Olympiad Geometry · Superchallenge problem

Cube $ABCDEFGH$ has edge $6$, with square $ABCD$ on the bottom and $E$, $F$, $G$, $H$ directly above $A$, $B$, $C$, $D$. A plane passes through vertex $A$, the midpoint of edge $\overline{FG}$ and the midpoint of edge $\overline{GH}$. It cuts the cube into two pieces. Find the volume of the smaller piece.