On the number of finite subgroups of a lattice

We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly with covolume. We obtain similar results for isotropy subgroups in lattices. Geometrically, this yields volume bounds for the number of strata in the natural stratification of a finite-volume locally symmetric orbifold.