When looking at finite group theory research, it seems to me that a lot of energy is devoted to determining the maximal subgroups of certain classes of groups. For example, the O'Nan Scott theorem gives a classification of the maximal subgroups of $S_n$. There is an entire book (link) devoted to studying the maximal subgroups of simple classical groups. I'm pretty sure there are many other examples.
Question: How does understanding the maximal subgroups of a group help us in understanding the structure of the group? Why should we care about maximal subgroups?
This question is of course a bit vague. What I'm looking for is motivation and examples of applications.
For example, I can see how the classification of finite simple groups (CFSG) is useful and interesting in group theory: every group is made up of simple groups (Jordan-Hölder), and CFSG is a very powerful tool for proving things. Many theorems have been proven with the following strategy:
Step 1: Prove that a minimal counterexample is a finite simple group.
Step 2: Check the theorem for each family of finite simple groups.
So the importance of CFSG is clear. What I don't understand currently is why a group theorist would care about maximal subgroups or classifying them.