  We investigate Zariski multiples of plane curves $Z_1, \dots, Z_N$ such that
each $Z_i$ is a union of a smooth quartic curve, some of its bitangents, and
some of its 4-tangent conics. We show that, for plane curves of this type, the
deformation types are equal to the homeomorphism types, and that the number of
deformation types grows as $O(d^{62})$ when the degree $d$ of the plane curves
tends to infinity.
