gr.group theory - What kind group can be realized as a Isometry group of some space?

Every group is the full group of isometries of a connected, locally connected, complete metric space:

de Groot, J. "Groups represented by homeomorphism groups."
Math. Ann. 138 (1959) 80–102.
MR119193
doi:10.1007/BF01369667

Being a group of symmetries is the same thing as being a group.

You may also be interested to know that every group is the full automorphism group of a graph, not just a subgroup. References for this and various refinements are given at the wikipedia page for Frucht's theorem.