Subgroups — C₃×#66366 #11757858

Order 972.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×A₄	C₃	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#1094	C₃	1	1	yes	maximal, characteristic, complemented
5	#66366	C₃	1	9	yes	maximal, direct product factor
6	C₃×AΣL(1,𝔽₂₇)	—	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	—	3	2	no	—
10	C₃²×A₄	—	3	1	no	—
11	C₃²×A₄	—	3	9	no	—
12	C₃²×A₄	—	3	1	no	—
13	C₃²×A₄	C₃²	1	3	yes	complemented
14	C₂²×3₋¹⁺²	—	3	2	no	—
15	C₂²×3₋¹⁺²	C₃²	1	3	yes	complemented
16	C₂²×C₃×C₉	—	3	1	no	—
17	C₂²⋊H(3,𝔽₃)	—	3	2	no	—
18	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	complemented
19	#1094	—	3	2	no	—
20	#1094	C₃²	1	3	yes	complemented
21	C₃×C₂²⋊C₉	—	3	1	no	—
22	C₃⁴	—	4	1	no	—
23	C₃×3₋¹⁺²	—	4	1	no	—
24	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
25	C₃×H(3,𝔽₃)	—	4	1	no	—
26	AΣL(1,𝔽₂₇)	—	4	9	no	—
27	C₂×C₃³	—	3	1	no	—
28	C₂×3₋¹⁺²	—	3	6	no	—
29	C₂×3₋¹⁺²	—	3	3	no	—
30	C₂×C₃×C₉	—	3	3	no	—
31	C₂²×C₃²	—	3	3	no	—
32	C₂²×C₃²	H(3,𝔽₃)	1	2	yes	—
33	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
34	C₂²×C₃²	C₃³	1	1	yes	characteristic
35	C₃×A₄	—	3	18	no	—
36	C₃×A₄	—	3	9	no	—
37	C₃×A₄	—	3	3	no	—
38	C₃×A₄	—	3	9	no	—
39	C₃×A₄	—	3	3	no	—
40	C₃×A₄	—	9	3	no	—
41	C₂²×C₉	—	3	3	no	—
42	C₂²⋊C₉	—	3	3	no	—
43	C₃³	—	4	3	no	—
44	C₃³	—	12	9	no	—
45	C₃³	—	12	2	no	—
46	C₃³	—	12	1	no	—
47	C₃³	—	12	1	no	—
48	C₃³	C₃×A₄	1	1	yes	characteristic
49	3₋¹⁺²	—	3	2	no	—
50	3₋¹⁺²	—	4	3	no	—
51	3₋¹⁺²	—	12	2	no	—
52	3₋¹⁺²	C₃×A₄	1	3	yes	complemented
53	H(3,𝔽₃)	—	4	3	no	—
54	H(3,𝔽₃)	—	12	2	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	12	1	no	—
57	C₂×C₃²	—	3	1	no	—
58	C₂×C₃²	—	3	2	no	—
59	C₂×C₃²	—	3	1	no	—
60	C₂×C₃²	—	9	3	no	—
61	C₂×C₉	—	3	9	no	—
62	C₂²×C₃	—	3	2	no	—
63	C₂²×C₃	—	3	1	no	—
64	C₂²×C₃	AΣL(1,𝔽₂₇)	1	3	yes	complemented
65	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	A₄	—	3	9	no	—
67	A₄	—	9	3	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	12	9	no	—
70	C₃²	—	12	18	no	—
71	C₃²	—	12	9	no	—
72	C₃²	—	12	3	no	—
73	C₃²	—	12	3	no	—
74	C₃²	—	36	3	no	—
75	C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
76	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₃²	C₃²×A₄	1	1	yes	characteristic
78	C₉	—	3	3	no	—
79	C₉	—	12	3	no	—
80	C₂×C₃	—	3	3	no	—
81	C₂×C₃	—	3	1	no	—
82	C₂×C₃	—	9	2	no	—
83	C₂×C₃	—	9	1	no	—
84	C₂²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic, complemented, Sylow
85	C₃	—	3	2	no	—
86	C₃	—	3	1	no	—
87	C₃	—	12	9	no	—
88	C₃	—	36	3	no	—
89	C₃	#66366	1	3	yes	direct product factor
90	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
91	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	C₃×AΣL(1,𝔽₂₇)
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃³×A₄	C₃	semidirect
2. C₂²×C₃×3₋¹⁺²	C₃	semidirect
3. C₃×C₂²⋊H(3,𝔽₃)	C₃	semidirect
4. C₃×#1094	C₃	semidirect
5. #66366	C₃	direct (N × H)
13. C₃²×A₄	C₃²	semidirect
15. C₂²×3₋¹⁺²	C₃²	semidirect
18. C₂²⋊H(3,𝔽₃)	C₃²	semidirect
20. #1094	C₃²	semidirect
24. C₃×3₋¹⁺²	A₄	semidirect
52. 3₋¹⁺²	C₃×A₄	semidirect
64. C₂²×C₃	AΣL(1,𝔽₂₇)	semidirect
84. C₂²	C₃×AΣL(1,𝔽₂₇)	semidirect
89. C₃	#66366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×A₄	3
C₂²×C₃×3₋¹⁺²	3
C₃×C₂²⋊H(3,𝔽₃)	3
C₃×#1094	3
#66366	3
C₃×AΣL(1,𝔽₂₇)	4

Subgroups — C3×#66366 #11757858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66366 #11757858