n dimensions is the symmetry group of an n-dimensional polytope which is the Cartesian product of regular polytopes, or a subgroup thereof.*

For example, the symmetries of an $m$-gonal anti-prism in 3-space are
contained in the symmetries of the $2m$-sided prism,
which is the 1-simplex $\times$ the regular $2m$-gon.

Since the regular polytopes are known in all dimensions, this would give
an easy way to obtain all finite point groups.
(at least in principle). Is this known? Or is this absurd? Is there an obvious counterexample?



  [1]: http://en.wikipedia.org/wiki/Point_group
  [2]: http://link.springer.com/article/10.1007%252FBF01457920
  [3]: http://link.springer.com/article/10.1007/BF01448910