Finite groups, where the number of subgroups is equal to the number of elements

Question

I am looking for finite groups $G$ such that the number of subgroups is equal to $|G|$. Examples are:

• the trivial group

• $\mathbb{Z}/2\mathbb{Z}$

• $S_3$

Does anyone know some more examples or can provide some insight? Are there any bounds on the number of subgroups (maybe using Sylow's theorems or whatever)?

Answer 1

Here is list of all examples $\leq 100$, found with GAP (EDIT: I added some more, but I didn't check for groups of order $128$). The notation $:$ means semidirect product.

$$\begin{array}{|c|c|} \text{StructureDescription} & \text{Order} \\ \hline \text{Trivial group} & 1 \\ \hline C_2 & 2 \\ \hline S_3 & 6 \\ \hline C_4 \times C_2 & 8 \\ \hline D_{28} & 28 \\ \hline C_6 \times S_3 & 36 \\ \hline (C_{10} \times C_2) : C_2 & 40 \\ \hline C_2 \times (C_5 : C_4) & 40 \\ \hline (C_3 \times Q_8) : C_2 & 48 \\ \hline ((C_3 \times C_3) : C_3) : C_2 & 54 \\ \hline C_6 \times A_4 & 72 \\ \hline C_2 \times ((C_4 \times C_4) : C_3) & 96 \\ \hline (C_5 \times C_5) : C_4 & 100 \\ \hline D_{104} & 104 \\ \hline S_3 \times D_{22} & 132 \\ \hline C_3 \times D_{48} & 144 \\ \hline (C_{40} \times C_2) : C_2 & 160 \\ \hline (C_5 \times (C_8 : C_2)) : C_2 & 160 \\ \hline ((C_2 \times (C_5 : C_4)) : C_2) : C_2 & 160 \\ \hline (C_4 \times (C_5 : C_4)) : C_2 & 160 \\ \hline (C_{40} \times C_2) : C_2 & 160 \\ \hline (C_8 \times D_{10}) : C_2 & 160 \\ \hline (C_2 \times (C_5 : Q_8)) : C_2 & 160 \\ \hline (C_2 \times (C_{11} : C_4)) : C_2 & 176 \\ \hline (C_{15} \times C_3) : C_4 & 180 \\ \hline \end{array}$$

Random related fact: the number of subgroups in the dihedral group $D_n$ of order $n$ is $\sigma(n/2) + d(n/2)$, where $\sigma$ is the sum of divisors function and $d$ is the divisor count function. Thus the dihedral group $D_n$ of order $n$ is an example for the problem when $$n = 2,\ 6,\ 28,\ 104,\ 260,\ 368,\ 1312,\ 17296,\ 24016,\ 69376,\ \ldots$$

I don't know if this sequence is infinite. For more terms, it is $2 \cdot$ $A083874$ from OEIS. Seems that really large examples exist, for example $9223653647124987904$ is in the sequence.