Subgroups — #127⋊H(3,𝔽₃) #11721354

Order 864.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²⋊(C₄²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#127	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#127⋊C₃	C₃	1	3	yes	maximal, complemented
4	C₂×C₂²⋊H(3,𝔽₃)	—	4	1	no	maximal
5	C₂×C₃²×D₄	—	3	1	no	—
6	C₂×C₃²×D₄	—	3	1	no	—
7	C₃²×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₄²⋊C₃	C₂×C₃	1	3	yes	complemented
9	C₂²⋊H(3,𝔽₃)	—	4	1	no	—
10	C₃×#127	—	3	1	no	—
11	C₃×#127	C₃²	1	1	yes	characteristic
12	#127⋊C₃	—	3	3	no	—
13	C₂³×C₃²	—	3	1	no	—
14	C₂³×C₃²	A₄	1	1	yes	characteristic
15	C₂×C₃×A₄	—	4	3	no	—
16	C₃²×D₄	—	6	2	no	—
17	C₃²×D₄	—	6	2	no	—
18	C₂×C₃²×C₄	—	3	1	no	—
19	C₂×H(3,𝔽₃)	—	16	1	no	—
20	C₂×C₃×D₄	—	3	1	no	—
21	C₂×C₃×D₄	—	3	3	no	—
22	C₂×C₃×D₄	—	3	3	no	—
23	C₂×C₃×D₄	—	3	1	no	—
24	C₃×C₄²	—	3	1	no	—
25	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
26	C₄²⋊C₃	—	3	3	no	—
27	C₂²×C₃²	—	6	1	no	—
28	C₂²×C₃²	—	6	1	no	—
29	C₂²×C₃²	—	6	2	no	—
30	C₂²×C₃²	C₂×A₄	1	1	yes	characteristic
31	C₃×A₄	—	4	3	no	—
32	C₃²×C₄	—	3	2	no	—
33	#127	H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
34	H(3,𝔽₃)	—	16	1	no	Sylow
35	C₂³×C₃	—	3	3	no	—
36	C₂³×C₃	—	3	1	no	—
37	C₂³×C₃	—	3	1	no	—
38	C₂³×C₃	C₃×A₄	1	1	yes	characteristic
39	C₂×A₄	—	12	3	no	—
40	C₃×D₄	—	6	2	no	—
41	C₃×D₄	—	6	2	no	—
42	C₃×D₄	—	6	6	no	—
43	C₃×D₄	—	6	6	no	—
44	C₂×C₃×C₄	—	3	3	no	—
45	C₂×C₃×C₄	—	3	1	no	—
46	C₂×C₃²	—	3	1	no	—
47	C₂×C₃²	—	4	1	no	—
48	C₂×C₃²	—	12	1	no	—
49	C₂×C₃²	—	16	3	no	—
50	C₂×D₄	—	3	1	no	—
51	C₂×D₄	—	3	1	no	—
52	C₄²	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
53	C₂²×C₃	—	3	1	no	—
54	C₂²×C₃	—	6	1	no	—
55	C₂²×C₃	—	6	3	no	—
56	C₂²×C₃	—	6	3	no	—
57	C₂²×C₃	—	6	6	no	—
58	C₂²×C₃	—	6	1	no	—
59	C₂²×C₃	—	6	2	no	—
60	C₂²×C₃	C₂×C₃×A₄	1	1	yes	characteristic
61	A₄	—	12	3	no	—
62	C₃×C₄	—	3	6	no	—
63	C₃×C₄	—	3	2	no	—
64	C₃²	—	16	3	no	—
65	C₃²	#127⋊C₃	1	1	yes	characteristic, complemented
66	C₂³	—	3	1	no	—
67	C₂³	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
68	D₄	—	6	2	no	—
69	D₄	—	6	2	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₃	—	3	3	no	—
72	C₂×C₃	—	3	1	no	—
73	C₂×C₃	—	4	1	no	—
74	C₂×C₃	—	12	1	no	—
75	C₂×C₃	—	12	3	no	—
76	C₂×C₃	—	12	1	no	—
77	C₂×C₃	—	48	3	no	—
78	C₂²	—	6	1	no	—
79	C₂²	—	6	1	no	—
80	C₂²	—	6	2	no	—
81	C₂²	C₂×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
82	C₄	—	3	2	no	—
83	C₃	—	3	1	no	—
84	C₃	—	48	3	no	—
85	C₃	C₃×#127⋊C₃	1	1	yes	characteristic
86	C₂	—	3	1	no	—
87	C₂	—	4	1	no	—
88	C₂	—	12	1	no	—

	subgroup	G/H
center	C₃	C₃×#127⋊C₃
commutator, cosocle	C₃×C₄²	C₂×C₃²
Frattini, socle	C₂²×C₃	C₂×C₃×A₄
Fitting	C₃²×#127	C₃
hypercenter	C₃²	#127⋊C₃
hypocenter	C₄²	C₂×H(3,𝔽₃)
LCS	G ▷ C₃×C₄² ▷ C₄²
UCS	C₁ ◁ C₃ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²⋊(C₄²⋊C₃)	C₂	semidirect
2. C₃²×#127	C₃	semidirect
3. C₃×#127⋊C₃	C₃	semidirect
7. C₃²×C₄²	C₂×C₃	semidirect
8. C₃×C₄²⋊C₃	C₂×C₃	semidirect
33. #127	H(3,𝔽₃)	semidirect
52. C₄²	C₂×H(3,𝔽₃)	semidirect
65. C₃²	#127⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²⋊(C₄²⋊C₃)	2
C₃²×#127	3
C₃×#127⋊C₃	3
C₂×C₂²⋊H(3,𝔽₃)	4

Subgroups — #127⋊H(3,𝔽3) #11721354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #127⋊H(3,𝔽₃) #11721354