Let r be a rational in (0,1]. There exists a finite group G which is the direct product of at most four metacyclic groups and whose proportion of normal subgroups is r. An analogous result holds for three other measures of "Hamiltonianess".
Ahearn, Stephen and Huber, Mark, "Finite Groups can be Arbitrarily Hamiltonian" (1994). Mathematical Sciences Technical Reports (MSTR). 121.