  We show that most of the genus-zero subgroups of the braid group
$\mathbb{B}_3$ (which are roughly the braid monodromy groups of the trigonal
curves on the Hirzebruch surfaces) are irrelevant as far as the Alexander
invariant is concerned: there is a very restricted class of \enquote{primitive}
genus-zero subgroups such that these subgroups and their genus-zero
intersections determine all the Alexander invariants. Then, we classify the
primitive subgroups in a special subclass. This result implies the known
classification of the dihedral covers of irreducible trigonal curves.
