The fourth industrial revolution, often termed as Industry 4.0 or Smart Manufacturing, is influencing all aspects of business management. Application of Industry 4.0 for quality management purpose is termed Quality 4.0. Lean Six Sigma (LSS) implementation process is also getting influenced by Quality 4.0. Use of Lean Six Sigma in Industry 4.0/Quality 4.0 can create differentiation in performance, and will improve competitiveness of organizations, besides making them future ready. Critical Success Factors (CSF) for Lean Six Sigma in Quality 4.0 set-up are recently established. However, the hierarchy of Critical Success Factors (CSF) for Lean Six Sigma under Quality 4.0 framework is not yet available. This paper establishes the hierarchy using multiple methodologies including ISM (Interpretive Structural Modelling) and TISM (Total Interpretive Structural Modelling), MICMAC (cross-impact matrix multiplication applied to classification), and Hierarchical Clustering. Managerial and theoretical implications of this study are presented from both Lean Six Sigma, and Information Technology perspectives.