  The Cohen-Lenstra-Martinet Heuristics gives a prediction of the distribution
of $\operatorname{Cl}_K[p^\infty]$ whne $K$ runs over $\Gamma$-fields and
$p\nmid|\Gamma|$. In this paper, we prove several results on the distribution
of ideal class groups for some $p||\Gamma|$, and show that the behaviour is
qualitatively different than what is predicted by the heuristics when
$p\nmid|\Gamma|$.We do this by using genus theory and the invariant part of the
class group to investigate the algebraic structure of the class group. For
general number fields, our result is conditional on a natural conjecture on
counting fields. For abelian or $D_4$-fields, our result is unconditional.
