Is it known a characterization of finite groups of order $n$ having exactly $n$ subgroups? 


A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}\_2$ with this property?



Finite groups of order $n$ having exactly $n$ subgroups
There are finite groups of order $n$ having exactly $n$ subgroups
for $n = 1$, $2$, $6$, $8$, $28$, $36$, $40$, $40$, $48$, $54$, $72$,
$\dots$, and this list is exhaustive for $n < 96$.
The structures of the groups of order $< 96$ which satisfy the condition 
are as follows:


* $1$,
* ${\rm C}\_2$,
* ${\rm S}\_3$,
* ${\rm C}\_4 \times {\rm C}\_2$,
* ${\rm D}\_{28}$,
* ${\rm C}\_6 \times {\rm S}\_3$,
* $({\rm C}\_{10} \times {\rm C}\_2) \rtimes {\rm C}\_2$,
* ${\rm C}\_2 \times ({\rm C}\_5 \rtimes {\rm C}\_4)$,
* $({\rm C}\_3 \times {\rm Q}\_8) \rtimes {\rm C}\_2$,
* $(({\rm C}\_3^2) \rtimes {\rm C}\_3) \rtimes {\rm C}\_2$, and
* ${\rm C}\_6 \times {\rm A}\_4$.


The [GAP](http://www.gap-system.org) [SmallGroups Library](http://www.gap-system.org/Packages/sgl.html) Id numbers of these groups are as follows:



```
[ [ 1, 1 ], [ 2, 1 ], [ 6, 1 ], [ 8, 2 ], [ 28, 3 ], [ 36, 12 ], 
  [ 40, 8 ], [ 40, 12 ], [ 48, 17 ], [ 54, 5 ], [ 72, 47 ] ]

```

The existence of a general non-trivial "characterization" of such groups
seems rather unlikely to me.


