Those of you who said 24 got it right. There are 6 faces that can be on the bottom and for each you can rotate the bottom face four times.
Now here is the connection to rotation axes. If you fix the center of any object in space then it may or may not be intuitively obvious to you that for any orientation, you can reach it by a single rotation about an appropriate axis through the center.
So what are the rotation axes for a cube. Everyone knows the three obvious ones, the lines through the centers of opposite faces. How many orientations do these axes account for. Each one allows for 4 positions, but the starting position is the same for each, so each axis only contribute 3 positions in addition to the starting one. The total is therefore 1 + 3 + 3 + 3 = 10, far short of the total of 24.
If, like me, you find it hard to visualize in 3 dimensions, this site shows the other rotation axes:
http://www.luc.edu/faculty/spavko1/minerals/prelims/plato/cube-main.htm. The site designates the above axes as C4.
There are two other types of axis. One goes through the long diagonal and are designated as C3 at the Web site. There are 4 of them. At the end of the diagonals, 3 faces come together, so each of these axes has 3 postions and contribute 4*2 =8 additional positions.
The other rotation axis goes through the centers of two “diagonally opposite” sides and are designated C2. The 12 sides of the cube can be divided up into 6 pairs of such sides and each axis has 2 positions, accounting for 6*1=6 new positions.
Adding the total for each type of axis gives 10 + 8 + 6 = 24 postions.