  We prove that if two germs of irreducible complex analytic curves at
$0\in\mathbb{C}^2$ have different sequence of characteristic exponents, then
there exists $0<\alpha<1$ such that those germs are not $\alpha$-H\"older
homeomorphic. For germs of complex analytic plane curves with several
irreducible components we prove that if any two of them are $\alpha$-H\"older
homeomorphic, for all $0<\alpha<1$, then there is a correspondence between
their branches preserving sequence of characteristic exponents and intersection
multiplicity of pair of branches. In particular, we recovery the sequence of
characteristic exponents of the branches and intersection multiplicity of pair
of branches are Lipschitz invariant of germs of complex analytic plane curves.
