It is well known that the superstring theory on certain compactifications leads to non-Abelian finite groups. Indeed, the torus comapactification gives the modular symmetry.
The modular group includes S3, A4, S4, and A5 as its congruence subgroups. However, there is a difference between the modular symmetry and the usual flavor symmetry. Coupling constants such as Yukawa couplings also transform non-trivially under the modular symmetry and are written as functions of the modulus called modular forms. The flavor structure of the mass matrices are essentially given by fixing the expectation value of the modulus, which is the only source of the breaking of the modular invariance. In this aspect, an attractive Ansatz was proposed by {Feruglio:2017spp}, where A4 was taken. Along with this work, S3 {Kobayashi:2018vbk} and S4 {Penedo:2018nmg} have been discussed as well as the numerical works in {Criado:2018thu, ours}. These approaches would make a bridge between flavor physics and underlying theory such as superstring theory from the viewpoint of flavor symmetries. In my talk, we present a brief review in this field, and then discuss the modular invariant neutrino mass matrix with numerical results of our comprehensive analyse.