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

The Deligne–Lusztig curves associated to the algebraic groups of type A22, B22, and G22 are classical examples of maximal curves over finite fields. The Hermitian curve Hq is maximal over Fqjavax.xml.bind.JAXBElement@77709645, for any prime power q, the Suzuki curve Sq is maximal over Fqjavax.xml.bind.JAXBElement@99c946, for q=22h+1, h≥1, and the Ree curve Rq is maximal over Fqjavax.xml.bind.JAXBElement@706d12f8, for q=32h+1, h≥0. In this paper we show that S8 is not Galois covered by H64. We also prove an unpublished result due to Rains and Zieve stating that R3 is not Galois covered by H27. Furthermore, we determine the spectrum of genera of Galois subcovers of H27, and we point out that some Galois subcovers of R3 are not Galois subcovers of H27.