Working memory, or the ability to hold and manipulate information in the mind, is a critical component of human intelligence and executive functioning. It is correlated with performance on various cognitive tasks, including measures of fluid intelligence, which encompasses reasoning and problem solving. We use a comprehensive set of classic working memory tasks to estimate the working memory capacity of large language models (LLMs). We find that in most cases, LLMs exceed normative human scores. However, we do not find that the increased capacity of working memory is associated with higher performance on other executive functioning tasks or problem solving benchmarks. These results suggest that LLMs may have deficits in attentional control and cognitive flexibility, which result in difficulties with inhibiting automatic responses and adapting to shifting information. Our findings suggest that reasoning models, although they often do not currently fully compensate for these deficits, may have the potential to do so in the future.