Each complex representation of a finite group is semisimple (i.e. decomposes into a direct sum of irreducible complex representations).
Question: What are the countable groups satisfying the same property?
Remark: We do not restrict to finite dimensional representations.
The characterization is two-way: all $\mathbb C$ representations are semisimple iff $G$ is finite.
(Corollary page 660 in Connell, Ian G. On the group ring. Canad. J. Math. 15 1963 650--685)
This is essentially what Maschke's theorem is about. I don't think finite dimensionality of representations comes into play.