E=5 because multiples of 5 end in 5 or 0
Alternate digits must be even, so the rest have to be odd.
Digits C and D have to go "odd,even", and make a number which is a multiple of 4. So D has to be 2 or 6.
Ditto position H- must be 2 or 6.
The only even numbers left are 4 and 8, and these must go in B and F.
Looking at the first three digits, whose digital root must be 3,6 or 9, there are 9 options for filling these given the conditions we've already worked out.
We tried each of these in turn and worked out the digital root up to F. This also has to be 3,6 or 9 to make it divisible by 6. So you can work out in each case whether D is 2 or 6.
From this we can see what H is as one of its options has been used.
Only two digits remain- we test whether either makes a multiple of 7 when put in position G.
Finding that one of these works, check that the first 8 digits are divisible by 8.
After this only one answer remains:
381654729

(Also, has anyone noticed the patterns this and other suggestions make on the calculator buttons? They are symmetrical or generally interesting.)
If you followed that, I'm impressed. :)

E=5 because multiples of 5 end in 5 or 0

Alternate digits must be even, so the rest have to be odd.

Digits C and D have to go "odd,even", and make a number which is a multiple of 4. So D has to be 2 or 6.

Ditto position H- must be 2 or 6.

The only even numbers left are 4 and 8, and these must go in B and F.

Looking at the first three digits, whose digital root must be 3,6 or 9, there are 9 options for filling these given the conditions we've already worked out.

We tried each of these in turn and worked out the digital root up to F. This also has to be 3,6 or 9 to make it divisible by 6. So you can work out in each case whether D is 2 or 6.

From this we can see what H is as one of its options has been used.

Only two digits remain- we test whether either makes a multiple of 7 when put in position G.

Finding that one of these works, check that the first 8 digits are divisible by 8.

After this only one answer remains:

381654729

(Also, has anyone noticed the patterns this and other suggestions make on the calculator buttons? They are symmetrical or generally interesting.)

If you followed that, I'm impressed. :)