  We consider finite group-actions on closed, orientable and nonorientable
3-manifolds M which preserve the two handlebodies of a Heegaard splitting of M
of some genus g > 1 (maybe interchanging the two handlebodies). The maximal
possible order of a finite group-action on a handlebody of genus g>1 is 12(g-1)
in the orientation-preserving case and 24(g-1) in general, and the maximal
order of a finite group preserving the Heegaard surface of a Heegaard splitting
of genus g is 48(g-1). This defines a hierarchy for finite group-actions on
3-manifolds which we discuss in the present paper; we present various manifolds
with an action of type 48(g-1) for small values of g, and in particular the
unique hyperbolic 3-manifold with such an action of smallest possible genus g =
6 (in strong analogy with the Euclidean case of the 3-torus which has such
actions for g = 3).
