  Since the recent discovery of plutonium-based superconductors such as
PuCoIn$_{\rm 5}$, systematic studies of the electronic properties for plutonium
compounds are providing insight into the itinerancy-localization crossover of
Pu 5$f$ electrons. We are particularly interested in understanding the Fermi
surface properties of PuIn$_{3}$ compound, which serves as the building block
for the PuCoIn$_{\rm 5}$ superconductor. Motivated by the first observation of
quantum oscillation and renewed interest in the de-Hass van-Alphen (dHvA)
measurements on PuIn$_{\rm 3}$, we study the Fermi surface (FS) topology and
the band dispersion in both paramagnetic and antiferromagnetic state of
PuIn$_{\rm 3}$, based on density functional theory with generalized gradient
approximation. We compare the results with its isostructural paramagnetic
compound PuSn$_{\rm 3}$. We show the detailed Fermi surfaces of compounds
PuIn$_{\rm 3}$ and PuSn$_{\rm 3}$ and conclude that the FS topology of an
antiferromagnetic PuIn$_3$ agrees better with dHvA measurements. In addition,
we show the magnetization of the antiferromagnetic order can alter the angle
dependence and values of the effective mass for the dHvA orbits. We point out
the accurate determination of the magnetic order orientation with respect to
the crystal orientation is crucial to advance the understanding of the
electronic structures of the PuIn$_{\rm 3}$ compound.
