Subgroups — #11720552

Order 864.

165 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#88538	C₂	1	2	yes	maximal
3	C₃×#819	—	3	1	no	maximal
4	C₃×#819	—	3	3	no	maximal
5	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₈×H(3,𝔽₃)	C₄	1	2	yes	—
7	#6180	—	2	2	no	—
8	C₂×C₃²×C₈	D₃	1	1	yes	characteristic
9	C₂×C₃²×C₈	D₃	1	3	yes	—
10	C₃×#208	—	3	2	no	—
11	C₃×#208	—	3	6	no	—
12	C₂²×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
13	C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
14	C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
15	C₃×#108	—	9	1	no	—
16	#819	—	3	1	no	—
17	#819	—	3	3	no	—
18	C₂×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
19	C₂×C₃²×C₄	C₂×D₃	1	3	yes	—
20	C₃²×C₈	Dic₃	1	2	yes	—
21	C₃²×C₈	Dic₃	1	6	yes	—
22	C₃×C₃⋊C₈	—	6	2	no	—
23	C₃×C₃⋊C₈	—	6	6	no	—
24	C₂×H(3,𝔽₃)	—	2	1	no	—
25	C₂×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
26	C₃×M₁₆	—	9	2	no	—
27	C₂×C₃×C₈	—	3	1	no	—
28	C₂×C₃×C₈	—	6	3	no	—
29	C₂×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
30	#208	—	3	2	no	—
31	#208	—	3	6	no	—
32	C₂²×C₃²	Dic₆	1	1	yes	characteristic
33	C₂²×C₃²	Dic₆	1	3	yes	—
34	C₃²×C₄	D₁₂	1	1	yes	characteristic
35	C₃²×C₄	D₁₂	1	3	yes	—
36	C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
37	C₃²×C₄	C₂×Dic₃	1	3	yes	—
38	#108	—	9	1	no	Sylow
39	H(3,𝔽₃)	#108	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃×C₄	—	3	1	no	—
41	C₂×C₃×C₄	—	6	3	no	—
42	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
43	C₃×C₈	—	3	2	no	—
44	C₃×C₈	—	6	6	no	—
45	C₃×C₈	—	18	2	no	—
46	C₃×C₈	C₃⋊Dic₃	1	2	yes	—
47	C₃⋊C₈	—	6	2	no	—
48	C₃⋊C₈	—	6	6	no	—
49	C₂×C₃²	—	2	1	no	—
50	C₂×C₃²	—	2	3	no	—
51	C₂×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
52	C₂×C₃²	C₄⋊Dic₃	1	3	yes	—
53	M₁₆	—	9	2	no	—
54	C₂×C₈	#272	1	1	yes	characteristic
55	C₂²×C₃	—	3	1	no	—
56	C₂²×C₃	—	6	3	no	—
57	C₂²×C₃	#634	1	1	yes	characteristic
58	C₃×C₄	—	3	1	no	—
59	C₃×C₄	—	3	1	no	—
60	C₃×C₄	—	6	3	no	—
61	C₃×C₄	—	6	3	no	—
62	C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
63	C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
64	C₃²	#819	1	1	yes	characteristic, complemented
65	C₃²	#819	1	3	yes	complemented
66	C₂×C₄	C₂×#272	1	1	yes	characteristic
67	C₈	—	18	2	no	—
68	C₈	#1084	1	2	yes	—
69	C₂×C₃	—	2	1	no	—
70	C₂×C₃	—	3	1	no	—
71	C₂×C₃	—	6	1	no	—
72	C₂×C₃	—	6	3	no	—
73	C₂×C₃	—	6	3	no	—
74	C₂×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
75	C₂²	#6229	1	1	yes	characteristic
76	C₄	C₄⋊#272	1	1	yes	characteristic
77	C₄	C₂×#1084	1	1	yes	characteristic
78	C₃	—	3	1	no	—
79	C₃	—	6	3	no	—
80	C₃	#63646	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	C₄⋊#1084	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₄⋊(C₃⋊D₃)
commutator	C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₈×H(3,𝔽₃)	C₂
socle	C₂×C₃	C₄⋊(C₃⋊Dic₃)
prime radical	C₂²×H(3,𝔽₃)	Q₈
cosocle	C₂×C₄×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×C₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#108
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
39. H(3,𝔽₃)	#108	semidirect
64. C₃²	#819	semidirect
65. C₃²	#819	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×H(3,𝔽₃)	2
#88538	2
C₃×#819	3