Modular Subgroups, Dessins d'Enfants and Elliptic K3 Surfaces

We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck's dessins d'enfants. In the particular case of the index 36 subgroups, the corresponding Calabi-Yau threefolds are identified, in analogy with the index 24 cases being associated with K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over P^1 as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.