I guess it all depends upon what you're assumed to know.

You can trivially check that the Sylow

-subgroup

is normal and

which evidently implies that

is solvable. That said,

is of order eight, and regardless of

's isomorphism type we know it's solvable--namely because it is a fourth-week-of-group theory result that the only groups of order

are the three abelian and

all of which are solvabe. So,

is solvable. Since solvability respects extensions we may conclude that

is solvable.

Of course, the above really shows that any group of order

with normal Sylow

-subgroup is solvable, which is consistent with the common theorem that NonCommAlg stated.