Hybrid membranes of lipids and A(x)B(y)A(z) triblock copolymers can possess better biocompatibility and mechanical stability. In this work, triblock copolymer conformations and stability of asymmetric membranes are explored by dissipative particle dynamics. The triblock copolymers in the membranes exhibit either the bridge or loop conformation. As hydrophobic B-blocks interact attractively with lipid heads, bridge-shaped copolymers are significantly inhibited and loop-shaped copolymers prefer to stay at the interface between hydrophilic and hydrophobic layers. This floating loop has a flattened conformation, consistent with the experimental findings. In contrast, for repulsive interactions between B-blocks and lipid heads, bridge-shaped copolymers are abundant and loop-shaped copolymers tend to plunge into the hydrophobic layer. This diving loop displays a random coil conformation. The asymmetric membrane in which the fractions of loop-shaped copolymers in the upper and lower leaflets are different is thermodynamically unstable. Two approaches are proposed to acquire kinetically stable asymmetric membranes. First, membrane symmetrization is arrested by eliminating bridge-shaped copolymers, which is achieved by B-block/lipid head attraction and B-block/lipid tail repulsion. Second, asymmetric triblock copolymers (x not equal z) are used to prevent the passage of the long A-block through the hydrophobic layer.