A subset of a compact Riemannian symmetric space is called an antipodal set if the geodesic symmetry at each point is the identity on the set. The cardinalities of antipodal sets are finite. An antipodal set whose cardinality attains the maximum of the cardinalities of antipodal sets is called a great antipodal set. In a symmetric R-space, a maximal antipodal set is a great antipodal set and a great antipodal set is unique up to the action of the identity component of the isometry group, moreover, a great antipodal set is an orbit of the Weyl group. On the other hand, we have not known much about antipodal sets of a compact Riemannian symmetric space which is not a symmetric R-space. In this talk we will present a classification of maximal antipodal subgroups in a quotient group of a compact Lie group of classical type and that in G_2. In many of these cases there exists a maximal antipodal subgroup but not a great antipodal subgroup. This talk is based on joint work with Hiroyuki Tasaki.