  We consider 2-local geometries and other subgroup complexes for sporadic
simple groups. For six groups, the fixed point set of a noncentral involution
is shown to be equivariantly homotopy equivalent to a standard geometry for the
component of the centralizer. For odd primes, fixed point sets are computed for
sporadic groups having an extraspecial Sylow p-subgroup of order p^3, acting on
the complex of those p-radical subgroups containing a p-central element in
their centers. Vertices for summands of the associated reduced Lefschetz
modules are described.
