Finite Group Databasebeta

← C₂×C₄×C₉ #612

Subgroups — C₂×C₄×C₉ #612

Order 72.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₄	C₃	1	1	yes	maximal, characteristic
4	C₂×C₉	C₄	1	2	yes	direct product factor
5	C₂×C₉	C₂²	1	1	yes	characteristic
6	C₂²×C₃	C₂×C₃	1	1	yes	characteristic
7	C₃×C₄	C₂×C₃	1	2	yes	—
8	C₉	C₂×C₄	1	1	yes	characteristic, direct product factor, Sylow
9	C₂×C₄	C₉	1	1	yes	characteristic, direct product factor, Sylow
10	C₂×C₃	C₃×C₄	1	2	yes	—
11	C₂×C₃	C₂²×C₃	1	1	yes	characteristic
12	C₂²	C₂×C₉	1	1	yes	characteristic
13	C₄	C₂×C₉	1	2	yes	direct product factor
14	C₃	C₂×C₃×C₄	1	1	yes	characteristic
15	C₂	C₄×C₉	1	2	yes	direct product factor
16	C₂	C₂²×C₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂²×C₃
socle, prime radical	C₂²×C₃	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄×C₉	C₂	direct (N × H)
4. C₂×C₉	C₄	direct (N × H)
8. C₉	C₂×C₄	direct (N × H)
9. C₂×C₄	C₉	direct (N × H)
13. C₄	C₂×C₉	direct (N × H)
15. C₂	C₄×C₉	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₉	2
C₄×C₉	2
C₂×C₃×C₄	3