  Systematic calculations of fission barriers allowing for triaxial deformation
are performed for even-even superheavy nuclei with charge number $Z=112-120$
using three classes of covariant density functional models. The softness of
nuclei in the triaxial plane leads to an emergence of several competing fission
pathes in the region of the inner fission barrier in some of these nuclei. The
outer fission barriers are considerably affected by triaxiality and octupole
deformation. General trends of the evolution of the inner and the outer fission
barrier heights are discussed as a function of the particle numbers.
