This is one of the problems that just gets hopelessly messy beyond a few small dimensions. The reason is that asking for all finite subgroups of isometries of Euclidean space is essentially the same as asking for all orthogonal representations of all finite groups, and since irreducible representations have dimension at most the square root of the order of the group, you have to use all groups of order up to at least n2 to find groups of isometries of Rn. A major problem in doing this is that there are huge numbers of nilpotent groups of order pn once n is larger than about 5; for example there are several hundred groups of order 64, all of whose irreducible representations have dimension at most 8. So my guess would be that classifying all groups of isometries in dimensions greater than about 10 will require a lot of obstinacy and a big computer.
(Added later) On checking the literature, I find that people classifying such subgroups usually make some simplifying assumptions, by only looking for ones that are irreducible, maximal, and that act on an integral lattice. With these extra simplifications one can get a bit further: the state of the art seems to be around 30 dimensions.