Commuting graph of the symmetric group S4

The smallest groups have star commuting graphs when condensed. In other words, absent the center they form cluster graphs. The first notable exception to this is the symmetric group $S_4$ which has twenty four elements. The commuting graph is highly ordered so we can first consider the commuting preorder of $S_4$ which forms the following tree when condensed:

This tree structure tells you a lot about the commuting graph of $S_4$ but it doesn't allow you to completely recover the commuting graph. To do that, we need one extra clique.

Commuting graph:
The commuting graph of $S_4$ is formed by the comparability of the preorder, plus an additional clique consisting of all the intermediate elements of the tree and the maximal element. This maximal commuting clique forms the normal Klein-four subgroup of $S_4$ which consists of all elements of $S_4$ that have the highest commuting degrees.

Commuting principal filters:
All commuting principal filters of a semigroup form subsemigroups. It follows that all commutative principal filters of $S_4$ form subgroups. The maximal commutative principal filters of $S_4$ come in three forms up to permutation group isomorphism:

• The cyclic group $C_3$ formed by any of the side elements of the commuting tree.
• The cyclic group $C_4$ formed by one of the two preorder predecessors of the intermediate element
• The non-normal group $C_2 \times C_2$ formed by the other of the two preorder predecessors of an intermediate element. Disjoint transpositions in $S_4$ like (0 1) and (2 3) are commuting equal, and form a principal filter with the intermediate element (0,1)(2,3) and the maximal identity element.

These commutativity principal filters and the non-normal klein four subgroup fully determine the commuting graph of $S_4$.