Maximal subgroups of the Coxeter group W(H4) and quaternions

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H₄) of order 14,400. Its derived subgroup is the largest finite subgroup W(H₄)/Z₂ of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups W(H₂) × W(H₂)] ⋊ Z₄ and W(H₃) × Z₂ possess non-crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H₄) with sets of quaternion pairs acting on the quaternionic root systems.