  Let $F$ be a field of characteristic $p$ at least 5. We study the Loewy
structures of Specht modules in the principal block of $F\Sigma_{3p}$. We show
that a Specht module in the block has Loewy length at most 4 and composition
length at most 14. Furthermore, we classify which Specht modules have Loewy
length 1, 2, 3, or 4, produce a Specht module having 14 composition factors,
describe the second radical layer and the socle of the reducible Specht
modules, and prove that if a Specht module corresponds to a partition that is
$p$-regular and $p$-restricted then the head of the Specht module does not
extend the socle.
