Subgroups — #11757859

Order 972.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#1094	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₃×A₄)⋊C₉	C₃	1	9	yes	maximal, complemented
6	#6893	—	4	1	no	maximal, Sylow
7	C₂×C₃×3₋¹⁺²	—	3	1	no	—
8	C₂²×C₃³	C₃²	1	1	yes	characteristic
9	C₃²×A₄	C₃²	1	3	yes	—
10	C₂²×3₋¹⁺²	—	3	3	no	—
11	C₂²×C₃×C₉	C₃²	1	3	yes	—
12	C₂²⋊H(3,𝔽₃)	—	3	3	no	—
13	#1094	—	3	3	no	—
14	C₃×C₂²⋊C₉	—	3	3	no	—
15	C₃×C₂²⋊C₉	C₃²	1	3	yes	—
16	C₃×C₂²⋊C₉	C₃²	1	3	yes	—
17	C₃×3₋¹⁺²	—	4	1	no	—
18	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
19	C₃×H(3,𝔽₃)	—	4	1	no	—
20	C₃²×C₉	—	4	1	no	—
21	C₃²⋊C₉	—	4	9	no	—
22	C₂×C₃³	—	3	1	no	—
23	C₂×3₋¹⁺²	—	3	9	no	—
24	C₂×C₃×C₉	—	3	3	no	—
25	C₂²×C₃²	—	3	3	no	—
26	C₂²×C₃²	H(3,𝔽₃)	1	3	yes	—
27	C₂²×C₃²	C₃³	1	1	yes	characteristic
28	C₃×A₄	—	3	3	no	—
29	C₃×A₄	—	3	9	no	—
30	C₂²×C₉	—	3	3	no	—
31	C₂²⋊C₉	—	3	3	no	—
32	C₂²⋊C₉	—	3	3	no	—
33	C₃³	—	4	3	no	—
34	C₃³	C₃×A₄	1	1	yes	characteristic
35	3₋¹⁺²	—	3	3	no	—
36	3₋¹⁺²	—	12	3	no	—
37	H(3,𝔽₃)	—	12	3	no	—
38	C₃×C₉	—	4	3	no	—
39	C₃×C₉	—	4	3	no	—
40	C₃×C₉	—	12	3	no	—
41	C₃×C₉	C₃×A₄	1	3	yes	—
42	C₂×C₃²	—	3	1	no	—
43	C₂×C₃²	—	3	3	no	—
44	C₂×C₃²	—	9	3	no	—
45	C₂×C₉	—	3	9	no	—
46	C₂²×C₃	—	3	3	no	—
47	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₂²×C₃	#737	1	3	yes	—
49	A₄	—	9	3	no	—
50	C₃²	—	3	3	no	—
51	C₃²	—	12	9	no	—
52	C₃²	—	12	3	no	—
53	C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
54	C₃²	C₃²×A₄	1	1	yes	characteristic
55	C₉	—	3	3	no	—
56	C₉	—	12	3	no	—
57	C₉	—	12	3	no	—
58	C₂×C₃	—	3	3	no	—
59	C₂×C₃	—	3	1	no	—
60	C₂×C₃	—	9	3	no	—
61	C₂²	#6893	1	1	yes	characteristic, complemented, Sylow
62	C₃	—	3	3	no	—
63	C₃	—	36	3	no	—
64	C₃	#66440	1	3	yes	—
65	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
66	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting	C₂²×C₃×3₋¹⁺²	C₃
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	#6893
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×3₋¹⁺²	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. C₃²×C₂²⋊C₉	C₃	semidirect
5. (C₃×A₄)⋊C₉	C₃	semidirect
18. C₃×3₋¹⁺²	A₄	semidirect
61. C₂²	#6893	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×3₋¹⁺²	3
C₃×C₂²⋊H(3,𝔽₃)	3
C₃×#1094	3
C₃²×C₂²⋊C₉	3
(C₃×A₄)⋊C₉	3
#6893	4