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Is there a way to calculate the total number of subgroups of a group?

I can imagine that for example if $G=D_n$ is the dihedral group or $G=S_n$ the symmetric group then there exists a formula to calculate the total number of subgroups.

The reason why I started to think about this question is because I was trying to find all subgroups of $D_4$ (the square).

And I found some but I want to prove that I found all of them.

So if the answer is no to the question above then I'd be equally happy with a way of being sure that given a collection of subgroups to determine that there cannot be more.

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You might be interested in this MO thread, which basically says the answer is no. However, it also contains the nice fact that any subgroup is generated by at most $\log_2(|G|)$ elements, which does limit the search space somewhat.

For dihedral groups, you can use a much more elementary counting argument: either the group is cyclic (and these are easily counted) or it contains a reflection. If we call one such reflection $r$... can you finish from here?

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The total number of subroups $D_{n}$ are $$\tau(n)+\sigma(n)$$ Where $\tau(n)$ is the number of divisors of n and $\sigma(n)$ is the sum of divisors of $n$

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