Torsion of elliptic curves in the compositum of Dihedral fields

Let E/Q be an elliptic curve. Let Q(G) denote the compositum of all fields with Galois group G over Q. We classify the possible torsion subgroups of E over Q(Dn) for n not divisible by 3 or 4, where D-n is the dihedral group on a regular n-gon. We also obtain upper bounds for the torsion of E over Q(G) for various semi-direct products G = F-q(zeta(p)) (sic) Z/pZ by way of classifications of torsion over Q(Z/pqZ).