Why are there at most 24 rotational symmetries of a cube? Proposition 2 of this article: "Symmetry Groups of Platonic Solids" claims that the rotational symmetry group of a cube is isomorphic to $S_4$.

Note that there can be at most 24 rotational symmetries of a cube. It is possible to couple the opposite vertices of the cube, because if one of the
  vertices in the pair moves, the other pair would move correspondingly to remain an opposite vertex. Since there are four such pairs of vertices, the maximum rotational symmetries is 24, provided that all the permutation of these four pairs of vertices can be found. Indeed, inspection shows that this can be done.

We can indeed find 24 rotational symmetries acting on the four main diagonals. However, why does this imply that there are at most 24 rotational symmetries of a cube? Specifically, there are 6 faces/8 vertices/12 edges, why don't we try to find more rotational symmetries acting on them? Furthermore, why don't we try to find more rotational symmetries acting on some other objects embedded in a cube? 
 A: One way to think about the rotation symmetries of a cube is a permutation of the diagonal lines passing through the cube, as shown below.

In this way, we can count the symmetries. 
If we choose an arbitrary diagonal, that diagonal has four diagonals it can be mapped to. If we choose another diagonal, there are only three choices for where it can be mapped to. Choosing another diagonal, there are two choices, and finally the last diagonal only has one place it can go. This gives us $4\cdot3\cdot2 = 24$ possible symmetries. 
A: Consider the set of oriented edges of the cube, so that each of the edges counts twice, once in one direction and once in the other. There are 24 of them.
Now a symmetry of the cube maps oriented edges to oriented edges, and is complete determined to what it does to one of them. If follows at once that there are at most 24 symmetries.
A: This book "Abstract Algebra - Theory and Applications" by Thomas Judson, 2016; Page 89 argues as follows:

A cube has 6 sides. If a particular side is facing upward, then there are four possible rotations of the cube that will preserve the upward-facing side. Hence, the order of the group is 6 · 4 = 24. 

