Subgroups — #126⋊H(3,𝔽₃) #11721351

Order 864.

114 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×#126	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#126⋊C₃	C₃	1	3	yes	maximal, complemented
4	C₂×C₂²⋊H(3,𝔽₃)	—	4	1	no	maximal
5	C₃²×C₄⋊C₄	—	3	1	no	—
6	C₃²×C₂²⋊C₄	—	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×#126	—	3	1	no	—
11	C₃×#126	C₃²	1	1	yes	characteristic
12	#126⋊C₃	—	3	3	no	—
13	C₂³×C₃²	A₄	1	1	yes	characteristic
14	C₂×C₃×A₄	—	4	3	no	—
15	C₂×C₃²×C₄	—	3	1	no	—
16	C₂×C₃²×C₄	—	3	1	no	—
17	C₂×H(3,𝔽₃)	—	16	1	no	—
18	C₃×C₄⋊C₄	—	3	1	no	—
19	C₃×C₄⋊C₄	—	3	3	no	—
20	C₃×C₂²⋊C₄	—	3	3	no	—
21	C₃×C₂²⋊C₄	—	3	1	no	—
22	C₃×C₄²	—	3	1	no	—
23	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
24	C₄²⋊C₃	—	3	3	no	—
25	C₂²×C₃²	—	6	1	no	—
26	C₂²×C₃²	C₂×A₄	1	1	yes	characteristic
27	C₃×A₄	—	4	3	no	—
28	C₃²×C₄	—	6	1	no	—
29	C₃²×C₄	—	6	1	no	—
30	#126	H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
31	H(3,𝔽₃)	—	16	1	no	Sylow
32	C₂³×C₃	—	3	1	no	—
33	C₂³×C₃	C₃×A₄	1	1	yes	characteristic
34	C₂×A₄	—	12	3	no	—
35	C₂×C₃×C₄	—	3	3	no	—
36	C₂×C₃×C₄	—	3	3	no	—
37	C₂×C₃×C₄	—	3	1	no	—
38	C₂×C₃×C₄	—	3	1	no	—
39	C₂×C₃²	—	3	1	no	—
40	C₂×C₃²	—	4	1	no	—
41	C₂×C₃²	—	16	3	no	—
42	C₄⋊C₄	—	3	1	no	—
43	C₂²⋊C₄	—	3	1	no	—
44	C₄²	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
45	C₂²×C₃	—	3	1	no	—
46	C₂²×C₃	—	6	1	no	—
47	C₂²×C₃	—	6	3	no	—
48	C₂²×C₃	C₂×C₃×A₄	1	1	yes	characteristic
49	A₄	—	12	3	no	—
50	C₃×C₄	—	6	3	no	—
51	C₃×C₄	—	6	3	no	—
52	C₃×C₄	—	6	1	no	—
53	C₃×C₄	—	6	1	no	—
54	C₃²	—	16	3	no	—
55	C₃²	#126⋊C₃	1	1	yes	characteristic, complemented
56	C₂³	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
57	C₂×C₄	—	3	1	no	—
58	C₂×C₄	—	3	1	no	—
59	C₂×C₃	—	3	3	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	4	1	no	—
62	C₂×C₃	—	12	1	no	—
63	C₂×C₃	—	48	3	no	—
64	C₂²	—	6	1	no	—
65	C₂²	C₂×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
66	C₄	—	6	1	no	—
67	C₄	—	6	1	no	—
68	C₃	—	3	1	no	—
69	C₃	—	48	3	no	—
70	C₃	C₃×#126⋊C₃	1	1	yes	characteristic
71	C₂	—	3	1	no	—
72	C₂	—	4	1	no	—

	subgroup	G/H
center	C₃	C₃×#126⋊C₃
commutator, cosocle	C₃×C₄²	C₂×C₃²
Frattini, socle	C₂²×C₃	C₂×C₃×A₄
Fitting	C₃²×#126	C₃
hypercenter	C₃²	#126⋊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×#126	C₃	semidirect
3. C₃×#126⋊C₃	C₃	semidirect
7. C₃²×C₄²	C₂×C₃	semidirect
8. C₃×C₄²⋊C₃	C₂×C₃	semidirect
30. #126	H(3,𝔽₃)	semidirect
44. C₄²	C₂×H(3,𝔽₃)	semidirect
55. C₃²	#126⋊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×#126	3
C₃×#126⋊C₃	3
C₂×C₂²⋊H(3,𝔽₃)	4

Subgroups — #126⋊H(3,𝔽3) #11721351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #126⋊H(3,𝔽₃) #11721351