  We study five-dimensional $N=4$ gauged supergravity coupled to five vector
multiplets with $SO(2)_D\times SO(3)\times SO(3)$ gauge group. There are four
supersymmetric $AdS_5$ vacua in the truncation to $SO(2)_{\textrm{diag}}$
invariant scalars. Two of these vacua preserve the full $N=4$ supersymmetry
with $SO(2)_D\times SO(3)\times SO(3)$ and $SO(2)_D\times
SO(3)_{\textrm{diag}}$ symmetries. These have an analogue in $N=4$ gauged
supergravity with $SO(2)\times SO(3)\times SO(3)$ gauge group. The other two
$AdS_5$ vacua preserve only $N=2$ supersymmetry with
$SO(2)_{\textrm{diag}}\times SO(3)$ and $SO(2)_{\textrm{diag}}$ symmetry. The
former has an analogue in the previous study of $SO(2)_D\times SO(3)$ gauge
group while the latter is a genuinely new $N=2$ $AdS_5$ vacuum. These vacua are
dual to $N=2$ and $N=1$ superconformal field theories (SCFTs) in four
dimensions with different flavour symmetries. We give the full scalar mass
spectra at all of the $AdS_5$ critical points which provide information on
conformal dimensions of the dual operators. Finally, we study holographic RG
flows interpolating between these $AdS_5$ vacua and find a new class of
solutions. In addition to the RG flows from the trivial $SO(2)_D\times
SO(3)\times SO(3)$ $N=4$ critical point, at the origin of the scalar manifold,
to all the other critical points, there is a family of RG flows from the
trivial $N=4$ critical point to the new $SO(2)_{\textrm{diag}}$ $N=2$ critical
point that pass arbitrarily close to the $SO(2)_D\times SO(3)_{\textrm{diag}}$
$N=4$ critical point.
