Some Ree and Suzuki curves are not Galois covered by the Hermitian curve

Abstract: The Deligne-Lusztig curves associated to the algebraic groups of type $^2A_2$, $^2B_2$, and $^2G_2$ are classical examples of maximal curves over finite fields. The Hermitian curve $\mathcal H_q$ is maximal over $\mathbb F_{q^2}$, for any prime power $q$, the Suzuki curve $\mathcal S_q$ is maximal over $\mathbb F_{q^4}$, for $q=2^{2h+1}$, $h\geq1$ and the Ree curve $\mathcal R_q$ is maximal over $\mathbb F_{q^6}$, for $q=3^{2h+1}$, $h\geq0$. In this paper we show that $\mathcal S_8$ is not Galois covered by $\mathcal H_{64}$. We also give a proof for an unpublished result due to Rains and Zieve stating that $\mathcal R_3$ is not Galois covered by $\mathcal H_{27}$. Furthermore, we determine the spectrum of genera of Galois subcovers of $\mathcal H_{27}$, and we point out that some Galois subcovers of $\mathcal R_3$ are not Galois subcovers of $\mathcal H_{27}$.