HMMT 2018 U35

In a wooden block shaped like a cube, all the vertices and edge midpoints are marked. The cube is cut along all possible planes that pass through at least four marked points. Let $N$ be the number of pieces the cube is cut into. Estimate $N$. An estimate of $E>0$ earns $\lfloor 20 \min (N / E, E / N)\rfloor$ points.