Is there an easy way to find the (number of) subgroups of a given group?

Question

Is there an easy way to find all subgroups of a given group? For example, say you had the dihedral group $D_{4}$ - is there a way to work out how many subgroups this has, or what they are, or can it only be done by trial and error or inspection?

Answer 1

I think the answer to the question "Is there an easy way to find all subgroups of a given finite group?" is no.

There are of course methods for doing this that are used by GAP and Magma, which are very effective for small and moderately large groups, but I would not describe them as easy. Also, as far as I am aware, none of the current implementations are complete algorithms, in the sense that they will not, even theoretically, work on arbitrarily large groups. The older methods required databases of perfect groups, which of course could only be complete up to some order.

The newer methods require knowledge of the maximal subgroups of all finite almost simple groups. This is still not completely known theoretically, although is is still a very active area of research. For a complete description, you would need to know (or understand) all irreducible modular representations of all almost simple groups. The maximal subgroups of the Monster sporadfic group are still not completely determined, but there are only a small number of uncertainties remaining.

Answer 2

Consider only abelian $p$-groups of a fixed order, say $p^n$.

Let $C=$cyclic group of order $p^n$ and $E=$ elementary abelian $p$-group of order $p^n$.

Let $G$ be any abelian $p$-group of order $p^n$. Then $$ 1=|\mbox{subgr. of order} p^k \mbox{ in C}| \leq |\mbox{subgr. of order} p^k \mbox{ in G}| \leq |\mbox{subgr. of order} p^k \mbox{ in E}| $$ The number of subgroups in cyclic or elementary abelian $p$-group are very well known.

In arbitrary abelian $p$-group, it is possible to describe by combinatorial argument the number of subgroups of a given order. But its complexity increases because of

(1) structure of $G$ and

(2) all the possible structures of subgroups of order $p^k$ in $G$.

So far, this is vague reason only for abelian $p$-groups. For non-abelian $p$-groups, it is further more difficult. (The study of non-abelian $p$-groups is especially concentrated on a specific class of groups of order $p^n$, rather than all the groups of order $p^n$.)

At last, I would say, there is no easy way to do the problem you posed.