  We prove the rationality of the exceptional W-algebras associated with the
simple Lie algebra $\mathfrak{sp}_4$ and subregular nilpotent elements, proving
a new particular case of a conjecture of Kac-Wakimoto. Moreover, we describe
the simple $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$-modules and compute
their characters. We also explicit the nontrivial action of the component group
on the set of these simple modules.
