We introduced a new concept, the mutual conic, in order to give a complete classification of the group $G$ generated by the images of two or three total sextactic points in the Jacobian $J_{\mathcal{C}}$ of a smooth projective plane quartic curve $\mathcal{C}$ (3-genus curves that are non-hyperelliptic), and we determined the geometric configuration of these points associated with each case. We supported the validity of the results with a variety of examples.