  Half-maximal, $\,\mathcal{N}=4\,$, sectors of $\,D=4\,$ $\,\mathcal{N}=8\,$
supergravity with a dyonic ISO(7) gauging are investigated. We focus on a
half-maximal sector including three vector multiplets, that arises as a certain
$\,\textrm{SO}(3)_{\textrm{R}}$-invariant sector of the full theory. We discuss
the embedding of this sector into the largest half-maximal sector of the
$\,\mathcal{N}=8\,$ supergravity retaining six vector multiplets. We also
provide its canonical $\,\mathcal{N}=4\,$ formulation and show that, from this
perspective, our model leads in its own right to a new explicit gauging of
$\,\mathcal{N}=4\,$ supergravity. Finally, expressions for the restricted
duality hierarchy are given and the vacuum structure is investigated. Five new
non-supersymmetric AdS vacua are found numerically. The previously known
$\,\mathcal{N}=2\,$ and $\,\mathcal{N}=3\,$ AdS vacua are also contained in our
$\,\mathcal{N}=4\,$ model. Unlike when embedded in previously considered
sectors with fewer fields, these vacua exhibit their full $\,\mathcal{N}=2\,$
and $\,\mathcal{N}=3\,$ supersymmetry within our $\,\mathcal{N}=4\,$ model.
