  Generically an almost complex structure has no symmetries at all, but there
exist symmetric structures. In this paper we describe how to guarantee that the
pseudogroup of local symmetries is small (finite-dimensional). It will be
indicated that a large symmetry pseudogroup (infinite-dimensional) is a
signature of some integrable structure, like a pseudoholomorphic foliation. We
are mostly concerned with almost complex structures in dimensions 4 and 6,
where we classify the sub-maximal symmetric structures, and we briefly discuss
the higher dimensions. For non-degenerate almost complex structures we give an
estimate on the dimension of the automorphism group.
