Subgroups — #88437

Order 432.

86 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	#634⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₃⋊(C₃⋊C₈))⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#3716	—	3	1	no	maximal
5	#3715	—	3	1	no	maximal
6	C₂×#272	—	2	1	no	—
7	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	(C₃⋊Dic₃)⋊C₃	—	2	1	no	—
9	#634	D₃	1	1	yes	characteristic, complemented
10	C₃×D₁₂	—	3	1	no	—
11	C₃×D₁₂	—	3	1	no	—
12	C₃×D₁₂	—	6	1	no	—
13	C₃×Dic₆	—	3	1	no	—
14	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
15	C₃×C₃⋊C₈	—	3	1	no	—
16	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	#272	—	4	1	no	—
18	#215	—	9	1	no	—
19	#214	—	9	1	no	—
20	#204	—	9	1	no	—
21	C₂×C₃×D₃	—	6	2	no	—
22	C₂×C₃×D₃	—	6	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	—	2	1	no	—
28	C₃×Dic₃	—	6	1	no	—
29	H(3,𝔽₃)	SD₁₆	1	1	yes	characteristic, complemented, Sylow
30	C₃×Q₈	—	9	1	no	—
31	C₃×D₄	—	9	1	no	—
32	D₁₂	—	3	1	no	—
33	D₁₂	—	3	1	no	—
34	D₁₂	—	6	1	no	—
35	Dic₆	—	3	1	no	—
36	Dic₆	—	18	1	no	—
37	C₃×C₈	—	9	1	no	—
38	C₃⋊C₈	—	3	1	no	—
39	C₃⋊C₈	—	18	1	no	—
40	C₂×C₃²	—	2	1	no	—
41	C₂×C₃²	D₁₂	1	1	yes	characteristic
42	C₂×C₃²	#67	1	1	yes	characteristic
43	C₃×D₃	—	12	2	no	—
44	C₃×D₃	—	12	1	no	—
45	C₃×D₃	—	12	1	no	—
46	SD₁₆	—	27	1	no	Sylow
47	C₂²×C₃	—	18	1	no	—
48	C₂×D₃	—	6	1	no	—
49	C₂×D₃	—	6	1	no	—
50	C₂×D₃	—	6	2	no	—
51	C₃×C₄	—	6	1	no	—
52	C₃×C₄	—	6	1	no	—
53	C₃×C₄	—	12	1	no	—
54	C₃×C₄	—	18	1	no	—
55	C₃×C₄	D₃²	1	1	yes	characteristic
56	Dic₃	—	6	1	no	—
57	Dic₃	—	18	1	no	—
58	C₃²	—	2	1	no	—
59	C₃²	#204	1	1	yes	characteristic, complemented
60	C₃²	#215	1	1	yes	characteristic, complemented
61	Q₈	—	9	1	no	—
62	D₄	—	9	1	no	—
63	C₈	—	9	1	no	—
64	C₂×C₃	—	6	1	no	—
65	C₂×C₃	—	6	1	no	—
66	C₂×C₃	—	12	1	no	—
67	C₂×C₃	—	36	1	no	—
68	C₂×C₃	#626	1	1	yes	characteristic
69	D₃	—	12	1	no	—
70	D₃	—	12	1	no	—
71	D₃	—	12	2	no	—
72	C₂²	—	18	1	no	—
73	C₄	—	18	1	no	—
74	C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
75	C₃	—	6	1	no	—
76	C₃	—	6	1	no	—
77	C₃	—	12	1	no	—
78	C₃	#3717	1	1	yes	characteristic
79	C₂	—	36	1	no	—
80	C₂	#6200	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6200
commutator, Fitting, cosocle	C₄×H(3,𝔽₃)	C₂²
Frattini	C₃×C₄	D₃²
socle	C₂×C₃	#626
prime radical	C₄⋊#272	C₂
hypercenter	C₄	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	SD₁₆
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. #634⋊C₃	C₂	semidirect
3. (C₃⋊(C₃⋊C₈))⋊C₃	C₂	semidirect
9. #634	D₃	semidirect
14. C₃⋊(C₃⋊C₈)	D₃	semidirect
29. H(3,𝔽₃)	SD₁₆	semidirect
59. C₃²	#204	semidirect
60. C₃²	#215	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
#634⋊C₃	2
(C₃⋊(C₃⋊C₈))⋊C₃	2
#3716	3
#3715	3