Layout design generation has recently gained significant attention due to its potential applications in various fields, including UI, graphic, and floor plan design. However, existing models face two main challenges that limits their adoption in practice. Firstly, the limited expressiveness of individual condition types used in previous works restricts designers' ability to convey complex design intentions and constraints. Secondly, most existing models focus on generating labels and coordinates, while real layouts contain a range of style properties. To address these limitations, we propose a novel framework, CoLay, that integrates multiple condition types and generates complex layouts with diverse style properties. Our approach outperforms prior works in terms of generation quality and condition satisfaction while empowering users to express their design intents using a flexible combination of modalities, including natural language prompts, layout guidelines, element types, and partially completed designs.