It is shown that the mod $3$ cohomology of a $1$-connected, homotopy associative mod $3$$H$-space that is rationally equivalent to the Lie group $E_6$ is isomorphic to that of $E_6$ as an algebra. Moreover, it is shown that the mod $3$ cohomology of a nilpotent, homotopy-associative mod $3$$H$-space that is rationally equivalent to $E_6$, and whose fundamental group localized at $3$ is non-trivial, is isomorphic to that of the Lie group $\Ad E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra. It is also shown that the mod $3$ cohomology of the universal cover of such an $H$-space is isomorphic to that of $E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra.