  Gaiotto and Witten found that one can construct 3d $\mathcal{N}=4$
Chern-Simons matter theories by using $\mathcal{N}=4$ SCFT whose momentum map
of global symmetries satisfy special condition. Usually, one uses free
hypermultiplet and twisted hypermultiplet, and more recently it was found that
strongly coupled theory such as 3d version of $T_N$ theory and Argyres-Douglas
matter can also be used. In this paper, we compute superconformal index of
these $\mathcal{N}=4$ theories and derive the Coulomb/Higgs limit. Our results
determine the moduli space of vacua, which is used to check various interesting
mirror symmetry involving CSM theory and usual $\mathcal{N}=4$ gauge theory.
