  We study supersymmetric $AdS_3\times \Sigma_2$ and $AdS_2\times \Sigma_3$
solutions, with $\Sigma_2=S^2,H^2$ and $\Sigma_3=S^3,H^3$, in five-dimensional
$N=4$ gauged supergravity coupled to five vector multiplets. The gauge groups
considered here are $U(1)\times SU(2)\times SU(2)$, $U(1)\times SO(3,1)$ and
$U(1)\times SL(3,\mathbb{R})$. For $U(1)\times SU(2)\times SU(2)$ gauge group
admiting two supersymmetric $N=4$ $AdS_5$ vacua, we identify a new class of
$AdS_3\times \Sigma_2$ and $AdS_2\times H^3$ solutions preserving four
supercharges. Holographic RG flows describing twisted compactifications of
$N=2$ four-dimensional SCFTs dual to the $AdS_5$ vacua to the SCFTs in two and
one dimensions dual to these geometries are numerically given. The solutions
can also be interpreted as supersymmetric black strings and black holes in
asymptotically $AdS_5$ spaces with near horizon geometries given by
$AdS_3\times \Sigma_2$ and $AdS_2\times H^3$, respectively. These solutions
broaden previously known black brane solutions including half-supersymmetric
$AdS_5$ black strings recently found in $N=4$ gauged supergravity. Similar
solutions are also studied in non-compact gauge groups $U(1)\times SO(3,1)$ and
$U(1)\times SL(3,\mathbb{R})$.
