[mapuxbob] Ultimate deep cut

Consider a twisty puzzle that has the motions of all the order-2 deep-cut puzzles on a cube: the 2x2 Rubik's cube, the Skewb, and the 24-cube.  The cuts on each face are an orthogonal cross and each quarter square is divided into 4 diagonally, so 16 isosceles right triangles per face.  Additionally there are cuts along the edges.

Also add all the additional turns that are possible using these cuts which preserve the shape of the cube after the turn.  I think these are: Dino cube vertex turn, very small vertex turn in the style of Pyraminx, Helicopter cube edge turn.

These extra cuts make it a 2 layer puzzle between opposite faces, a 4 layer puzzle between opposite edges, and a 6 layer puzzle between opposite corners.

This seems very difficult to build mechanically so is best done virtually.  Virtual can prohibit jumbling moves which alter the cube shape.  Virtual can provide visibility to the internal pieces, i.e., the "super super" or "real" version of the cube.  How many internal pieces are there?

Inspired by 2 xscreensaver hacks, one demonstrating various twisty puzzles and the other the Lament cube.