Evidence for nonaxial γ deformations has been widely found in collective rotational states. The γ deformation has led to very interesting characteristics of nuclear motions, such as wobbling, chiral band, and signature inversion in rotational states. There is an interesting question; why the nonaxial γ deformation is not favored in the ground states of even-even (e-e) nuclei. The quest for stable triaxial shapes in the ground states of e-e nuclei, with a maximum triaxial deformation of \left| \gamma \right| ≈ 30°, is still a major theme in nuclear structure. In the present work, we use the cranked Woods-Saxon (WS) shell model to investigate possible triaxial shapes in ground and collective rotational states. Total-Routhian-surface calculations by means of the pairing-deformation-frequency self-consistent cranked shell model are carried out for even-even germanium and selenium isotopes, in order to search for possible triaxial deformations of nuclear states. Calculations are performed in the lattice of quadrupole (β₂, γ) deformations with the hexadecapole β₄ variation. In fact, at each grid point of the quadrupole deformation (β₂, γ) lattice, the calculated energy is minimized with respect to the hexadecapole deformation β₄. The shape phase transition from triaxial shape in ⁶⁴Ge, oblate shape in ⁶⁶Ge, again through triaxiality, to prolate deformations is found in germanium isotopes. In general, the Ge and Se isotopes have γ-soft shapes, resulting in significant dynamical triaxial effect. There is no evidence in the calculations pointing toward rigid triaxiality in ground states. The triaxiality of \gamma = - 30^\circ for the ground and collective rotational states, that is the limit of triaxial shape, is found in ^{64, 74}Ge. One should also note that the depth of the triaxial minimum increases with rotational frequency increasing in these two nuclei. The present work focuses on the possible triaxial deformation of N = Z nucleus ⁶⁴Ge. Single-particle level diagrams can give a further understanding of the origin of the triaxiality. Based on the information about single-particle levels obtained with the phenomenological Woods-Saxon (WS) potential, the mechanism of triaxial deformation in N = Z nucleus ⁶⁴Ge is discussed, and caused surely by a deformed γ≈30° shell gap at Z(N) = 32. At N = 34, however, an oblate shell gap appears, which results in an oblate shape in ⁶⁶Ge (N = 34). With neutron number increasing, the effect from the N = 34 oblate gap decreases, and hence the deformations of heavier Ge isotopes change toward the triaxiality (or prolate).