  A new set of lattice-models based on the hexagonal $\sqrt{N}\times\sqrt{N}$
super-cells of the well-known honeycomb lattice with single-hole defect
(HL-D-1/2N) are proposed to realize the nontrivial isolated flat-bands. Through
performing both tight-binding and density functional theory calculations, we
demonstrate that the experimentally realized graphitic carbon nitride (Adv.
Mater., 22, 1004, 2010; Nat. Commun., 9, 3366, 2018), the HL-D-1/8 based
C$_4$N$_3$, is a perfect system to host such flat bands. For the flat
high-energy P-6m2 C$_4$N$_3$ structure, it displays the ferromagnetic
half-metallicity which is not related to the isolated flat bands. However, the
P-6m2 C$_4$N$_3$ structure is dynamically unstable. Using a structure searching
method based on group and graph theory, we find that a new corrugated Pca21
C4N3 structure has the lowest energy among all known C$_4$N$_3$ structures.
This Pca21 C$_4$N$_3$ structure is an intrinsic ferromagnetic
half-semiconductor (Tc$\approx$241 K) with one semiconducting spin-channel
(1.75 eV) and one insulating spin-channel (3.64 eV), which is quite rare in the
two-dimensional (2D) systems. Its ferromagnetic semiconducting property
originates from the isolated p$_z$-state flat-band as the corrugation shift the
flat band upward to the Fermi level. Interestingly, this Pca21 C$_4$N$_3$
structure is found to be piezoelectric and ferroelectric, which makes
C$_4$N$_3$ an unusual transition-metal-free 2D multiferroic.
