Subgroups and Subspaces of Z/4Z x Z/4Z x Z/4Z

The elements of G = Z/4Z x Z/4Z x Z/4Z can be considered either as elements of an additive group or as points in a vector space. If we think of G as a group, we can look at the set of subgroups of G. Ordering those subgroups by inclusion, we get the partially ordered set described by the diagram below and to the left. The lowest vertex of the cube-like structure corresponds to the trivial group containing only (0,0,0) and the uppermost vertex corresponds to the whole group.
subgroup lattice

If we consider G as a six dimensional vector space over Z/2Z, we can study the subspaces invariant under the action of matrices in GL6(Z/2Z) with the following Jordan form:
0 1 0 0 0 0
0 0 0 0 0 0
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 0 0

The partially ordered set obtained by ordering these subspaces by inclusion is shown above on the right. The two sets look surprisingly similar. The real surprise is that they differ in just four places, indicated in the pictures by colored edges in the upper right corner.

An interesting feature of the subgroup poset is its symmetry; shown below is a projection of the poset lattice along its axis of 180 degree rotational symmetry.
symmetry of subgroup lattice

More information about these sets can be found on Lynne Butler's undergraduate research page. A diagram of the subgroup poset of Z/4Z x Z/2Z x Z/2Z is also available.

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