If a knot is viewed as a pair, (S3, K), its symmetry group is defined to be the group of diffeomorphisms of the pair, modulo the normal subgroup generated by those diffeomorphisms that are isotopic to the identity.
For a hyperbolic knot, the symmetry group is always finite, given by the group of isometries of the unique complete hyperbolic structure on the complement. For torus knots, the symmetry group is always of order 2.
The table gives the symmetry group for all prime knots listed. Dn denotes the dihedral group with 2n elements, and Zn denotes the cyclic group (so D1 = Z2). The data is taken from Knotscape, which itself is using Snappea to compute the values. We thank Ryan Budney for correcting our original description of the symmetry group. Details can be found in Kawauchi's book, "A Survey of Knot Theory."