X a X e X i X o X u X

In the above there are 6 positions marked with X's in which 
we can insert consonants q, t, n.  

We can choose 3 of those 6 positions to insert one consonant
each in C(6,3) ways.          Example:   a t e i n o u q

We can choose 2 of those 6 positions to insert two consonants 
in the leftmost one and one consonant in the rightmost one
in C(6,2) ways                Example:   a e i q n o t u 

We can choose 2 of those 6 positions to insert one consonant 
in the leftmost one and two consonants in the rightmost one
in C(6,2) ways                Example:   n a e i t q o u

We can choose 1 of those 6 positions to insert all three
consonants in C(6,1) ways     Example:   a e i n q t o u 

So that's C(6,3)+C(6,2)+C(6,2)+C(6,1) = 20+15+15+6 = 56 ways
to choose places to insert consonants.

For each of those 56 ways to insert 3 consonants, the consonants
can be arranged in 3! or 6 ways.

Answer 56·6 = 336 ways.

Edwin