Classify groups all of whose subgroups are nested We consider groups $G$ for which taking intersections of subgroups does not produce "new" subgroups. Let us define:


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*We say that a group $G$ has the strong nesting-subgroups property if for any subgroups $H$ and $K$ of $G$, either $H \subset K$ or $K \subset H$.

*And we say $G$ has the weak nesting-subgroups property if for any subgroups $H$ and $K$ such that $H \cap K \ne \{ e\}$ we have either $H \subset K$ or $K \subset H$.
PROBLEM: Classify all groups $G$ with the strong nesting-subgroups property, and classify all groups with the weak nesting-subgroups property.
If the general problem is too hard, solutions of special cases like when $G$ is finite and/or abelian etc. would also be interesting.
This was inspired by the thread Must subgroups sharing a common element be nested in each other?

Update: From the comments to the question it is clear that the thread found and linked by lhf covers all cases with the strong property (these groups must be abelian since for arbitrary elements $a,b\in G$ the subgroups $\langle a\rangle$ and $\langle b\rangle$ are inside each other, so $ab=ba$). Therefore we are only interested in (partial or full) answers to the weak nesting-subgroups property part.
 A: Suppose that $G$ is a group havinng the weak nesting property. If $N$ is a nontrivial normal subgroup of $G$ then $N$ has the weak nesting property and $G/N$ has the strong nesting property, so $G/N$ must be cyclic of prime power order.
Suppose that $G$ is a finite nonabelian simple group, let $M$ be a maximal subgroup of $G$, and consider the (faithful) permutation representation of $G$ on the set $\Omega$ of cosets $M$ in $G$. Since Frobenius groups are not simple, there exist distinct $\alpha,\beta \in \Omega$ for which the 2-point stabilizer in this action is nontrivial. This 2-point stabilizer $G_{\alpha\beta}$  is contained in two distinct point stabilizers $G_\alpha$ and $G_\beta$ (their distinctness follows from the primitivity of the action), so $G$ does not have the weak intersection property.
So it follows by induction that a finite group with this property is solvable. We can take $N$ to be a minimal normal subgroup and then, since $N$ has the property, $N$ is either cyclic of prime order $p$ or elementary abelian of order $p^2$. Since $G/N$ is cyclic of prime power order, I expect this gives enough information to write down a completer description of all of the possible structures. But I'll leave it at that for now!
Added later: Verret has now shown in comments below that the only finite groups with the weak intersection property are cyclic groups of prime power order, $Z_p^2$ with $p$ prime and, for distinct primes $p,q$,  $Z_{pq}$, $Z_p \rtimes Z_q$ with nontrivial action (so $q\mid p-1$), and $Z_p^2 \rtimes Z_q$ with irreducible action (so $q \mid p+1$).
