We study higher-symmetric periodic structures with hexagonal lattice arrangement. We define three ways to synthesize a glide symmetry in a hexagonal lattice. If the periodic motifs have sufficient symmetry, the three resulting structures are related through a rotation. Furthermore, we define two ways of synthesizing a mirrored half-turn symmetry in a hexagonal lattice. Again, if the periodic motifs have sufficient symmetry, the two resulting structures are related through a rotation. We analyze a structure with mirrored half-turn symmetry and compare its response to that of a periodic structure with conventional hexagonal symmetry. The mirrored half-turn-symmetric structure provides a higher effective refractive index for the same geometrical dimensional. As a result, the higher-symmetric structure is useful for devices in future wireless applications.