What Happens When These 8 Planes Slice a Cube? 🤔

Описание: This problem about dissecting a cube was the final question in the 2003 Australian Mathematics Competition (Intermediate Division):
For each vertex of a cube, a plane is constructed through the three vertices which are neighbours of that vertex. Into how many parts do these eight planes dissect the cube? 
Give it a try, I think you'll find it surprisingly difficult to visualize!