Multiplicative groups avoiding a fixed group

We know that any finite abelian group G appears as a subgroup of infinitely many multiplicative groups [Formula: see text] (the abelian groups of size [Formula: see text] that are the multiplicative groups of units in the rings [Formula: see text]). It seems to be less well appreciated that G appears as a subgroup of almost all multiplicative groups [Formula: see text]. We exhibit an asymptotic formula for the counting function of those integers whose multiplicative group fails to contain a copy of G, for all finite abelian groups G (other than the trivial one-element group).