Subgroups — #11757811

Order 972.

139 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66440	C₃	1	3	yes	maximal, complemented
2	C₃×C₉⋊A₄	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×#733	C₃	1	1	yes	maximal, characteristic, complemented
4	#66366	C₃	1	3	yes	maximal, complemented
5	#66365	C₃	1	3	yes	maximal, complemented
6	#66359	C₃	1	1	yes	maximal, characteristic, complemented
7	#66358	C₃	1	1	yes	maximal, characteristic, complemented
8	#6913	—	4	1	no	maximal, Sylow
9	C₂×#733	—	3	1	no	—
10	C₃²×A₄	C₃²	1	1	yes	characteristic
11	C₂²×3₋¹⁺²	C₃²	1	3	yes	—
12	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
13	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
14	#1094	C₃²	1	3	yes	—
15	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
16	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
17	C₉⋊A₄	—	3	3	no	—
18	C₉⋊A₄	—	3	6	no	—
19	C₉×A₄	—	3	3	no	—
20	#737	—	4	3	no	—
21	C₃×3₋¹⁺²	—	4	1	no	—
22	#733	—	4	1	no	—
23	#733	A₄	1	1	yes	characteristic, complemented
24	#732	—	4	1	no	—
25	#731	—	4	3	no	—
26	AΣL(1,𝔽₂₇)	—	4	3	no	—
27	C₂×3₋¹⁺²	—	3	3	no	—
28	C₂×C₃×C₉	—	3	1	no	—
29	C₂²×C₃²	C₃³	1	1	yes	characteristic
30	C₃×A₄	—	3	3	no	—
31	C₃×A₄	—	3	1	no	—
32	C₃×A₄	—	9	1	no	—
33	C₂²×C₉	—	3	3	no	—
34	C₂²×C₉	—	3	1	no	—
35	C₂²⋊C₉	—	3	3	no	—
36	C₂²⋊C₉	—	3	1	no	—
37	C₂²⋊C₉	H(3,𝔽₃)	1	3	yes	—
38	C₃³	—	4	1	no	—
39	3₋¹⁺²	—	4	3	no	—
40	3₋¹⁺²	—	12	3	no	—
41	3₋¹⁺²	—	12	6	no	—
42	3₋¹⁺²	C₃×A₄	1	3	yes	—
43	H(3,𝔽₃)	—	4	3	no	—
44	C₃×C₉	—	4	1	no	—
45	C₃×C₉	—	4	1	no	—
46	C₃×C₉	—	12	3	no	—
47	C₃×C₉	C₃×A₄	1	1	yes	characteristic
48	C₂×C₃²	—	3	1	no	—
49	C₂×C₉	—	9	1	no	—
50	C₂×C₉	—	9	3	no	—
51	C₂²×C₃	—	3	1	no	—
52	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	A₄	—	9	3	no	—
54	A₄	—	9	1	no	—
55	C₃²	—	12	3	no	—
56	C₃²	—	12	1	no	—
57	C₃²	—	36	1	no	—
58	C₃²	C₃²×A₄	1	1	yes	characteristic
59	C₉	—	3	3	no	—
60	C₉	—	3	1	no	—
61	C₉	—	4	3	no	—
62	C₉	—	12	1	no	—
63	C₉	—	12	3	no	—
64	C₂×C₃	—	3	1	no	—
65	C₂×C₃	—	9	1	no	—
66	C₂²	#6913	1	1	yes	characteristic, complemented, Sylow
67	C₃	—	3	1	no	—
68	C₃	—	36	1	no	—
69	C₃	—	36	3	no	—
70	C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
71	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	A₄×H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×#733	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
hypercenter	#733	A₄
hypocenter	C₂²	#6913
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

Complements

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

N	H ≅ G/N	type
1. #66440	C₃	semidirect
2. C₃×C₉⋊A₄	C₃	semidirect
3. C₂²×#733	C₃	semidirect
4. #66366	C₃	semidirect
5. #66365	C₃	semidirect
6. #66359	C₃	semidirect
7. #66358	C₃	semidirect
23. #733	A₄	semidirect
66. C₂²	#6913	semidirect

Maximal subgroups

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

subgroup	index
#66440	3
C₃×C₉⋊A₄	3
C₂²×#733	3
#66366	3
#66365	3
#66359	3
#66358	3
#6913	4