Let G be a compact group and let L(G) be the space of all closed subgroups of G with Chabauty topology. It is well known that L(G) is compact. The main result of this paper is the following one. Let G be a compact abelian group. The space L(G) is dyadic (i.e. L(G) is a continuous image of the Cantor cube) iff the weight of G is less then \(\aleph_ 2\). The paper contains also a description of all compact abelian groups G such that the space L(G) contains isolated points. A note. The reviewer has now proved that the Souslin number of the space L(G) is countable for any compact group G. The main result of the paper under review is true for all compact groups.