  We explore the transport properties of a periodically modulated
$\alpha$-$\mathcal{T}_3$ lattice in the presence of a perpendicular magnetic
field. The effect of the Berry phase on electrical conductivity oscillation,
so-called Weiss oscillation, caused by the modulation induced non-zero drift
velocity of charge carriers is investigated. Employing linear response theory
within the low temperature regime, we analyze Weiss oscillation as a function
of the external magnetic field for both electrically and magnetically modulated
$\alpha$-$\mathcal{T}_3$ lattice numerically as well as analytically. The Berry
phase makes this hexagonal lattice structure behave differently than other
two-dimensional fermionic systems. It causes a significant valley polarization
in magnetoconductivity. Most interestingly, the combined effect of both
modulations breaks the particle-hole symmetry and causes a smooth transition
from even (odd) to odd (even) filling fraction corresponding to the density of
states peaks by means of the Berry phase.
