Must two groups with all subgroups of the same order be isomorphic?

Question

Let $G,H$ be finite groups such that there's bijection $\varphi$ from the subgroups of $G$ into the subgroups of $H$ satisfying:

$|g_1|=|\varphi(g_1)|$;
$g_1 < g_2$ if and only if $\varphi(g_1)<\varphi(g_2)$;
$g_1 \triangleleft g_2$ if and only if $\varphi(g_1)\triangleleft\varphi(g_2)$.

Is it always true that $G\cong H$?

This question arises if you consider two Galois extensions $F_1, F_2$ of $K$. If all intermediate extensions $K\subseteq E_1\subseteq F_1$ and $K\subseteq E_2\subseteq F_2$ look the same as $K$-vector fields then $gal(F_1 |K)\cong gal(F_2|K)$?

Without 3, the answer is definitely no. I'm pretty sure the answer remains "no" even if we add 3. — Arturo Magidin, Oct 03 '23 at 22:16
No, I am, sorry! Still, it's worth checking if the order-$16$ example still works here. — Misha Lavrov, Oct 04 '23 at 01:35
Here is another example. It satisfies the stronger condition that corresponding proper subgroups are isomorphic, and the normal subgroups map to normal subgroups. But I am not sure it satisfies 3, either. — Arturo Magidin, Oct 04 '23 at 01:41
@MishaLavrov According to the link in my other comment, there shouldn't be counterexamples of order dividing 64. — Arturo Magidin, Oct 04 '23 at 01:42

score 1 · Accepted Answer · answered Oct 04 '23 at 15:41

1

Jack Schmidt posted this answer saying that the two groups of order 243, $G = \mathrm{SmallGroup}(243, 19)$ and $H = \mathrm{SmallGroup}(243, 20)$ in the GAP small groups library satisfy these conditions and are not isomorphic.

answered Oct 04 '23 at 15:41

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Must two groups with all subgroups of the same order be isomorphic?

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