  We classify subgroups of $\textrm{SL}(2,\mathbb{Z})$ up to conjugacy, which
occur as monodromy groups of elliptically fibered K3 surfaces following a
general strategy proposed by Bogomolov and Tschinkel. The essential step is the
factorisation of the functional invariant $j$ with second factor a Belyi
function of maximal possible degree and the classification of the corresponding
subgroups $\bar\Gamma$ of $\textrm{PSL}(2,\mathbb{Z})$ using dessins d'enfants.
