We call 4-hedrite a 4-valent graph with three 2-gons, two 3-gons and 4-gons The classification of possible point groups was obtained in

M. Deza, M.Dutour and M.Shtogrin, "4-valent plane graphs with 2-, 3- or 4-gonal faces", satellite conference (of congress of ISM in Beijing) on algebra and combinatorics 73--97.