Subgroups — #88421

Order 432.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3713	—	3	1	no	maximal
5	#3673	—	3	1	no	maximal
6	C₂×D₃×C₉	—	2	1	no	—
7	C₂×C₃×D₉	—	2	1	no	—
8	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×D₁₂	—	3	1	no	—
10	C₃×D₁₂	D₃	1	1	yes	characteristic
11	C₃⋊(C₃⋊C₈)	—	3	1	no	—
12	D₄×C₉	—	3	1	no	—
13	D₃₆	D₃	1	1	yes	characteristic, complemented
14	C₉⋊C₈	—	3	1	no	—
15	C₉⋊C₈	—	6	1	no	—
16	C₂×C₃×C₉	D₄	1	1	yes	characteristic
17	D₃×C₉	—	4	1	no	—
18	C₃×D₉	—	4	1	no	—
19	#213	—	9	1	no	—
20	#213	—	9	1	no	—
21	C₂×C₃×D₃	—	2	1	no	—
22	C₂×C₃×D₃	—	6	1	no	—
23	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
24	C₂²×C₉	—	6	1	no	—
25	C₂×D₉	—	2	1	no	—
26	C₄×C₉	—	2	1	no	—
27	C₄×C₉	C₂×D₃	1	1	yes	characteristic
28	C₃×C₉	D₈	1	1	yes	characteristic, complemented, Sylow
29	C₃×D₄	—	3	1	no	—
30	C₃×D₄	—	9	1	no	—
31	D₁₂	—	3	1	no	—
32	D₁₂	D₉	1	1	yes	characteristic, complemented
33	C₃⋊C₈	—	9	1	no	—
34	C₃⋊C₈	—	9	1	no	—
35	C₃⋊C₈	—	18	1	no	—
36	C₂×C₃²	#67	1	1	yes	characteristic
37	C₃×D₃	—	4	1	no	—
38	C₃×D₃	—	12	1	no	—
39	C₂×C₉	—	2	1	no	—
40	C₂×C₉	—	12	1	no	—
41	C₂×C₉	#67	1	1	yes	characteristic
42	D₉	—	4	1	no	—
43	D₈	—	27	1	no	Sylow
44	C₂²×C₃	—	6	1	no	—
45	C₂²×C₃	—	18	1	no	—
46	C₂×D₃	—	2	1	no	—
47	C₂×D₃	—	6	1	no	—
48	C₃×C₄	—	2	1	no	—
49	C₃×C₄	C₂×D₉	1	1	yes	characteristic
50	C₃×C₄	D₃²	1	1	yes	characteristic
51	C₃²	#213	1	1	yes	characteristic
52	C₉	—	2	1	no	—
53	C₉	#213	1	1	yes	characteristic, complemented
54	D₄	—	3	1	no	—
55	D₄	—	9	1	no	—
56	C₈	—	27	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	12	1	no	—
59	C₂×C₃	—	36	1	no	—
60	C₂×C₃	#611	1	1	yes	characteristic
61	C₂×C₃	#625	1	1	yes	characteristic
62	D₃	—	4	1	no	—
63	D₃	—	12	1	no	—
64	C₂²	—	6	1	no	—
65	C₂²	—	18	1	no	—
66	C₄	D₃×D₉	1	1	yes	characteristic
67	C₃	—	2	1	no	—
68	C₃	#3673	1	1	yes	characteristic, complemented
69	C₃	#3713	1	1	yes	characteristic
70	C₂	—	12	1	no	—
71	C₂	—	36	1	no	—
72	C₂	#6194	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6194
commutator, Fitting, cosocle	C₃×C₄×C₉	C₂²
Frattini	C₃×C₄	D₃²
socle	C₂×C₃²	#67
hypercenter	C₄	D₃×D₉
hypocenter	C₃×C₉	D₈
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₉×D₁₂	C₂	semidirect
2. C₃×D₃₆	C₂	semidirect
3. C₃⋊(C₉⋊C₈)	C₂	semidirect
13. D₃₆	D₃	semidirect
28. C₃×C₉	D₈	semidirect
32. D₁₂	D₉	semidirect
53. C₉	#213	semidirect
68. C₃	#3673	semidirect

Maximal subgroups

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

subgroup	index
C₉×D₁₂	2
C₃×D₃₆	2
C₃⋊(C₉⋊C₈)	2
#3713	3
#3673	3