We aim to graphically analyze the depth of conceptual understanding behind the Force Concept Inventory (FCI) responses of students, focusing on three questions (questions 1, 15, and 28). In our study, we created and implemented subquestions to clarify and quantify the students' reasoning steps in reaching their responses to the original FCI questions. We used alluvial diagrams to visualize the responses of students on the subquestions and original FCI questions, representing the transition rates across the question options as flows. The flows represent common patterns of consistency and inconsistency in the students' conceptual understandings across the original FCI question and the subquestions. By combining subquestions and alluvial diagrams as we did, we are able to efficiently analyze the depth of understanding for a given situation and to visualize the existence of a specific misconception. For example, we found that even among those who correctly answered FCI question 15 about Newton's third law, many students incorrectly answered a corresponding subquestion asking about whether Newton's third law holds at the end of the interaction between the two objects. This misconception turned out to be fairly robust, as these same students had a similar answering pattern for question 28. Alluvial diagrams can be used for a broad range of applications, including analysis of multiple-tier tests, testlets (e.g., FCI questions 8 through 11), tests for representational consistency (e.g., a representational variant of the FCI), and so on.