Possible Candidates for SUSY E$_6$ GUT with an Intermediate Scale

We study the possibility of an intermediate scale existing in supersymmetric E$_6$ grand unified theories. The intermediate scale is demanded to be around $10^{12}$ GeV so that neutrinos can obtain masses suitable for explaining the experimental data on the deficit of solar neutrinos with the Mikheev-Smirnov-Wolfenstein solution and the existence of hot dark matter. We require that at the intermediate scale, a certain symmetry breakdown to the Standard Model symmetry. We show that only a few E$_6$ subgroups are likely to be realized as the intermediate symmetry, though there are many candidates for the intermediate symmetry in E$_6$ GUT.