Summary: For a smooth quartic plane curve \(C\) we show an existence of a family of Galois closure curves \(\varphi:S\to C\), where \(S\) is a nonsingular projective surface and \(\varphi^{-1}(P)\) is isomorphic to the Galois closure curve \(C_P\) for a general point \(P\in C\). Moreover we determine the types of singular fibers. As a corollary we can say that \(C_p\) is not isomorphic to \(C_Q\) if \(P\) is close to \(Q\).