  Inspired by the observation of the extremely large anomalous Hall effect in
the absence of applied magnetic fields or uniform magnetization in
CoNb$_3$S$_6$ [Nature Comm. 9, 3280 (2018); Phys. Rev. Research 2, 023051
(2020)], we perform a first-principles study of this and related compounds of
the $M$Nb$_3$S$_6$ type with different transition metal $M$ ions to determine
their magnetic orders and the anomalous Hall conductivity (AHC). We find that
non-coplanar antiferromagnetic ordering is favored relative to collinear or
coplanar order in the case of $M$=Co, Fe and Ni, while ferromagnetic ordering
is favored in MnNb$_3$S$_6$ at low temperatures. The AHC in these materials
with non-coplanar spin ordering can reach about $e^2/h$ per crystalline layer,
while being negligible for coplanar and collinear cases. We also find that the
AHC depends sensitively on doping and reaches a maximum for intermediate values
of the local spin exchange potential between 0.3 and 0.8 eV. Our AHC results
are consistent with the reported Hall measurements in CoNb$_3$S$_6$ and suggest
a possibility of similarly large anomalous Hall effects in related compounds.
