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High spin states in the neutron-rich 108Ru nucleus have been studied through measuring prompt γγ-ray coincidences following the spontaneous fission of 252Cf with the Gammasphere detector array. The yrast band has been confirmed. The one-phonon γ-vibrational band and the two-quasiparticle band based on the 5- level have been extended up to 13+ and 15-, respectively. In addition, two levels at 1644.8kev and 1826.5kev excitation energies are newly identified and proposed to be the members of a two-phonon γ-vibrational band. It is shown that the 108Ru nucleus has triaxial deformation with parametersβ2 ～ 0.29 and γγ = -22° from the total Routhian surface calculations. The observed band crossing in the yrast band is due to the alignment of a pair of h11/2 neutrons according to the cranked shell model calculations. The possible configuration for the two-quasiparticle band has been discussed.