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As a thought experiment consider the case of the familar 4x4x4 cube with a 2x2x2 cube embedded inside it, instead of the usual mechanism. I'll call this the "Interdimensional 4x4x4 cube" for lack of a better name. Now clearly if we turn the slices of the 4x4x4 cube it would have an effect on the internal 2x2x2 cube. Now moving the slice adjacent to the U face and moving the slice adjacent to the R face this would be the equivalent of turning the internal 2x2x2's U face and R face.

My question is: Is it possible to reach all the positions of the internal 2x2x2 without having any constraints on the 4x4x4 cube? How many positions are there?