The Enhanced Hasse diagram of the subgroup structure of the symmetries of the cube.

 This one took some work. Let me explain it.

 

The numbers along the left side give the order of the subgroups. There is one subgroup of order 1, nine subgroups of order 2, four subgroups of order 3, four subgroups of order 4, four subgroups of order 6, three subgroups of order 8, and only one subgroup of order 12 and one of order 24. 

 

Circles indicate normal subgroups. There are four, and their orders are 1, 4, 12 and 24.

 

Squares indicate subgroups that are not normal, and the ovals are subnormal.

 

Of the nine subgroups of order 2, three are generated by even permutations and six by odd permutations. The even permutation subgroups each connect to two subgroups of order 4. Each one is matched directly to one of the subnormal subgroups above it, while all are subgroups of the normal subgroup.

The six odd permutations generate subgroups of one of the order 8 subgroups and two of the order 6 subgroups. This creates a lot of traffic in the middle of the diagram, so I assigned a color of the rainbow to each of the six order 2 subgroups and all the arrows leading up to the higher levels from a single order 2 subgroup are the same color. For example, the first odd permutation order 2 subgroup is connected with red arrows to the second order 8 subgroup and the to the first and third order 6 subgroups.

 

None of these six subgroups are normal or even subnormal.

 

My next task is explaining the symmetries of the dodecahedron, a group that is isomorphic to the symmetries of the icosahedron. It is a group of order 60 and will be represented as a subgroup of S12. The subgroup structure is much more complex than the group we have been working on, and I'm not sure I'm up to making an enhanced Hasse diagram of it.

 

Time will tell.