  A function field over a finite field is called maximal if it achieves the
Hasse-Weil bound. Finding possible genera that maximal function fields achieve
has both theoretical interest and practical applications to coding theory and
other topics. As a subfield of a maximal function field is also maximal, one
way to find maximal function fields is to find all subfields of a maximal
function field. Due to the large automorphism group of the Hermitian function
field, it is natural to find as many subfields of the Hermitian function field
as possible. In literature, most of papers studied subfields fixed by subgroups
of the decomposition group at one point (usually the point at infinity). This
is because it becomes much more complicated to study the subfield fixed by a
subgroup that is not contained in the decomposition group at one point.
  In this paper, we study subfields of the Hermitian function field fixed by
subgroups that are not contained in the decomposition group of any point except
the cyclic subgroups. It turns out that some new maximal function fields are
found.
