  We have investigated the modular binary octahedral group $2O$ as a flavor
symmetry to explain the structure of Standard Model. The vector-valued modular
forms in all irreducible representations of this group are constructed. We have
classified all possible fermion masses models based on the modular binary
octahedral group $2O$. A comprehensive numerical analysis is performed, and we
present some benchmark quark/lepton masses models in well agreement with the
experimental data. Notably we find a minimal modular invariant model for
leptons and quarks, which is able to explain simultaneously the masses and
mixing parameters of both quarks and leptons in terms of 14 real free
parameters including the modulus $\tau$. The fermion mass hierarchies around
the vicinity of the modular fixed points are explored.
