Let $G$ be a connected, simply connected, semi-simple linear algebraic group over an algebraically closed field $k$. Then $G$ acts on its Lie algebra under the adjoint action. We will consider the nilpotent orbits under this action. The geometry of the nilpotent orbit closures has been studied extensively, in particular the question whether or not these orbits have normal closure. If $k$ is of characteristic zero, then this question has been answered for all nilpotent orbits when $G$ is of type $A$, $B$, $C$ or $D$. In 2003 Eric Sommers classified the nilpotent orbits with normal closure when $G$ is of type $E_6$ and the characteristic of $k$ is zero. In this thesis we prove that this classification remains valid if the characteristic of $k$ is good for $G$.