I have the group of order $108$, $$G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$$ obtained from an algorithm in GAP, but I need to prove that it has a $3$-embedded subgroup (a subgroup $H\leq G$ such that $p\mid \ \vert H\vert $ and for any $x\in G-H$, $p \nmid\ \vert H\cap {}^xH \vert$ where $^{x}H=xHx^{-1}$). Does anyone have an idea of how to attack it without using GAP algorithms?
If it helps, it's the group with Id $[108,17]$.