The inclined flame front break in the filtration gas combustion of a lean methane (CH4)/air mixture during propagation downstream in an inert porous medium was investigated using a two-temperature and two-dimensional numerical model. The development of flame inclination was studied by focusing on the flame front break, which splits into two or three waves. The essence of the flame front instability and inclined flame front break mechanisms was discussed. The physical parameters, such as flame structure, gas temperature distributions, and effects of system properties on flame characteristics, were studied in detail. The results show that the flame front break easily takes place in higher Lewis number mixtures, higher inlet velocity, lower equivalence ratio, and larger heat loss regimes. Higher inlet velocity and lower equivalence ratios result in higher flame instability in the combustion process.