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The isomorphism classes of hypermaps of a given genus g [less than or equal to] 6 and a given number d of darts are enumerated. The hypermaps of a given genus g are distinguished up to orientation preserving isomorphisms. The obtained results depend on recent progress in counting rooted hypermaps, in particular, by P. Zograf, M. Kazarian, A. Giorgetti, and T. Walsh. These results can be interpreted as an enumeration of conjugacy classes of subgroups in a free Fuchsian group of rank two with a genus restriction. Bibliography: 36 titles., UDC 519.175.3, 525.162.6 1. Introduction An oriented map is a 2-cell decomposition of a closed orientable surface with a fixed global orientation. Oriented hypermaps are generalizations of oriented maps. While [...]