The 4^3 doesn't have a supergroup in the sense of the 3^3 --
the orientations of the ceter cubies are determined by their positions.

However, there is one fairly natural adjunct group that people might
try thinking about and solving. A 4^3 shows 24 center cubies,
24 edge cubies, and eight corner cubies. But if it were really
a solid cube chopped up by parallel slices, it would have eight
more cubies buried inside. Call them stomach cubies. The eight
stomach cubies form a 2^3 buried in the 4^3. They move when you
twist slices. Can people come up with tools to frob the
stomach cubies without disturbing the visible cubies? What is
the order of the adjunct group?

--- Allan