The structural, electronic and elastic properties of common intermetallic compounds in FeTiCoNiVCrMnCuAI system high entropy alloy were investigated by the first principles calculation. The calculation results of formation enthalpy and cohesive energy show that FeTi, Fe2Ti, AlCrFe2, Co2Ti, AlMn2V and Mn2Ti phases may form in the formation process of the alloy. Further studies show that FeTi, FezTi, AlCrFe2, Co2Ti and AlMn2V phases with higher shear modulus and elastic modulus would be excellent strengthening phases in high entropy alloy and would improve the hardness of the alloy. In addition, the partial density of states was investigated for revealing the bonding mode, and the analyses on the strength of p-d hybridization also reveal the underlying mechanism for the elastic properties of these compounds.