Exercise
What are the orders of subfields of a field of order 64?
2, 64.
2, 4, 8, 64
2, 4, 8, 16, 32, 64
No. The fixed field of x -> x4 has order 4.
Correct.
No, if there were a subfield of order 32,
then 64 would be a power of 32, a contradiction.