Generalized Hecke group H p,∞ (λ) is generated by X(z) = −(z − λ p ) −1and Y (z) = −(z + λ) −1 where λ p = 2 cos πp , p ≥ 2 integer and λ ≥ 2 .Extended generalized Hecke group H p,∞ (λ) is obtained by adding thereection R(z) = 1/z to the generators of generalized Hecke groupH p,∞ (λ). In this paper, we study the commutator subgroups of ex-tended generalized Hecke groups H p,∞ (λ). Also, we determine thepower subgroups of generalized Hecke groups H p,∞ (λ) and extendedgeneralized Hecke groups H p,∞ (λ) .