Groups of order $4p^2$ I am interested about a Fact (if it is right) of the structure of groups of order $4p^2$. Let $G$ be a nonabelian groups of order $4p^2$, classify all this groups.
 A: It is worth pointing out a nice Theorem of J. Brodkey, which may not be very well known (although it is treated in I.M. Isaacs' book). In your case, the interesting case (as already indicated in comments) is when $p=3$. If there is more than one Sylow $3$-subgroup, then there must be $4$ Sylow $3$-subgroups. Let $P$ and $Q$ be distinct Sylow $3$-subgroups of $G.$ Then $\langle P,Q \rangle$ has more than one Sylow $3$-subgroup, so must have $4$ Sylow $3$-subgroups and must have order divisible by $36,$ so must be all of $G.$ Also, $PQ$ ( which is a set, not necessarily a subgroup) has cardinality $\frac{|P| |Q|}{|P \cap Q|}$, but must have size at most $36$. Hence $|P \cap Q| =3.$  But $P$ and $Q$ are both Abelian, so $P \cap Q$ is in the center of $\langle P,Q \rangle = G.$ Thus a group of order $36$
which has more than one Sylow $3$-subgroup has center of order divisible by $3.$
The general result of Brodkey is that if a group $H$ has an Abelian Sylow $p$-subgroup $P,$ then there is a Sylow $p$-subgroup $Q$ of $H$ (possibly $Q = P$) such that $P \cap Q \lhd H.$
