  We find a large class of new supersymmetric $AdS_5$ black strings from
five-dimensional $N=4$ gauged supergravity coupled to five vector multiplets
with $SO(2)_D\times SO(3)\times SO(3)$ gauge group. These solutions have near
horizon geometries of the form $AdS_3\times \Sigma^2$ for $\Sigma^2$ being a
two-sphere ($S^2$) or a hyperbolic space ($H^2$). There are four supersymmetric
$AdS_5$ vacua with $N=4$ and $N=2$ supersymmetries. By performing topological
twists along $\Sigma^2$ with $SO(2)\times SO(2)_{\textrm{diag}}$ and
$SO(2)_{\textrm{diag}}$ gauge fields, we find a number of $AdS_3\times
\Sigma^2$ fixed points describing near horizon geometries of black strings in
asymptotically $AdS_5$ spaces. Most of the solutions take the form of
$AdS_3\times H^2$ with only one being $AdS_3\times S^2$ preserving
$SO(2)_{\textrm{diag}}$ symmetry. We also give the corresponding black string
solutions interpolating between asymptotically locally $AdS_5$ vacua and the
near horizon $AdS_3\times \Sigma^2$ geometries. There are a number of solutions
flowing from one, two or three $AdS_5$ vacua to an $AdS_3\times \Sigma^2$ fixed
point. These solutions can also be considered as holographic RG flows across
dimensions from $N=2$ and $N=1$ SCFTs in four dimensions to two-dimensional
SCFTs with $N=(2,0)$ or $N=(0,2)$ supersymmetry.
