Let $G$ be the group $SO(n)$, $SU(n)$ or $Sp(n)$.

Let $F_m$ be the collection of finite subgroups of $G$ such that its order is bounded by $m$. Two elements in $F_m$ are identified if they are conjugate in $G$.

Can we prove that $\#F_m \le C(n,m)$ for some finite number $C(n,m)$?