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In this paper we deal with the problem of classifying the genera of quotientcurves $\mathcal{H}_q/G$, where $\mathcal{H}_q$ is the$\mathbb{F}_{q^2}$-maximal Hermitian curve and $G$ is an automorphism group of$\mathcal{H}_q$. The groups $G$ considered in the literature fix either a pointor a triangle in the plane ${\rm PG}(2,q^6)$. In this paper, we give a completelist of genera of quotients $\mathcal{H}_q/G$, when $G \leq {\rmAut}(\mathcal{H}_q) \cong {\rm PGU}(3,q)$ does not leave invariant any point ortriangle in the plane. As a result, the classification of subgroups $G$ of${\rm PGU}(3,q)$ satisfying this property is given up to isomorphism.

Quotients of the Hermitian curve from subgroups of ${\rm PGU}(3,q)$ without fixed points or triangles