All finite groups are involved in the Mapping Class Group
  Let $\Gamma_g$ denote the orientation-preserving Mapping Class Group of the genus $g\geq 1$ closed orientable surface. In this paper we show that for fixed $g$, every finite group occurs as a quotient of a finite index subgroup of $\Gamma_g$. 
Gregor Masbaum and Alan W. Reid
math.GT