Based on first-principles calculations, we predict a new two-dimensional ferromagnetic material that exhibits exotic Fermi surface topology. We show that monolayer hexagonal indium carbide ({em h}-InC) is thermodynamically and dynamically stable, and it energetically favors the ferromagnetic ordering of spins. The perfectly planar geometry in two dimensions, together with ferromagnetism, gives rise to a unique opportunity to encounter intriguing electronic properties, captured in the Fermi surface and band topology. We show that multiple nodal lines coexist in momentum space, accompanied by the electron and hole pockets that touch each other linearly at the nodal lines. Inclusion of spin-orbit coupling enriches the magnetic and electronic properties of {em h}-InC. Spin-orbit coupling leads to an easy-plane type magnetocrystalline anisotropy, and the nodal lines can be tuned into topological nodal points, contingent upon the magnetization direction. Symmetry analysis and a tight-binding model are provided to explain the nodal structure of the bands. Our findings suggest {em h}-InC as a new venue for supporting carbon-based magnetism and exotic band topology in two dimensions.
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