Cactus groups arise from the Deligne–Mumford compactification of the moduli space of stable real genus-zero curves with n+1 marked points: the pure cactus group is its fundamental group, while the full cactus group is naturally related to the corresponding quotient orbifold. They also appear as analogues of braid groups in the theory of coboundary categories. Recently, Genevois initiated a systematic study of cactus groups from the viewpoint of geometric group theory, showing that they admit geometric actions on CAT(0) cube complexes and are virtually cocompact special. In this talk, using the geometry of CAT(0) cube complexes, we will discuss the structure of subgroups of cactus groups. At first, we determine the possible maximal rank of a free abelian subgroup. Secondly, we classify which finite groups can be realized inside cactus groups. Both results resolve conjectures of Genevois. As an application of the first result, we show that cactus groups are not quasi-isometric to any right-angled Artin group. The first result is joint work with Jan Kim.