  We consider $\alpha$-$T_3$ lattice illuminated by intense circularly
polarized radiation in terahertz regime. We present quasienergy band structure,
time-averaged energy spectrum and time-averaged density of states of
$\alpha$-$T_3$ lattice by solving the Floquet Hamiltonian numerically. We
obtain exact analytical expressions of the quasienergies at the Dirac points
for all values of $\alpha$ and field strength. We find that the quasienergy
band gaps at the Dirac point decrease with increase of $\alpha$. Approximate
forms of quasienergy and band gaps at single and multi-photon resonant points
are derived using rotating wave approximation. The expressions reveal a stark
dependence of quasienergy on the Berry phase of the charge carrier. The quasi
energy flat band remains unaltered in presence of radiation for dice lattice
($\alpha=1$). However, it acquires a dispersion in and around the Dirac and
even-photon resonant points when $0<\alpha<1$. The valley degeneracy and
electron-hole symmetry in the quasienergy spectrum are broken for $0<\alpha<1$.
Unlike graphene, the mean energy follows closely the linear dispersion of the
Dirac cones till near the single-photon resonant point in dice lattice. There
are additional peaks in the time-averaged density of states at the Dirac point
for $0 < \alpha \leq 1$.
