Subgroups — #11721396

Order 864.

195 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂×AGL(1,ℤ/9))⋊C₄	C₂	1	3	yes	maximal
3	(3₋¹⁺²⋊C₄)⋊C₄	C₂	1	3	yes	maximal, complemented
4	C₃×#876	—	3	1	no	maximal
5	#63438	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
7	C₂×C₄×3₋¹⁺²	C₂²	1	3	yes	—
8	C₂×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
9	C₃²×C₄²	D₃	1	1	yes	characteristic
10	C₃×(C₂×D₃)⋊C₄	—	3	3	no	—
11	C₃×Dic₃⋊C₄	—	3	3	no	—
12	C₄²×C₉	—	2	1	no	—
13	C₄²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
14	(C₂×D₉)⋊C₄	C₂×C₃	1	3	yes	—
15	Dic₉⋊C₄	C₂×C₃	1	3	yes	complemented
16	C₂²×3₋¹⁺²	C₂³	1	1	yes	characteristic
17	C₂×AGL(1,ℤ/9)	—	2	3	no	—
18	C₄×3₋¹⁺²	—	2	3	no	—
19	3₋¹⁺²⋊C₄	—	2	3	no	—
20	C₃×#126	—	9	1	no	—
21	#876	—	3	1	no	—
22	C₂²×C₃×D₃	—	3	1	no	—
23	C₂×C₃²×C₄	C₂×D₃	1	3	yes	—
24	C₂×C₃×Dic₃	—	3	3	no	—
25	C₂²×D₉	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₄×C₉	—	2	3	no	—
27	C₂×C₄×C₉	C₂²×C₃	1	3	yes	—
28	C₂×Dic₉	C₂²×C₃	1	3	yes	—
29	C₂×3₋¹⁺²	#41	1	3	yes	—
30	AGL(1,ℤ/9)	—	4	1	no	—
31	C₃×C₄⋊C₄	—	9	3	no	—
32	C₃×C₂²⋊C₄	—	9	3	no	—
33	C₃×C₄²	—	6	1	no	—
34	C₃×C₄²	C₃×D₃	1	1	yes	characteristic
35	(C₂×D₃)⋊C₄	—	3	3	no	—
36	Dic₃⋊C₄	—	3	3	no	—
37	C₂²×C₃²	C₂²×D₃	1	1	yes	characteristic
38	C₂×C₃×D₃	—	6	3	no	—
39	C₃²×C₄	—	2	3	no	—
40	C₃×Dic₃	—	6	3	no	—
41	C₂²×C₉	—	2	1	no	—
42	C₂²×C₉	C₂³×C₃	1	1	yes	characteristic
43	C₂×D₉	—	2	3	no	—
44	C₄×C₉	—	2	6	no	—
45	C₄×C₉	—	2	3	no	—
46	Dic₉	—	2	3	no	—
47	#126	—	9	1	no	Sylow
48	3₋¹⁺²	#126	1	1	yes	characteristic, complemented, Sylow
49	C₂³×C₃	—	9	1	no	—
50	C₂²×D₃	—	3	1	no	—
51	C₂×C₃×C₄	—	6	3	no	—
52	C₂×C₃×C₄	—	9	3	no	—
53	C₂×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
54	C₂×Dic₃	—	3	3	no	—
55	C₂×C₃²	#235	1	3	yes	—
56	C₃×D₃	—	12	1	no	—
57	C₂×C₉	—	2	3	no	—
58	C₂×C₉	C₃×#41	1	3	yes	—
59	D₉	—	4	1	no	—
60	C₄⋊C₄	—	9	3	no	—
61	C₂²⋊C₄	—	9	3	no	—
62	C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
63	C₂²×C₃	—	6	1	no	—
64	C₂²×C₃	—	18	3	no	—
65	C₂²×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
66	C₂×D₃	—	6	3	no	—
67	C₃×C₄	—	2	3	no	—
68	C₃×C₄	—	6	3	no	—
69	C₃×C₄	—	18	3	no	—
70	Dic₃	—	6	3	no	—
71	C₃²	#876	1	1	yes	characteristic
72	C₉	—	2	1	no	—
73	C₉	C₃×#126	1	1	yes	characteristic, complemented
74	C₂³	—	9	1	no	—
75	C₂×C₄	—	9	3	no	—
76	C₂×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
77	C₂×C₃	—	6	3	no	—
78	C₂×C₃	—	36	1	no	—
79	C₂×C₃	C₃×#235	1	3	yes	—
80	D₃	—	12	1	no	—
81	C₂²	—	18	3	no	—
82	C₂²	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
83	C₄	—	2	3	no	—
84	C₄	—	18	3	no	—
85	C₃	—	6	1	no	—
86	C₃	C₃×#876	1	1	yes	characteristic
87	C₂	—	36	1	no	—
88	C₂	#3697⋊C₃	1	3	yes	—

	subgroup	G/H
center	C₂²	C₂²×AGL(1,ℤ/9)
commutator, cosocle	C₂²×C₉	C₂³×C₃
Frattini, socle	C₂²×C₃	C₂²×C₃×D₃
Fitting	C₄²×3₋¹⁺²	C₂
prime radical	C₂²×AGL(1,ℤ/9)	C₂²
hypercenter	C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#126
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄²×3₋¹⁺²	C₂	semidirect
3. (3₋¹⁺²⋊C₄)⋊C₄	C₂	semidirect
5. #63438	C₃	semidirect
13. C₄²×C₉	C₂×C₃	semidirect
15. Dic₉⋊C₄	C₂×C₃	semidirect
48. 3₋¹⁺²	#126	semidirect
62. C₄²	AGL(1,ℤ/9)	semidirect
73. C₉	C₃×#126	semidirect

Maximal subgroups

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

subgroup	index
C₄²×3₋¹⁺²	2
(C₂×AGL(1,ℤ/9))⋊C₄	2
(3₋¹⁺²⋊C₄)⋊C₄	2
C₃×#876	3
#63438	3