We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [B. Ho and S. Nelson, Matrices and finite quandles, Homology Homotopy Appl.7(1) (2005) 197–208.], automorphism groups of quandles (up to isomorphisms) of order less than or equal to 5 were given. With the help of the software Maple, we compute the inner and automorphism groups of all seventy three quandles of order six listed in the appendix of [S. Carter, S. Kamada and M. Saito, Surfaces in 4-Space, Encyclopaedia of Mathematical Sciences, Vol. 142, Low-Dimensional Topology, III (Springer-Verlag, Berlin, 2004)]. Since computations of automorphisms of quandles relate to the problem of classification of quandles, we also describe an algorithm implemented in C for computing all quandles (up to isomorphism) of order less than or equal to nine.