Subgroups — #88434

Order 432.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#272	C₂	1	1	yes	maximal, characteristic
2	(C₃⋊(C₃⋊C₈))⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₃⋊(C₃⋊C₈))⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₃⋊C₈	—	3	1	no	maximal
5	D₃×C₃⋊C₈	—	3	1	no	maximal
6	C₂×#272	C₄	1	1	yes	characteristic
7	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	#1084	C₄	1	1	yes	characteristic
9	C₃×C₄×D₃	—	3	1	no	—
10	C₃×C₄×D₃	—	3	1	no	—
11	C₃×C₄×D₃	—	6	1	no	—
12	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
13	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
14	C₃×C₃⋊C₈	—	3	1	no	—
15	C₃×C₃⋊C₈	—	3	1	no	—
16	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
17	#272	C₈	1	2	yes	complemented
18	C₂×C₃⋊C₈	—	9	1	no	—
19	D₃×C₈	—	9	1	no	—
20	D₃×C₈	—	9	1	no	—
21	C₂×C₃×D₃	—	3	1	no	—
22	C₂×C₃×D₃	—	3	1	no	—
23	C₂×C₃×D₃	—	6	1	no	—
24	C₃²×C₄	—	2	1	no	—
25	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
27	C₃×Dic₃	—	3	1	no	—
28	C₃×Dic₃	—	3	1	no	—
29	C₃×Dic₃	—	6	1	no	—
30	H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃×C₄	—	9	1	no	—
32	C₄×D₃	—	3	1	no	—
33	C₄×D₃	—	3	1	no	—
34	C₄×D₃	—	6	1	no	—
35	C₃×C₈	—	9	1	no	—
36	C₃×C₈	—	9	1	no	—
37	C₃⋊C₈	—	3	1	no	—
38	C₃⋊C₈	—	3	1	no	—
39	C₃⋊C₈	—	9	1	no	—
40	C₃⋊C₈	—	18	1	no	—
41	C₂×C₃²	—	2	1	no	—
42	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
43	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
44	C₃×D₃	—	3	2	no	—
45	C₃×D₃	—	3	2	no	—
46	C₃×D₃	—	6	2	no	—
47	C₂×C₈	—	27	1	no	Sylow
48	C₂²×C₃	—	9	1	no	—
49	C₂×D₃	—	3	1	no	—
50	C₂×D₃	—	3	1	no	—
51	C₂×D₃	—	6	1	no	—
52	C₃×C₄	—	3	1	no	—
53	C₃×C₄	—	6	1	no	—
54	C₃×C₄	—	9	1	no	—
55	C₃×C₄	—	12	1	no	—
56	C₃×C₄	D₃²	1	1	yes	characteristic
57	Dic₃	—	3	1	no	—
58	Dic₃	—	3	1	no	—
59	Dic₃	—	6	1	no	—
60	C₃²	—	2	1	no	—
61	C₃²	D₃×C₈	1	1	yes	characteristic, complemented
62	C₃²	D₃×C₈	1	1	yes	characteristic, complemented
63	C₂×C₄	—	9	1	no	—
64	C₈	—	9	1	no	—
65	C₈	—	9	1	no	—
66	C₂×C₃	—	3	1	no	—
67	C₂×C₃	—	6	1	no	—
68	C₂×C₃	—	9	2	no	—
69	C₂×C₃	—	12	1	no	—
70	C₂×C₃	#624	1	1	yes	characteristic
71	D₃	—	3	2	no	—
72	D₃	—	3	2	no	—
73	D₃	—	6	2	no	—
74	C₂²	—	9	1	no	—
75	C₄	—	9	1	no	—
76	C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
77	C₃	—	3	1	no	—
78	C₃	—	6	1	no	—
79	C₃	—	12	1	no	—
80	C₃	#3710	1	1	yes	characteristic
81	C₂	—	9	2	no	—
82	C₂	#6199	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, hypocenter	H(3,𝔽₃)	C₂×C₈
Frattini	C₃×C₄	D₃²
Fitting, cosocle	C₄×H(3,𝔽₃)	C₂²
socle	C₂×C₃	#624
prime radical	C₂×#272	C₄
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. (C₃⋊(C₃⋊C₈))⋊C₃	C₂	semidirect
3. (C₃⋊(C₃⋊C₈))⋊C₃	C₂	semidirect
12. C₃⋊(C₃⋊C₈)	D₃	semidirect
13. C₃⋊(C₃⋊C₈)	D₃	semidirect
17. #272	C₈	semidirect
30. H(3,𝔽₃)	C₂×C₈	semidirect
61. C₃²	D₃×C₈	semidirect
62. C₃²	D₃×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#272	2
(C₃⋊(C₃⋊C₈))⋊C₃	2
D₃×C₃⋊C₈	3