  We give a classification of the $p$--local stable homotopy type of $BG$,
where $G$ is a finite group, in purely algebraic terms. $BG$ is determined by
conjugacy classes of homomorphisms from $p$--groups into $G$. This
classification greatly simplifies if $G$ has a normal Sylow $p$--subgroup; the
stable homotopy types then depends only on the Weyl group of the Sylow
$p$--subgroup. If $G$ is cyclic mod $p$ then $BG$ determines $G$ up to
isomorphism. The last class of groups is important because in an appropriate
Grothendieck group $BG$ can be written as a unique linear combination of
$BH$'s, where $H$ is cyclic mod $p$.
