  The subgroup lattice of a group is a great source of information about the
structure of the group itself. The aim of this paper is to use a similar tool
for studying profinite groups. In more detail, we study the lattices of closed
or open subgroups of a profinite group and its relation with the whole group.
We show, for example, that procyclic groups are the only profinite groups with
a distributive lattice of closed or open subgroups, and we give a sharp
characterization of profinite groups whose lattice of closed (or open)
subgroups satisfies the Dedekind modular law; we actually give a precise
description of the behaviour of modular elements of the lattice of closed
subgroups. We also deal with the problem of carrying some structural
information from a profinite group to another one having an isomorphic lattice
of closed (or open) subgroups. Some interesting consequences and related
results concerning decomposability and the number of profinite groups with a
given lattice of closed (or open) subgroups are also obtained.
