Subgroups — #11757825

Order 972.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	#66436	C₃	1	3	yes	maximal, complemented
3	#66368	C₃	1	3	yes	maximal, complemented
4	#66359	C₃	1	2	yes	maximal, complemented
5	#66358	C₃	1	1	yes	maximal, characteristic, complemented
6	#66357	C₃	1	3	yes	maximal, complemented
7	#6908	—	4	1	no	maximal, Sylow
8	C₂×C₃²×C₉	—	3	1	no	—
9	C₂²×C₃³	C₃²	1	1	yes	characteristic
10	C₂²×C₃×C₉	—	3	1	no	—
11	C₂²×C₃×C₉	—	3	2	no	—
12	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
13	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
14	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
15	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
16	#1094	C₃²	1	6	yes	—
17	C₉⋊A₄	—	3	6	no	—
18	C₉×A₄	—	3	3	no	—
19	#737	—	4	3	no	—
20	C₃²×C₉	A₄	1	1	yes	characteristic, complemented
21	#733	—	4	2	no	—
22	#732	—	4	1	no	—
23	#731	—	4	3	no	—
24	AΣL(1,𝔽₂₇)	—	4	3	no	—
25	C₂×C₃³	—	3	1	no	—
26	C₂×C₃×C₉	—	3	6	no	—
27	C₂×C₃×C₉	—	3	3	no	—
28	C₂×C₃×C₉	—	3	1	no	—
29	C₂×C₃×C₉	—	3	1	no	—
30	C₂×C₃×C₉	—	3	1	no	—
31	C₂²×C₃²	—	3	1	no	—
32	C₂²×C₃²	—	3	3	no	—
33	C₂²×C₃²	C₃³	1	1	yes	characteristic
34	C₃×A₄	—	3	3	no	—
35	C₂²×C₉	—	3	1	no	—
36	C₂²×C₉	—	3	1	no	—
37	C₂²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
38	C₂²×C₉	H(3,𝔽₃)	1	2	yes	—
39	C₂²⋊C₉	—	3	6	no	—
40	C₃³	C₃×A₄	1	1	yes	characteristic
41	3₋¹⁺²	—	4	6	no	—
42	3₋¹⁺²	—	12	6	no	—
43	H(3,𝔽₃)	—	4	3	no	—
44	C₃×C₉	—	3	2	no	—
45	C₃×C₉	—	3	1	no	—
46	C₃×C₉	—	12	3	no	—
47	C₃×C₉	C₃×A₄	1	1	yes	characteristic
48	C₃×C₉	C₃×A₄	1	1	yes	characteristic
49	C₃×C₉	C₃×A₄	1	1	yes	characteristic
50	C₂×C₃²	—	3	3	no	—
51	C₂×C₃²	—	3	9	no	—
52	C₂×C₃²	—	3	1	no	—
53	C₂×C₉	—	3	3	no	—
54	C₂×C₉	—	3	3	no	—
55	C₂×C₉	—	3	2	no	—
56	C₂×C₉	—	3	1	no	—
57	C₂²×C₃	—	3	3	no	—
58	C₂²×C₃	—	3	1	no	—
59	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	A₄	—	9	3	no	—
61	C₃²	—	3	3	no	—
62	C₃²	—	3	1	no	—
63	C₃²	—	12	3	no	—
64	C₃²	C₃²×A₄	1	1	yes	characteristic
65	C₉	—	3	1	no	—
66	C₉	—	3	1	no	—
67	C₉	—	12	6	no	—
68	C₉	C₂²⋊H(3,𝔽₃)	1	2	yes	—
69	C₉	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
70	C₂×C₃	—	3	9	no	—
71	C₂×C₃	—	3	3	no	—
72	C₂×C₃	—	3	1	no	—
73	C₂²	#6908	1	1	yes	characteristic, complemented, Sylow
74	C₃	—	3	3	no	—
75	C₃	—	3	1	no	—
76	C₃	—	36	3	no	—
77	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
78	C₂	—	3	1	no	—

	subgroup	G/H
center	C₉	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃²×C₉	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
hypercenter	C₃²×C₉	A₄
hypocenter	C₂²	#6908
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₉	C₃	semidirect
2. #66436	C₃	semidirect
3. #66368	C₃	semidirect
4. #66359	C₃	semidirect
5. #66358	C₃	semidirect
6. #66357	C₃	semidirect
20. C₃²×C₉	A₄	semidirect
73. C₂²	#6908	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₉	3
#66436	3
#66368	3
#66359	3
#66358	3
#66357	3
#6908	4