In order to clarify the unsteady behavior and the quenching mechanism of the hydrogen-air premixed flame at a small circular hole made in the cooled plate put in the large container, the numerical calculation was carried out by using the detailed chemical kinetics. In this calculation, the stationary premixed gas is filled in the container at the start, and then the mixture is ignited at the end face and the flame propagates to the hole. At the hole, the flow is induced by the flame propagation and the heat is taken from the cooled wall, so that the flame front changes unsteadily and has the curvature. When the flame front has the curvature, the effects of non-unity Lewis number of the hydrogen fuel lean or rich mixtures appear in the quenching process. The relationship between the quenching diameter and the equivalence ratio are shown for the different thickness of the cooled plate. Moreover, the unsteady flame behavior and the critical quenching process are shown for the fuel lean and rich mixtures, and the quenching mechanism is examined so that the effects of the cooling and the Lewis number are charified.