# Union of subgroups is subgroup

I am doing an exercise where I am asked to prove or disprove the statement:

If $G$ is a group, and $H_1,H_2,H_3$ and $H_1 \cup H_2 \cup H_3$ are subgroups of $G$ then $\exists i,j$ with $i \neq j$ such that $H_i \subset H_j$.

In the same exercise, I've already prove that this statement is true in the case union of two subgroups is a subgroup. I have no idea how to show this statement holds for this case and I couldn't think of a counterexample. Any suggestions would be appreciated.

Theorem (Bruckheimer, Bryan and Muir) A group is the union of three proper subgroups if and only if it has a quotient isomorphic to $C_2 \times C_2$.
The proof appeared in the American Math. Monthly $77$, no. $1 (1970)$. The theorem seems to be proved earlier by the Italian mathematician Gaetano Scorza, I gruppi che possono pensarsi come somma di tre loro sottogruppi, Boll. Un. Mat. Ital. $5 (1926), 216-218$.
There is a nice overview paper by Mira Bhargava, Groups as unions of subgroups, The American Mathematical Monthly, $116$, no. $5, (2009)$.
Hint: Klein's four element group ($\Bbb Z_2\times\Bbb Z_2$).