Let G be a connected, simply connected, simple complex algebraic group and let ǫ be a primitive ℓ-th root of 1, ℓ odd and 3 ∤ ℓ if G is of type G2. In this talk we show how to construct and determine all quantum subgroups of the quantum group Oǫ(G), i.e. all Hopf algebra quotients of the quantized coordinate algebra of G. Then the question of isomorphism between these Hopf algebras is considered. Finally, we will use these results to show the existence of infinitely many non-isomorphic Hopf algebras of the same dimension, presented as extensions of finite quantum groups by finite groups. These results are part of joint works with N. Andruskiewitsch: Quantum subgroups of a simple quantum group at roots of 1, arXiv:0707.0070v1 and Extensions of finite quantum groups by finite groups, arXiv:math/0608647v6.