How to find and count the subgroups of $D_4\times\mathbb{Z}/6\mathbb{Z}$? In general, are there some hints to study these kind of problems? In particular I am interested in the case of semidirect products..
How to count and find the subgroups of a specific group
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abstract-algebra
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0Naively thinking, I would try to find subgroup of each constituents and then see if their product is a subgroup of the original one. I would love to see a general method too. I am not aware of any yet. – 2017-01-11
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1Are you counting subgroups up to isomorphism? – 2017-01-11
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0@Omnomnomnom I would like to know the exact number of subgroups for each possibility – 2017-01-11
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0I'll take that to mean that you are *not* counting up to isomorphism, which is to say that two isomorphic but non-equal subgroups should be counted separately. – 2017-01-11
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0@HumbleStudent I think this is good if I am interested in counting the subgroups up to isomorphism, but if I want the exact number this is not enough.. for example in $\mathbb{Z}\times\mathbb{Z}$ there are three subgroups isomorphism to $\mathbb{Z}$. And with a semidirect product I don't know if it is a good idea.. – 2017-01-11
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0It seems like it would be easy to count the abelian subgroups (most of which are cyclic), and the non-abelian subgroups must contain $D_4$. I don't know if that idea generalizes, though. – 2017-01-11