  We study non-supersymmetric truncations of $\omega$-deformed ${\cal N}=8$
gauged supergravity that retain a $U(1)$ gauge field and three scalars, of
which two are neutral and one charged. We construct dyonic domain-wall and
black hole solutions with AdS$_4$ boundary conditions when only one (neutral)
scalar is non-vanishing, and examine their behavior as the magnetic field and
temperature of the system are varied. In the infrared the domain-wall solutions
approach either dyonic AdS$_2 \times \mathbb{R}^2$ or else Lifshitz-like,
hyperscaling violating geometries. The scaling exponents of the latter are
$z=3/2$ and $\theta = -2$, and are independent of the $\omega$-deformation. New
$\omega$-dependent AdS$_4$ vacua are also identified. We find a rich structure
for the magnetization of the system, including a line of metamagnetic
first-order phase transitions when the magnetic field lies in a particular
range. Such transitions arise generically in the $\omega$-deformed theories.
Finally, we study the onset of a superfluid phase by allowing a fluctuation of
the charged scalar field to condense, spontaneously breaking the abelian gauge
symmetry. The mechanism by which the superconducting instability ceases to
exist for strong magnetic fields is different depending on whether the field is
positive or negative. Finally, such instabilities are expected to compete with
spatially modulated phases.
