  The purpose of this paper is to list the refined Humbert invariants for a
given automorphism group of a curve $C/K$ of genus 2 over an algebraically
closed field $K$ with characteristic $0$. This invariant is an algebraic
generalization of the (usual) \textit{Humbert invariant}. It is a positive
definite quadratic form associated to the curve $C$, and it encodes many
geometric properties of the curve. The paper has a special interest in the
cases where $Aut(C)\simeq D_4$ or $D_6$. In these cases, several applications
of the main results are discussed, including the curves with elliptic subcovers
of a given degree.
