We study the Higgs self couplings of a supersymmetric $E_6$ model that has two Higgs doublets and two Higgs singlets. The lightest scalar Higgs boson in the model may be heavier than 112 GeV, at the one-loop level, where the negative results for the Higgs search at the LEP2 experiments are taken into account. The contributions from the top and scalar top quark loops are included in the radiative corrections to the one-loop mass of the lightest scalar Higgs boson, in the effective potential approximation. The effect of the Higgs self couplings may be observed in the production of the lightest scalar Higgs bosons in $e^+e^-$ collisions at the International Linear Collider (ILC) via double Higgs-strahlung process. For the center of mass energy of 500 GeV with the integrated luminosity of 500 fb$^{-1}$ and the efficiency of 20 %, we expect that at least 5 events of the lightest scalar Higgs boson may be produced at the ILC via double Higgs-strahlung process.
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