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We study embedding of non-commuting $Z_2$ and $Z_m$, $m\geq 3$ symmetries indiscrete subgroups (DSG) of $U(3)$ and analytically work out the mixingpatterns implied by the assumption that $Z_2$ and $Z_m$ describe the residualsymmetries of the neutrino and the charged lepton mass matrices respectively.Both $Z_2$ and $Z_m$ are assumed to be subgroups of a larger discrete symmetrygroup $G_f$ possessing three dimensional faithful irreducible representation.The residual symmetries predict the magnitude of a column of the leptonicmixing matrix $U_{\rm PMNS}$ which are studied here assuming $G_f$ as the DSGof $SU(3)$ designated as type C and D and large number of DSG of $U(3)$ whichare not in $SU(3)$. These include the known group series $\Sigma(3n^3)$,$T_n(m)$, $\Delta(3n^2,m)$, $\Delta(6n^2,m)$ and $\Delta'(6n^2,j,k)$. It isshown that the predictions for a column of $|U_{\rm PMNS}|$ in these groupseries and the C and D types of groups are all contained in the predictions ofthe $\Delta(6N^2)$ groups for some integer $N$. The $\Delta(6N^2)$ groupstherefore represent a sufficient set of $G_f$ to obtain predictions of theresidual symmetries $Z_2$ and $Z_m$.

Residual $Z_2$ symmetries and leptonic mixing patterns from finite discrete subgroups of $U(3)$