  We study the superconformal index of S-fold theories by using AdS/CFT
correspondence. It has been known that the index in the large $N$ limit is
reproduced as the contribution of bulk Kaluza-Klein modes. For finite $N$
D3-branes wrapped around the non-trivial cycle in
$\boldsymbol{S}^5/\mathbb{Z}_k$, which corresponds to Pfaffian-like operators,
give the corrections of order $q^N$ to the index. We calculate the finite $N$
corrections by analyzing the fluctuations of wrapped D3-branes. Comparisons to
known results show that our formula correctly reproduces the corrections up to
errors of order $q^{2N}$.
