Broadly speaking, many physicists value intuition in their work, and many instructors hope their students develop intuition (while possibly being wary of their initial, unrefined intuitions). These considerations are especially relevant in quantum mechanics, a subject many see as counterintuitive because it is removed from classical everyday experience. Do students consider quantum mechanics intuitive, how does this affect their approach to the subject, and what does "intuitive" mean to them? We investigate these questions through a mixed-methods approach within the context of one upper-division quantum mechanics class at an R1 university. We find that most students in this population expect to have little intuition for quantum mechanics, so many consider it more unintuitive than counterintuitive. We also find that students use the word intuitive to refer to a number of distinct ideas. Overall, students have a diverse set of perspectives on intuition and its role in studying quantum mechanics. This study lays groundwork for additional research into students' views on intuition in physics and informs how we can address intuition as educators. Quantum instructors should be aware of their students' perspectives on intuition, and can integrate the different ways students perceive intuition into their lessons.