Allow to answer this in greater detail, and describe all the important products of groups. It is one of the marvels of Rubik's cube that all these products arise in a very natural fashion when one investigates it. Here is a little picture to describe the situation. So M and N appear both as subgroups and quotients of the direct product.
Direct products are very simple.
GC is the stabilizer of the edges, i.e., the subgroup of those elements that do not permute the edges at all, and thus operates only on the corners. GE is the stabilizer of the corners, i.e., the subgroup of those elements that do not permute the corners, and thus operates only on the edges. A picture for this situation would be identical to the picture for the direct product, since the only real difference to the direct product is that H need not be normal. Let G be the subgroup of S generated by the six face turns. Let C be the subgroup of S generated by the rotations of the entire cube, which has size 24.