  It is know that any Neumann subgroup of the modular group is maximal
non-parabolic. The question arises as to whether these are the only maximal
non-parabolic subgroups. The wild class of maximal, non-parabolic, not Neumann
subgroups of the modular group was constructed by Brenner and Lyndon. The new
construction of such a class is presented. Those groups are obtained as
subgroups of elements of positive determinant of any Neumann subgroup of the
extended modular group (the notion which we introduce in the paper) and in this
sens they are as "close" as possible to Neumann subgroups of the modular group.
