  In this article we give a classification of the sub-groups in PSL(2,Z) and of
the conjugacy classes of these sub-groups by the mean of an combinatorial
invariant: some trivalent diagrams (dotted or not). We give explicit formulae
enabling to count the number of isomorphism classes of these structures and of
some of their variations, as function of the number of their arcs. Until now,
the counting of non-dotted diagrams was an open problem, for it gives also the
number of unrooted combinatorial maps, triangular or general respectively. The
article ends with the description of a high performance algorithm to enumerate
those structures witch is built upon an unexpected factoring of the cycle index
series of the considered combinatorial species.
