Numbers greater than 6000 can be 4-digit or 5-digit.

5-digit numbers: All arrangements of 5 digits $= 5! = 120$.

4-digit numbers $> 6000$: The leading digit must be $6, 7,$ or $8$ (3 choices). Remaining 3 positions filled from 4 remaining digits: $3 \times 4 \times 3 \times 2 = 72$.

Total $= 120 + 72 = \mathbf{192}$