We discuss convex l-subgroups of an l-group G in their role as direct summands, not so much of G as of each other. This is done by writing A ≥dB for subgroups A, B to mean that A is a direct summand of B, and studying the properties of the resulting poset. It is shown to be a hypolattice, that is, to have local lattice properties in a certain sense. However it need not be a lattice; and there may exist meets of pairs of elements, outside the hypolattice structure. It need not be conditionally complete even when G is conditionally complete. We look also at the map which sends a subgroup to its lattice-closure; the lattice-closed subgroups also form a hypolattice. Our main result asserts that this hypolattice is conditionally complete if G is. The paper ends with some examples and counter examples in C(X).