Gluing of posets, and lattices of subgroups

We discuss a notion of gluing for arbitrary posets which extends the standard one for lattices. We show that the gluing of two sublattices along a suitable intersection often is a lattice. This is not always the case, though; we amend an imprecise statement in the literature on this point. Next we consider some instances of the problem of determining when the gluing of some given lattices gives rise to the lattice of all subgroups of a group, and for which groups does this happen. Our main result answers these problem for repeated gluing of finite lattices whose maximal chains have length 2.