  Based on recent studies regarding high-temperature (high-$T_c$) La-Y ternary
hydrides (e.g., $P{\bar{1}}$-La$_2$YH$_{12}$, $Pm{\bar{3}}m$-LaYH$_{12}$, and
$Pm{\bar{3}}m$-(La,Y)H$_{10}$ with a maximum $T_c \sim 253$ K), we examined the
phase and structural stabilities of the (LaH$_6$)(YH$_6$)$_y$ series as
high-$T_c$ ternary hydride compositions using a genetic algorithm and $\it ab$
$\it initio$ calculations. Our evaluation showed that the
$Pm\bar{3}m$-LaYH$_{12}$ reported in the previous study was unstable during
decomposition into $R\bar{3}c$-LaH$_{6}$ + $Im\bar{3}m$-YH$_{6}$. We also
discovered new crystal structures, namely $Cmmm$-LaYH$_{12}$ ($y=1$),
$R\bar{3}c$-LaYH$_{12}$ ($y=1$), $Cmmm$-LaY$_3$H$_{24}$ ($y=3$), and
$R\bar{3}$-LaY$_3$H$_{24}$ ($y=3$), showing stability against such
decomposition. While $R\bar{3}c$ ($y=1$) and $R\bar{3}$ ($y=3$) did not exhibit
superconductivity owing to the extremely low density of states at the Fermi
level, $Cmmm$ phases exhibited a $T_{c}$ of approximately 140~K at around
200~GPa owing to the extremely high electron--phonon coupling constant
($\lambda$ = 1.876 for LaYH$_{12}$). By the twice longer stacking for
$Cmmm$-LaY$_3$H$_{24}$, the coupling constant increased owing to the chemical
pressure of Y, leading to a slightly increased $T_{c}$.
