With the vast expansion of content on social media platforms, analyzing and comprehending online discourse has become increasingly complex. This paper introduces LLMTaxo, a novel framework leveraging large language models for the automated construction of taxonomy of factual claims from social media by generating topics from multi-level granularities. This approach aids stakeholders in more effectively navigating the social media landscapes. We implement this framework with different models across three distinct datasets and introduce specially designed taxonomy evaluation metrics for a comprehensive assessment. With the evaluations from both human evaluators and GPT-4, the results indicate that LLMTaxo effectively categorizes factual claims from social media, and reveals that certain models perform better on specific datasets.