In a rectangular parallelepiped, $AB=4$, $BC=3$, and $CG=6$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?

1. Write $x=AB$, $y=BC$, and $z=CG$. The base $BCHE$ is a rectangle with side lengths $y$ and $BE=\sqrt{x^2+z^2}$, so its area is $y\sqrt{x^2+z^2}$.
2. The perpendicular distance from $M$ to plane $BCHE$ is $\dfrac{xz}{\sqrt{x^2+z^2}}$. Thus the pyramid volume is
   \[\frac13\cdot y\sqrt{x^2+z^2}\cdot\frac{xz}{\sqrt{x^2+z^2}}=\frac{xyz}{3}.\]
3. Substituting $x=4$, $y=3$, and $z=6$ gives volume $\boxed{24}$ cubic units.