  In this paper, we compute the unit groups and the $2$-class numbers of the
Fr\"ohlich's triquadratic fields $\KK=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q})$,
where $p$ and $q$ are two prime numbers such that ($p\equiv 1 \pmod8$ and
$q\equiv 3 \pmod4$) or ($p\equiv 5$ or $3 \pmod8$ and $q\equiv 3 \pmod4$).
Furthermore, we determine some families of the fields $\KK$ whose $2$-class
groups are trivial or cyclic non trivial, and some other families with
$2$-class groups isomorphic to the Klein group.
