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 Hqis maximal over Fq2, for any prime power q, the Suzuki curve Sqis maximal over Fq4, for q=22h+1, h≥1, and the Ree curve Rqis maximal over Fq6, for q=32h+1, h≥0. In this paper we show that S8is not Galois covered by H64. We also prove an unpublished result due to Rains and Zieve stating that R3is not Galois covered by H27.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 R3are not Galois subcovers of H27.