Subgroups — #11721722

Order 864.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	#88542	C₂	1	3	yes	maximal
3	#88540	C₂	1	3	yes	maximal, complemented
4	C₃×#876	—	3	2	no	maximal
5	C₃×#876	—	3	2	no	maximal
6	C₂²×#272	C₂²	1	1	yes	characteristic
7	C₂×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
8	C₂×#1084	C₂²	1	3	yes	—
9	C₃²×C₄²	D₃	1	2	yes	complemented
10	C₃²×C₄²	D₃	1	2	yes	complemented
11	C₃×(C₂×D₃)⋊C₄	—	3	6	no	—
12	C₃×(C₂×D₃)⋊C₄	—	3	6	no	—
13	C₃×Dic₃⋊C₄	—	3	6	no	—
14	C₃×Dic₃⋊C₄	—	3	6	no	—
15	C₂²×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
16	C₂×#272	—	2	3	no	—
17	C₄×H(3,𝔽₃)	—	2	3	no	—
18	#1084	—	2	3	no	—
19	C₃×#126	—	9	1	no	—
20	#876	—	3	2	no	—
21	#876	—	3	2	no	—
22	C₂²×C₃×D₃	—	3	2	no	—
23	C₂²×C₃×D₃	—	3	2	no	—
24	C₂×C₃²×C₄	C₂×D₃	1	6	yes	—
25	C₂×C₃²×C₄	C₂×D₃	1	6	yes	—
26	C₂×C₃×Dic₃	—	3	6	no	—
27	C₂×C₃×Dic₃	—	3	6	no	—
28	C₂×H(3,𝔽₃)	#41	1	3	yes	—
29	#272	—	4	1	no	—
30	C₃×C₄⋊C₄	—	9	3	no	—
31	C₃×C₂²⋊C₄	—	9	3	no	—
32	C₃×C₄²	—	3	2	no	—
33	C₃×C₄²	—	6	2	no	—
34	C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
35	(C₂×D₃)⋊C₄	—	3	6	no	—
36	(C₂×D₃)⋊C₄	—	3	6	no	—
37	Dic₃⋊C₄	—	3	6	no	—
38	Dic₃⋊C₄	—	3	6	no	—
39	C₂²×C₃²	C₂²×D₃	1	2	yes	—
40	C₂²×C₃²	C₂²×D₃	1	2	yes	—
41	C₂×C₃×D₃	—	6	6	no	—
42	C₂×C₃×D₃	—	6	6	no	—
43	C₃²×C₄	—	2	6	no	—
44	C₃²×C₄	—	2	6	no	—
45	C₃×Dic₃	—	6	6	no	—
46	C₃×Dic₃	—	6	6	no	—
47	#126	—	9	1	no	Sylow
48	H(3,𝔽₃)	#126	1	1	yes	characteristic, complemented, Sylow
49	C₂³×C₃	—	9	1	no	—
50	C₂²×D₃	—	3	2	no	—
51	C₂²×D₃	—	3	2	no	—
52	C₂×C₃×C₄	—	3	6	no	—
53	C₂×C₃×C₄	—	6	6	no	—
54	C₂×C₃×C₄	—	9	3	no	—
55	C₂×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
56	C₂×Dic₃	—	3	6	no	—
57	C₂×Dic₃	—	3	6	no	—
58	C₂×C₃²	#235	1	6	yes	—
59	C₂×C₃²	#235	1	6	yes	—
60	C₃×D₃	—	12	2	no	—
61	C₃×D₃	—	12	2	no	—
62	C₄⋊C₄	—	9	3	no	—
63	C₂²⋊C₄	—	9	3	no	—
64	C₄²	#272	1	1	yes	characteristic, complemented
65	C₂²×C₃	—	3	2	no	—
66	C₂²×C₃	—	6	2	no	—
67	C₂²×C₃	—	18	3	no	—
68	C₂²×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
69	C₂×D₃	—	6	6	no	—
70	C₂×D₃	—	6	6	no	—
71	C₃×C₄	—	2	3	no	—
72	C₃×C₄	—	6	6	no	—
73	C₃×C₄	—	6	6	no	—
74	C₃×C₄	—	18	3	no	—
75	Dic₃	—	6	6	no	—
76	Dic₃	—	6	6	no	—
77	C₃²	#876	1	2	yes	complemented
78	C₃²	#876	1	2	yes	complemented
79	C₂³	—	9	1	no	—
80	C₂×C₄	—	9	3	no	—
81	C₂×C₄	C₂×#272	1	3	yes	—
82	C₂×C₃	—	3	6	no	—
83	C₂×C₃	—	6	6	no	—
84	C₂×C₃	—	36	1	no	—
85	C₂×C₃	#3828	1	3	yes	—
86	D₃	—	12	2	no	—
87	D₃	—	12	2	no	—
88	C₂²	—	18	3	no	—
89	C₂²	C₂²×#272	1	1	yes	characteristic
90	C₄	—	2	3	no	—
91	C₄	—	18	3	no	—
92	C₃	—	3	2	no	—
93	C₃	—	6	2	no	—
94	C₃	#64085	1	1	yes	characteristic
95	C₂	—	36	1	no	—
96	C₂	#88740	1	3	yes	—

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂²×H(3,𝔽₃)	C₂³
Fitting	C₄²×H(3,𝔽₃)	C₂
prime radical	C₂²×#272	C₂²
hypercenter	C₃×C₄²	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#126
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

Complements

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

N	H ≅ G/N	type
1. C₄²×H(3,𝔽₃)	C₂	semidirect
3. #88540	C₂	semidirect
9. C₃²×C₄²	D₃	semidirect
10. C₃²×C₄²	D₃	semidirect
48. H(3,𝔽₃)	#126	semidirect
64. C₄²	#272	semidirect
77. C₃²	#876	semidirect
78. C₃²	#876	semidirect

Maximal subgroups

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

subgroup	index
C₄²×H(3,𝔽₃)	2
#88542	2
#88540	2
C₃×#876	3