Let's call the plane orthogonal to and passing through the midpoint of a segment $\overline{AB}$ the "medial plane" of $\overline{AB}$. There exist three different kinds of medial planes: the medial planes of the edges of $C$ (there are 3 different such planes); the medial planes of the diagonals of the faces of $C$ (there are 6 of such planes); the medial planes of the diagonals of $C$ (3 more planes). They divide $C$ into triangular pyramids, all of them having the center of $C$ as vertex. Moreover they divide each face of $C$ into triangles, that serve as bases for the pyramids. It's easy to see that each face is divided in exactly 16 triangles, so that the total number of pieces is $16 \times 6 = 96$.