Cut a $4 \times 4 \times 4$ cube into 64 $1 \times 1 \times 1$ small cubes, then dye 16 of the $1 \times 1 \times 1$ small cubes red, requiring that among any 4 small cubes parallel to any edge, exactly 1 small cube is dyed red. The number of different dyeing methods is $\qquad$ (dyeing methods that are the same after rotation are also considered different dyeing methods).