[en] The possible existence of islands of shell-stabilized superheavy nuclei has been an inspiring problem in heavy-ion physics for almost four decades. For review papers. The expectation that in the near future experimental progress will access this region is a strong motivation to investigate the shell structure of superheavy nuclei within the self-consistent nuclear mean-field models. The long-lived superheavy elements can exist in a variety of shapes, including spherical, axial and triaxial configurations that has been studied in the state-of-the-art non-relativistic nuclear energy density functional theory. The shape effects are expected to affect a-decay lifetimes by changing the Q_α values or by hindering the transitions involving parent and daughter nuclei with different shapes. Therefore, such shape effects play very important role in the stability of superheavy nuclei. It was found that the interplay between the pronounced oblate gaps at Z = 120 and N = 172, 178, and the large prolate gaps at Z = 114, 116 and N = 174, 176 that determines the microscopy of observed shape coexistence and shape-transitional behavior in this region. In view of these facts, it is essential and interesting to examine this phenomena in a covariant density functional theory that gives different prediction for the magic numbers in superheavy nuclei. In this work, we make a systematic investigation of the properties of Z = 116 isotopes with the triaxial relativistic mean-field theory using point-coupling effective interactions of both PGF1 and PC-PK1. The shell structure and effects of triaxial deformation on observable are studied in details. (authors)