  We show that the $p$-group complex of a finite group $G$ is homotopy
equivalent to a wedge of spheres of dimension at most $n$ if $G$ contains a
self-centralising normal subgroup $H$ which is isomorphic to a group of Lie
type and Lie rank $n$ in characteristic $p$. If in addition every order-$p$
element of $G$ induces an inner or field automorphism on $H$, the $p$-group
complex of $G$ is $G$-homotopy equivalent to a spherical complex obtained from
the Tits building of $H$.
  We also prove that the reduced Euler characteristic of the $p$-group complex
of a finite group $G$ is non-zero if $G$ has trivial $p$-core and $H$ is a
self-centralising normal subgroup of $G$ which is a group of Lie type (in any
characteristic), except possibly when $p=2$ and $H=A_n(4^a)$ ($n\geq 2$) or
$E_6(4^a)$. In particular, we conclude that the Euler characteristic of the
$p$-group complex of an almost simple group does not vanish for $p\geq 7$.
