The euclidean bianchi group

The Bianchi Groups Γ_d are PSL₂(0_d) where 0_d is the ring of integers in the quadratic imaginary number field Q√-d with d a positive square-free rational integer. If d=1,2,3,7,11 0_d has a Euclidean algorithm and the corresponding groups are called the Euclidean Bianchi Groups. The group Γ₁ is the Picard Group has been studied independently. Here the subgroup structure of the remaining Euclidean Bianchi Groups is investigated. We show that they have a unique normal subgroup of index in if(n,6)=1 among other results. We also classify the abelian and nilpotent subgroups and discuss the structure of both congruence and non-congruence subgroups.