Subgroups — C₃×C₃⋊(#126⋊C₃) #11723051

Order 864.

132 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	C₃×#876	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#126⋊C₃	—	3	1	no	maximal
4	C₃⋊(#126⋊C₃)	C₃	1	3	yes	maximal, direct product factor
5	C₃×D₃×A₄	—	4	1	no	maximal
6	C₃²×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×(C₂×D₃)⋊C₄	—	3	1	no	—
8	C₃×Dic₃⋊C₄	—	3	1	no	—
9	C₃×C₄²⋊C₃	—	2	1	no	—
10	C₃×C₄²⋊C₃	—	2	3	no	—
11	C₃×C₄²⋊C₃	D₃	1	1	yes	characteristic, complemented
12	C₃×C₄²⋊C₃	C₂×C₃	1	3	yes	complemented
13	C₃²×A₄	—	4	1	no	—
14	C₃×#126	—	3	1	no	—
15	#876	C₃²	1	1	yes	characteristic, complemented
16	#126⋊C₃	—	3	3	no	—
17	C₂²×C₃×D₃	A₄	1	1	yes	characteristic
18	C₂×C₃×A₄	—	12	1	no	—
19	D₃×A₄	—	4	3	no	—
20	C₂×C₃²×C₄	—	3	1	no	—
21	C₂×C₃×Dic₃	—	3	1	no	—
22	C₃²×D₃	—	16	1	no	—
23	C₃×C₄⋊C₄	—	9	1	no	—
24	C₃×C₂²⋊C₄	—	9	1	no	—
25	C₃×C₄²	—	2	1	no	—
26	C₃×C₄²	C₃×D₃	1	1	yes	characteristic, complemented
27	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic, complemented
28	(C₂×D₃)⋊C₄	—	3	1	no	—
29	Dic₃⋊C₄	—	3	1	no	—
30	C₄²⋊C₃	—	2	3	no	—
31	C₄²⋊C₃	C₃×D₃	1	3	yes	complemented
32	C₂²×C₃²	C₂×A₄	1	1	yes	characteristic
33	C₂×C₃×D₃	—	6	1	no	—
34	C₃×A₄	—	4	1	no	—
35	C₃×A₄	—	4	3	no	—
36	C₃×A₄	—	8	3	no	—
37	C₃×A₄	—	8	1	no	—
38	C₃²×C₄	—	6	1	no	—
39	C₃×Dic₃	—	6	1	no	—
40	#126	—	3	1	no	Sylow
41	C₃³	—	16	1	no	Sylow
42	C₂³×C₃	—	3	1	no	—
43	C₂²×D₃	C₃×A₄	1	1	yes	characteristic
44	C₂×A₄	—	12	3	no	—
45	C₂×C₃×C₄	—	3	1	no	—
46	C₂×C₃×C₄	—	3	1	no	—
47	C₂×C₃×C₄	—	6	1	no	—
48	C₂×C₃×C₄	—	9	1	no	—
49	C₂×Dic₃	—	3	1	no	—
50	C₂×C₃²	—	3	1	no	—
51	C₂×C₃²	—	48	1	no	—
52	C₃×D₃	—	4	1	no	—
53	C₃×D₃	—	16	3	no	—
54	C₄⋊C₄	—	9	1	no	—
55	C₂²⋊C₄	—	9	1	no	—
56	C₄²	C₃²×D₃	1	1	yes	characteristic, complemented
57	C₂²×C₃	—	2	1	no	—
58	C₂²×C₃	—	18	1	no	—
59	C₂²×C₃	D₃×A₄	1	1	yes	characteristic
60	C₂²×C₃	C₂×C₃×A₄	1	1	yes	characteristic
61	C₂×D₃	—	6	1	no	—
62	A₄	—	4	3	no	—
63	A₄	—	8	3	no	—
64	C₃×C₄	—	6	1	no	—
65	C₃×C₄	—	6	1	no	—
66	C₃×C₄	—	6	2	no	—
67	C₃×C₄	—	18	1	no	—
68	Dic₃	—	6	1	no	—
69	C₃²	—	16	1	no	—
70	C₃²	—	16	3	no	—
71	C₃²	—	32	3	no	—
72	C₃²	—	32	1	no	—
73	C₃²	#126⋊C₃	1	1	yes	characteristic, complemented
74	C₂³	—	3	1	no	—
75	C₂×C₄	—	3	1	no	—
76	C₂×C₄	—	9	1	no	—
77	C₂×C₃	—	3	1	no	—
78	C₂×C₃	—	3	1	no	—
79	C₂×C₃	—	6	1	no	—
80	C₂×C₃	—	12	1	no	—
81	C₂×C₃	—	48	3	no	—
82	D₃	—	4	1	no	—
83	C₂²	—	18	1	no	—
84	C₂²	C₃×D₃×A₄	1	1	yes	characteristic
85	C₄	—	6	1	no	—
86	C₄	—	18	1	no	—
87	C₃	—	2	1	no	—
88	C₃	—	16	3	no	—
89	C₃	—	32	3	no	—
90	C₃	C₃⋊(#126⋊C₃)	1	1	yes	characteristic, direct product factor
91	C₃	C₃×#126⋊C₃	1	1	yes	characteristic, complemented
92	C₂	—	3	1	no	—
93	C₂	—	12	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(#126⋊C₃)
commutator, cosocle, hypocenter	C₃×C₄²	C₂×C₃²
Frattini	C₂²	C₃×D₃×A₄
Fitting	C₃²×C₄²	C₂×C₃
socle	C₂²×C₃²	C₂×A₄
LCS	G ▷ C₃×C₄²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃²×C₄²⋊C₃	C₂	semidirect
2. C₃×#876	C₃	semidirect
4. C₃⋊(#126⋊C₃)	C₃	direct (N × H)
6. C₃²×C₄²	C₂×C₃	semidirect
11. C₃×C₄²⋊C₃	D₃	semidirect
12. C₃×C₄²⋊C₃	C₂×C₃	semidirect
15. #876	C₃²	semidirect
26. C₃×C₄²	C₃×D₃	semidirect
27. C₃×C₄²	C₂×C₃²	semidirect
31. C₄²⋊C₃	C₃×D₃	semidirect
56. C₄²	C₃²×D₃	semidirect
73. C₃²	#126⋊C₃	semidirect
90. C₃	C₃⋊(#126⋊C₃)	direct (N × H)
91. C₃	C₃×#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
C₃×#876	3
C₃×#126⋊C₃	3
C₃⋊(#126⋊C₃)	3
C₃×D₃×A₄	4

Subgroups — C3×C3⋊(#126⋊C3) #11723051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊(#126⋊C₃) #11723051