jmc

First, we look at $△MDN.$ We know that $DM=4,$ $DN=2,$ and $\angle MDN=60^{\circ }$ (because $△EDF$ is equilateral). Since $DM:DN=2:1$ and the contained angle is $60^{\circ },$ $△MDN$ is a $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangle. Therefore, $MN$ is perpendicular to $DF,$ and $MN=\sqrt{3}DN=2\sqrt{3}.$

Next, we calculate $CP.$ We know that $QC=8$ and $\angle QCP=60^{\circ }.$ Since $MN\perp DF,$ plane $MNPQ$ is perpendicular to plane $BCDF.$ Since $QP||MN$ (they lie in the same plane $MNPQ$ and in parallel planes $ACB$ and $DEF$), $QP\perp CB.$

Therefore, $△QCP$ is right-angled at $P$ and contains a $60^{\circ }$ angle, so is also a $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangle. It follows that $$CP=\frac{1}{2}(CQ)=\frac{1}{2}(8)=4$$and $QP=\sqrt{3}CP=4\sqrt{3}.$

Then, we construct. We extend $CD$ downwards and extend $QM$ until it intersects the extension of $CD$ at $R.$ (Note here that the line through $QM$ will intersect the line through $CD$ since they are two non-parallel lines lying in the same plane.) $△RDM$ and $△RCQ$ share a common angle at $R$ and each is right-angled ($△RDM$ at $D$ and $△RCQ$ at $C$), so the two triangles are similar. Since $QC=8$ and $MD=4,$ their ratio of similarity is $2:1.$ Thus, $RC=2RD,$ and since $CD=16,$ $DR=16.$ Similarly, since $CP:DN=2:1,$ when $PN$ is extended to meet the extension of $CD,$ it will do so at the same point $R.$ Finally, we calculate the volume of $QPCDMN.$ The volume of $QPCDMN$ equals the difference between the volume of the triangular -based pyramid $RCQP$ and the volume of the triangular-based pyramid $RDMN.$

We have $$[△CPQ]=\frac{1}{2}(CP)(QP)=\frac{1}{2}(4)(4\sqrt{3})=8\sqrt{3}$$and $$[△DNM]=\frac{1}{2}(DN)(MN)=\frac{1}{2}(2)(2\sqrt{3})=2\sqrt{3}.$$The volume of a tetrahedron equals one-third times the area of the base times the height. We have $RD=16$ and $RC=32.$ Therefore, the volume of $QPCDMN$ is $$\frac{1}{3}(8\sqrt{3})(32)-\frac{1}{3}(2\sqrt{3})(16)=\frac{256\sqrt{3}}{3}-\frac{32\sqrt{3}}{3}=\boxed{\frac{224\sqrt{3}}{3}}.$$