A solid cube has side length $3$ inches. A $2$-inch by $2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? $\textbf{(A)}\ 7 \qquad$ $ \textbf{(B)}\ 8 \qquad$ $ \textbf{(C)}\ 10 \qquad$ $ \textbf{(D)}\ 12 \qquad$ $ \textbf{(E)}\ 15$