Let the side of the large cube = $x$

Number of smaller cubes along one edge = 4 (since the side of the small cube is 1/4th of the large cube)

Total number of smaller cubes = $4^3 = 64$

Small cubes with only one painted face are on the center of each face of the large cube.

Number of small cubes on one face = $2 \cdot 2 = 4$

Total number of small cubes with only one face painted = $4 \cdot 6 = 24$.