  We introduce a class of countable groups by some abstract group-theoretic
conditions. It includes linear groups with finite amenable radical and finitely
generated residually finite groups with some non-vanishing $\ell^2$-Betti
numbers that are not virtually a product of two infinite groups. Further, it
includes acylindrically hyperbolic groups. For any group $\Gamma$ in this class
we determine the general structure of its possible lattice embeddings, i.e. of
all compactly generated, locally compact groups that contain $\Gamma$ as a
lattice. This leads to a precise description of possible non-uniform lattice
embeddings of groups in this class. Further applications include the
determination of possible lattice embeddings of fundamental groups of closed
manifolds with pinched negative curvature.
