The evolution of structure property along the fission path in the superheavy nucleus 256 Sg is predicted through the multi-dimensional potential-energy(or Routhian)-surface calculations, in which the phenomenological deformed Woods-Saxon potential is adopted. Calculated nuclear deformations and fission barriers for 256 106 Sg 150 and its neighbors, e.g., 258,260 Sg, 254 Rf and 252 No are presented and compared with other theoretical results. A series of energy maps and curves are provided and used to evaluate the corresponding shape-instability properties, especially in the directions of triaxial γ and different hexadecapole deformations (e.g., α 40 , α 42 and α 44 ). It is found that the triaxial deformation may help the nucleus bypass the first fission-barrier of the axial case. After the first minimum in the nuclear energy surface, the fission pathway of the nucleus can be affected by γ and hexadecapole deformation degrees of freedom. In addition, microscopic single-particle structure, pairing and Coriolis effects are briefly investigated and discussed.