  A p-local finite group consists of a finite p-group S, together with a pair
of categories which encode ``conjugacy'' relations among subgroups of S, and
which are modelled on the fusion in a Sylow p-subgroup of a finite group. It
contains enough information to define a classifying space which has many of the
same properties as p-completed classifying spaces of finite groups. In this
paper, we examine which subgroups control this structure. More precisely, we
prove that the question of whether an abstract fusion system F over a finite
p-group S is saturated can be determined by just looking at smaller classes of
subgroups of S. We also prove that the homotopy type of the classifying space
of a given p-local finite group is independent of the family of subgroups used
to define it, in the sense that it remains unchanged when that family ranges
from the set of F-centric F-radical subgroups (at a minimum) to the set of
F-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion
systems, analogous to p-constrained finite groups, and prove that they in fact
all arise from groups.
