The lower bound you get is actually the exact number of reachable different-looking configurations. I'd write it as
$$ \frac{8! \cdot 24! \cdot 12!}{2} \cdot 3^7\cdot 2^{11}\cdot \frac{24!}{4!^6} \cdot \frac{24!}{4!^6} $$
The first factor counts the number of reachable permutations of the non-center cubies, ignoring orientation. The two middle ones count the number of possible orientations for the corners and central edges. Finally the last ones count the number of distinguishable states for each of the two sets of 24 movable center panels.
In order to see that this many different arrangements are actually possible, we can observe:
Of course the 8 corners are all different, as are the 12 central edges.
The 24 non-central edge pieces are all different, when we take into account that the outer end of such a piece stays the outer end no matter how we twist the cube. So seen from that end of the edge it has a distinct left and right face, and the colors of these two faces determine which piece it is.
So the two denominators of $4!^6$ for the center panels is all that is needed to take into account states that look the same but actually have physical pieces switched around. (By ignoring the middle centers completely in the count, we're implicitly taking account of the fact that they look the same in all four orientations).
So all that is needed is to convince ourselves that we can actually get to each of the positions we have counted, starting with a solved cube.
First get the $24!$ factor done with by moving the non-middle edges around such that each of them is next to the corner piece we want them to end up next to, and on the right side of it. During this phase the corner pieces themselves make no net movement. We don't care what happens to the middle edges or center pieces in this phase, so we can treat the cube as a 4×4×4 one with an irrelevant middle layer sandwiched into each dimension. Getting the non-center edges into place can be done with the 4×4×4 combination for flipping two neighbor edges together, and appropriate adaptations of that. (Here "adaptation" means "do something that brings the two edges you want to swap next to each other, not caring about anything else, then swap them, and finally do the initial something backwards").
Next get the the corners and the middle edges into the desired positions. If we work with only the two middle ones of the four cuts in each dimension, the cube will work like a plain old 3×3×3, and the non-middle edges will stay next to the corners we've just anchored them to. So the number of choices we have in this phase is exactly the number of positions on a 3×3×3 cube, namely $\frac{8!\cdot 12!}{2}3^7 2^{11}$.
In the first two phases we haven't cared about getting the center panels scrambled, but now finally handle the two factors of $\frac{24!}{4!^6}$ one by one by arranging each group of 24 center tiles as we want them. For each such group, there's a combination that cyclically permutes three centers pieces on different sides without touching anything else on the cube. Again, appropriate adaptations of this will allow us to do anything -- officially 3-cycles are not enough to do everything, but in pinch we can make the 3-cycle look like a plain swap of two pieces by choosing two of the three participants to be positions that are already of the same color, and plain swaps are (of course) enough to reach any desired pattern.
There's some tiny amount of group theory hidden in the talk about which permutations can be built up from which pieces, but it doesn't need to be phrased in those terms because the point is just that transpositions are enough to do everything. We don't need to tell that the "adaptations" of the basic operations I describe would be called conjugations by a group theorist, and we certainly don't need to mention that, say, the basic 3-cycle of center panels is constructed as a commutator.
(Note that this would be horribly slow as an actual solution algorithm, since it's optimized for making clear in a uniform way that all the states we've counted can be produced reachable).
