Subgroups — #89076

Order 432.

560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#6312	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
5	#6308	C₂	1	1	yes	maximal, characteristic, complemented
6	#3828	—	3	13	no	maximal
7	C₂²×C₃³	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
9	C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₂×C₃²×C₄	D₃	1	13	yes	complemented
13	#638	—	3	26	no	—
14	C₄⋊(C₃⋊D₃)	—	3	13	no	—
15	C₄×C₃⋊D₃	—	3	26	no	—
16	#634	—	3	13	no	—
17	C₂×C₃³	—	2	1	no	—
18	C₂×C₃³	C₂³	1	1	yes	characteristic
19	C₃⋊(C₃⋊D₃)	—	2	2	no	—
20	#235	—	9	13	no	—
21	C₂²×C₃²	C₂×D₃	1	13	yes	—
22	C₂×C₃⋊D₃	—	3	26	no	—
23	C₃²×C₄	C₂×D₃	1	13	yes	—
24	C₃²×C₄	C₂×D₃	1	13	yes	—
25	C₃⋊Dic₃	—	3	26	no	—
26	C₃³	#41	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃×C₄	C₃⋊D₃	1	13	yes	complemented
28	#67	—	9	26	no	—
29	D₁₂	—	9	13	no	—
30	C₄×D₃	—	9	26	no	—
31	Dic₆	—	9	13	no	—
32	C₂×C₃²	—	2	13	no	—
33	C₂×C₃²	C₂²×D₃	1	13	yes	—
34	C₃⋊D₃	—	6	26	no	—
35	#41	—	27	1	no	Sylow
36	C₂²×C₃	C₂×C₃⋊D₃	1	13	yes	—
37	C₂×D₃	—	9	26	no	—
38	C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
39	C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
40	Dic₃	—	9	26	no	—
41	C₃²	#235	1	13	yes	complemented
42	Q₈	—	27	1	no	—
43	D₄	—	27	2	no	—
44	D₄	—	27	1	no	—
45	C₂×C₄	—	27	2	no	—
46	C₂×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
47	C₂×C₃	—	2	13	no	—
48	C₂×C₃	C₂²×C₃⋊D₃	1	13	yes	—
49	D₃	—	18	26	no	—
50	C₂²	—	27	2	no	—
51	C₂²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
52	C₄	—	27	2	no	—
53	C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
54	C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
55	C₃	#3828	1	13	yes	complemented
56	C₂	—	2	1	no	—
57	C₂	—	54	2	no	—
58	C₂	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂×C₃³	C₂³
Frattini	C₂	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₄	C₂
hypercenter	C₂×C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#41
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×C₄	C₂	semidirect
2. #6312	C₂	semidirect
3. C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
4. C₄×C₃⋊(C₃⋊D₃)	C₂	semidirect
5. #6308	C₂	semidirect
12. C₂×C₃²×C₄	D₃	semidirect
26. C₃³	#41	semidirect
27. C₂×C₃×C₄	C₃⋊D₃	semidirect
41. C₃²	#235	semidirect
46. C₂×C₄	C₃⋊(C₃⋊D₃)	semidirect
55. C₃	#3828	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×C₄	2
#6312	2
C₄⋊(C₃⋊(C₃⋊D₃))	2
C₄×C₃⋊(C₃⋊D₃)	2
#6308	2
#3828	3