Classification of finite subgroups of generalized cactus groups

We classify the finite groups that embed into generalized cactus groups $C_W$ associated with arbitrary Coxeter systems $(W,S)$. Every finite subgroup is conjugate to a subgroup of $C_{W_T}$ for some spherical subset $T\subseteq S$. This reduces the classification to explicit lists of finite groups for the finite irreducible Coxeter systems. For types $B$, $D$, $E$, $F$, and $H$, we analyze cube stabilizers to obtain these lists. For type $I_2(p)$, we determine $C_{W(I_2(p))}$ directly from its defining presentation. We explicitly realize every listed group as a subgroup of the corresponding generalized cactus group. When $S$ is finite, we determine the largest order and all possible orders of finite subgroups of $C_W$. In type $A$, we prove directly that a finite group embeds into the ordinary cactus group $J_n$ if and only if it embeds into a Sylow $2$-subgroup of the symmetric group $S_n$. We also give weighted embedding criteria in types $A$ and $B$ and counterexamples to Genevois’s conjecture on finite subgroups of cactus groups.