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@christianp Oh, nice idea! The subgroup structure looks pretty complicated: https://en.wikipedia.org/wiki/Rubik%27s_Cube_group#Subgroups

The one for the centre of the group would be interesting: the centre consists of {e, superflip} where superflip leaves all pieces in the correct positions but rotates the edges through 180 degrees: https://en.wikipedia.org/wiki/Superflip

@christianp I think I'd actually flip your idea, Galois-connection stylee, and have each cube be invariant under the subgroup it represents. So the {e, superflip} cube has the usual face and corner pieces, and each edge is a rotation-invariant pattern of the two colours that normally live on that edge.

@christianp I'd want to lay them out in a pattern showing the subgroup inclusion structure, perhaps a set of little pedestals with arrows between them.

@pozorvlak I wonder if this is feasible to do online, before making a physical version

@christianp I'd definitely want to at least enumerate the subgroups before I started buying cubes! Do you know how many there are? Wikipedia gives the structure as $({\mathbb{Z}}_3^7\times {\mathbb{Z}}_2^{11})⋊((A_8\times A_{12})⋊{\mathbb{Z}}_2)$ - if you went down that far you'd want ten cubes (the whole group, ${\mathbb{Z}}_3^7\times {\mathbb{Z}}_2^{11}$, ${\mathbb{Z}}_3^7$, etc), but most of those components have lots of subgroups of their own.

@pozorvlak oh, there are thousands and thousands, so you could only hope to pick a sample of them