We are in the contest of the classification of all groups of order $2^3$.
We know that a group $G$ with a unique maximal subgroup is necessarely cyclic.
1)Then my teacher said that the dual proposition is "almost" true: a group $G$ with a unique minimal subgroup is a $p$-group and if $p\neq2$ it's also cyclic. Is this true? After this he said that there is a unique counterexample to this (he spoke about $Q_8$, but it seems it respects the dual proposition: $\langle-1\rangle$ is its unique minimal subgroup). So what is this group that makes exception?
2)Then he said: suppose $G$ s.t. $H\unlhd G\;\;\forall\;\;H\le G$, then either $G$ is abelian, or $G$ admits $Q_8$ as a direct factor (i.e. $\exists A\le G$ s.t.$G=Q_8\times A$): why? Finally, from this it follows that the only two groups of order $2^3$ are $D_4$ or $Q_8$. WHY? I'm struggling on this. I found on the web some useful pdf that explain why the only two groups of order $2^3$ are these two... but it was made only with computations, and I'm really interested in understand this in the way my teacher did: it seems smarter, mathematically better.
If someone can help me I would be really grateful. I'm aware my questions could seem a little bit confusing, but this is the best can I did with my notes. Thank you all.