  In this paper, we consider elliptic curves induced by rational Diophantine
quadruples, i.e. sets of four nonzero rationals such that the product of any
two of them plus 1 is a perfect square. We show that for each of the groups
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/k\mathbb{Z}$ for $k = 2, 4, 6, 8$,
there are infinitely many rational Diophantine quadruples with the property
that the induced elliptic curve has this torsion group. We also construct
curves with moderately large rank in each of these four cases.
