In this paper, the D₃ dihedral logistic map of fractional order is introduced. The map presents a dihedral symmetry D₃. It is numerically shown that the construction and interpretation of the bifurcation diagram versus the fractional order requires special attention. The system stability is determined and the problem of hidden attractors is analyzed. Furthermore, analytical and numerical results show that the chaotic attractor of integer order, with D₃ symmetries, looses its symmetry in the fractional-order variant.