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I need a method to find all subgroups from any finite group. For example, the subgroups of $D_4 $ (of order $4$): $\{(1), (1 \ 2 \ 3 \ 4), (1 \ 3)(2 \ 4), (1 \ 4 \ 3 \ 2)\}$.

Okay, I can understand this but the others .. how we can get them? Is there any theorem or something?

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  • $\begingroup$ Your group "$D_4$" appears to be $S_4.$ The group $S_4$ has $4!=24$ elements, while $D_4$ has $8$ or $16$ (depending on which version of dihedral notation is being used). $\endgroup$ – Cameron Buie Dec 12 '15 at 15:13
  • $\begingroup$ @CameronBuie: I think that you've rushed a bit. The OP asks for a general algorithm for finding all the subgroups of any finite given group. $D_4$ is not central to his question. Could you please reconsider your close vote? $\endgroup$ – Alex M. Dec 12 '15 at 15:51
  • $\begingroup$ @AlexM.: I'm aware. I outline a general algorithm in my answer there, though it happens to coincide with finding all subgroups of $D_4.$ $\endgroup$ – Cameron Buie Dec 12 '15 at 15:56
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    $\begingroup$ Then there's this post and this post, too. $\endgroup$ – Cameron Buie Dec 12 '15 at 16:17