  Let $\Gamma$ be a discrete group. When $\Gamma$ is nonamenable, the reduced
and full group $C$*-algebras differ and it is generally believed that there
should be many intermediate $C$*-algebras, however few examples are known. In
this paper we give new constructions and compare existing constructions of
intermediate group $C$*-algebras for both generic and specific groups $\Gamma$.
