Shell structures have gained significant attention in various fields, including architectural design, art, and aerospace engineering, due to their aesthetic appeal and exceptional structural efficiency. In recent years, topology optimization has been applied to shell structures to improve stiffness-to-weight ratios. However, current methods often focus on simple, single-patch shells, and for more complex models, existing approaches typically divide the structure into multiple patches, which are analyzed using coupling techniques. While some research has explored the use of T-splines for topology optimization of complex shell structures, the process remains relatively intricate. This paper presents a novel and straightforward topology optimization framework based on subdivision surfaces and isogeometric analysis (IGA) for Kirchhoff-Love shell structures. The proposed method represents both geometry in the CAD model, displacement in the analysis model, and density in the design model using subdivision surfaces. Each vertex in the subdivision mesh is assigned displacement and density coefficients, allowing for a direct and efficient optimization process. Subdivision surfaces naturally accommodate topologically complex structures, including those with genus, while also ensuring at least C^1-continuity, thus eliminating the need for traditional coupling operations. The integration of subdivision surfaces and IGA provides a powerful tool for the design of thin shell structures, offering both efficiency and flexibility. The effectiveness of the proposed framework is demonstrated through several numerical examples, which confirm its robustness and computational efficiency in optimizing complex shell structures.