AMC 12 2012a Problem 22

Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P =\bigcup_{j=1}^{k}p_{j}$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$?

\textbf{(A)}\ 8 \textbf{(B)}\ 12 \textbf{(C)}\ 20 \textbf{(D)}\ 23 \textbf{(E)}\ 24