As others have done, this is easier by analyzing the individual cubelets. I’m assuming throughout that this is a standard Rubik’s cube which has gotten into whatever state it is by the usual twisting of faces, i.e. it hasn’t been taken apart and put back together in such a way that the cube is not longer solvable.

There is only one possibility for each of the center cubes; it is not possible to move these.

I can see seven of the eight corner cubes, and can deduce they are all correctly placed. Since it is impossible to twist a single corner cube by standard Rubiks manipulation without eventually twisting another cube in the exact opposite direction, we must conclude the eighth, unseen corner cube is placed properly (If you have a cube handy, try the following manipulation on a solved cube: F-, B, B, F, T, F-, B, B, F, where “F” indicates “rotate the front face clockwise”, “F-” = “rotate the front face counter-clockwise”. When completed, two of the corner cubelets on the top of the cube will have rotated; one 1/3 clockwise, one 1/3 counterclockwise).

The edges, then, are the only possible cubelets on the unseen sides that can be out of position. Six of these are partially unseen, three are completely unseen. The six partially unseen can be grouped in pairs with like color showing; these are each either correctly placed or swapped, and so there are 2*2*2=8 possibilities.

The three completely unseen edges, I think, can be placed in any of the three unseen edge positions, making for six possibilities. Finally, each of these edge piece can be in one of two orientations, but as with the corner pieces, you can’t flip one edge cubelet without flipping another in standard Rubik’s moves, limiting the total number of orientations to 4 (all 3 are correctly oriented, or only one of the three is correctly oriented). Thus, for the unseen edges, there are 6*4=24 possible orientations.

Multiply 6*4*8 = 192. It’s a back-of-the-envelope analysis, so I’d appreciate any clarifications/corrections. My guess is there is some other interaction between the position of the edge cubelets and their orientation which could limit this number further, but I don’t know what that could be.