Modular symmetry and non-Abelian discrete flavor symmetries in string compactification

We study the modular symmetry in magnetized D-brane models on T². Non-Abelian flavor symmetry D₄ in the model with magnetic flux $M=2$ (in a certain unit) is a subgroup of the modular symmetry. We also study the modular symmetry in heterotic orbifold models. The T²/Z₄ orbifold model has the same modular symmetry as the magnetized brane model with $M=2$, and its flavor symmetry D₄ is a subgroup of the modular symmetry.