What are the possible values of a six digit positive integer N, where all digits of N are different, and 4*N has the same six digits as N?

Prove that there are no others.

(In reply to

Let me count the ways by Steve Herman)

Have reviewed Charlie's list, I see that my assertion that N = 0 mod 9 was incorrect.

N can also = 3 or 6 mod 9.

So the total number of viable candidates was about 6,720, about three times what I thought.

64, a little less that 1% of these, actually worked.