There are 2! ways to put the 5 and 7 in 3rd and 4th places.
That leaves 3 even digits to put in the other 3 places.
That can be done in 3! ways.

So the number of ways to put 5 and 7 in 3rd and 4th places is 2!3!

There are 2!=2 ways to put the 4 and 8 in 3rd and 4th places.
Then the 6 must go 5th.  That's 1! way 
That leaves the 5 and 7 to go in front in 2! ways

So the number of ways to put 4 and 8 in those places is 2!1!2!

So the total number of ways is 2!3!+2!1!2! = 2*6+2*1*2 = 12+4 = 16 ways.

Here are all 16 ways:

 1.  46578
 2.  46758
 3.  48576
 4.  48756
 5.  57486
 6.  57846
 7.  64578
 8.  64758
 9.  68574
10.  68754
11.  75486
12.  75846
13.  84576
14.  84756
15.  86574
16.  86754

Edwin