We call 2n the class of 3-valent plane graphs whose faces are 2-gons (necesarily three 2-gons) or 6-gons.

All those graphs come by Goldberg-Coxeter construction of the basic graph with 2 vertices and three edges.

So, their possible point groups are D3 and D3h.

B. Grunbaum and J. Zaks, "The existence of certain planar maps", Discrete Mathematics, 10 (1974) 93--115.