Subgroups — #88789

Order 432.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
4	#3753	—	3	1	no	maximal
5	#3713	—	3	4	no	maximal
6	C₂×C₃×C₃⋊D₃	—	2	1	no	—
7	C₂×C₃²×D₃	—	2	1	no	—
8	C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₃²×D₄	—	3	1	no	—
10	C₄⋊(C₃⋊D₃)	D₃	1	1	yes	characteristic, complemented
11	C₃×D₁₂	—	3	4	no	—
12	C₃×D₁₂	D₃	1	4	yes	complemented
13	C₃⋊(C₃⋊C₈)	—	3	1	no	—
14	C₃⋊(C₃⋊C₈)	—	3	4	no	—
15	C₃⋊(C₃⋊C₈)	—	6	4	no	—
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	C₃×C₃⋊D₃	—	4	1	no	—
18	C₃²×D₃	—	4	1	no	—
19	#213	—	9	4	no	—
20	#213	—	9	1	no	—
21	C₂²×C₃²	—	6	1	no	—
22	C₂×C₃⋊D₃	—	2	1	no	—
23	C₂×C₃×D₃	—	2	4	no	—
24	C₂×C₃×D₃	—	6	4	no	—
25	C₃²×C₄	—	2	4	no	—
26	C₃²×C₄	C₂×D₃	1	4	yes	—
27	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
28	C₃³	D₈	1	1	yes	characteristic, complemented, Sylow
29	C₃×D₄	—	3	4	no	—
30	C₃×D₄	—	9	1	no	—
31	D₁₂	—	3	4	no	—
32	D₁₂	C₃⋊D₃	1	1	yes	characteristic, complemented
33	C₃⋊C₈	—	9	4	no	—
34	C₃⋊C₈	—	9	1	no	—
35	C₃⋊C₈	—	18	4	no	—
36	C₂×C₃²	—	2	4	no	—
37	C₂×C₃²	—	12	1	no	—
38	C₂×C₃²	#67	1	4	yes	—
39	C₂×C₃²	#67	1	1	yes	characteristic
40	C₃⋊D₃	—	4	1	no	—
41	C₃×D₃	—	4	4	no	—
42	C₃×D₃	—	12	4	no	—
43	D₈	—	27	1	no	Sylow
44	C₂²×C₃	—	6	4	no	—
45	C₂²×C₃	—	18	1	no	—
46	C₂×D₃	—	2	1	no	—
47	C₂×D₃	—	6	4	no	—
48	C₃×C₄	—	2	4	no	—
49	C₃×C₄	D₃²	1	4	yes	—
50	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
51	C₃²	—	2	4	no	—
52	C₃²	#213	1	4	yes	complemented
53	C₃²	#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	4	no	—
58	C₂×C₃	—	12	4	no	—
59	C₂×C₃	—	36	1	no	—
60	C₂×C₃	#625	1	4	yes	—
61	C₂×C₃	#638	1	1	yes	characteristic
62	D₃	—	4	1	no	—
63	D₃	—	12	4	no	—
64	C₂²	—	6	1	no	—
65	C₂²	—	18	1	no	—
66	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
67	C₃	—	2	4	no	—
68	C₃	#3713	1	4	yes	complemented
69	C₃	#3753	1	1	yes	characteristic, complemented
70	C₂	—	12	1	no	—
71	C₂	—	36	1	no	—
72	C₂	#6290	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6290
commutator, Fitting, cosocle	C₃³×C₄	C₂²
Frattini, hypercenter	C₄	D₃×C₃⋊D₃
socle	C₂×C₃³	D₄
hypocenter	C₃³	D₈
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₄⋊(C₃⋊D₃)	C₂	semidirect
2. C₃²×D₁₂	C₂	semidirect
3. C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
10. C₄⋊(C₃⋊D₃)	D₃	semidirect
12. C₃×D₁₂	D₃	semidirect
28. C₃³	D₈	semidirect
32. D₁₂	C₃⋊D₃	semidirect
52. C₃²	#213	semidirect
53. C₃²	#213	semidirect
68. C₃	#3713	semidirect
69. C₃	#3753	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄⋊(C₃⋊D₃)	2
C₃²×D₁₂	2
C₃⋊(C₃⋊(C₃⋊C₈))	2
#3753	3
#3713	3