Subgroups — #11757454

Order 972.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
4	#6856	—	4	1	no	maximal, Sylow
5	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
6	C₂²×C₃³	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₃²×A₄	—	3	1	no	—
9	C₂²×H(3,𝔽₃)	—	3	3	no	—
10	C₂²⋊H(3,𝔽₃)	—	3	3	no	—
11	C₃×C₂²⋊C₉	—	3	2	no	—
12	C₃×H(3,𝔽₃)	—	4	1	no	—
13	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic, complemented
14	C₃²⋊C₉	—	4	2	no	—
15	C₂×C₃³	—	3	3	no	—
16	C₂×C₃³	—	3	1	no	—
17	C₂×H(3,𝔽₃)	—	3	9	no	—
18	C₂²×C₃²	—	3	1	no	—
19	C₂²×C₃²	—	3	3	no	—
20	C₂²×C₃²	—	3	1	no	—
21	C₂²×C₃²	—	3	2	no	—
22	C₂²×C₃²	—	9	1	no	—
23	C₂²×C₃²	—	9	2	no	—
24	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×A₄	—	3	3	no	—
26	C₃×A₄	—	9	1	no	—
27	C₃×A₄	—	9	2	no	—
28	C₂²⋊C₉	—	9	2	no	—
29	C₃³	—	3	1	no	—
30	C₃³	—	12	1	no	—
31	C₃³	C₃×A₄	1	1	yes	characteristic
32	H(3,𝔽₃)	—	3	3	no	—
33	H(3,𝔽₃)	—	12	3	no	—
34	C₃×C₉	—	12	2	no	—
35	C₂×C₃²	—	3	1	no	—
36	C₂×C₃²	—	3	3	no	—
37	C₂×C₃²	—	3	9	no	—
38	C₂×C₃²	—	9	2	no	—
39	C₂×C₃²	—	9	1	no	—
40	C₂×C₃²	—	9	6	no	—
41	C₂×C₃²	—	9	3	no	—
42	C₂²×C₃	—	9	1	no	—
43	C₂²×C₃	—	9	3	no	—
44	C₂²×C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
45	C₂²×C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
46	C₂²×C₃	#732	1	2	yes	—
47	A₄	—	9	3	no	—
48	C₃²	—	3	2	no	—
49	C₃²	—	3	1	no	—
50	C₃²	—	3	1	no	—
51	C₃²	—	3	3	no	—
52	C₃²	—	9	2	no	—
53	C₃²	—	9	1	no	—
54	C₃²	—	12	3	no	—
55	C₃²	—	36	2	no	—
56	C₃²	—	36	1	no	—
57	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
58	C₉	—	36	2	no	—
59	C₂×C₃	—	3	2	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	3	1	no	—
62	C₂×C₃	—	9	3	no	—
63	C₂×C₃	—	9	9	no	—
64	C₂²	#6856	1	1	yes	characteristic, complemented, Sylow
65	C₃	—	9	1	no	—
66	C₃	—	9	3	no	—
67	C₃	—	36	3	no	—
68	C₃	#66362	1	2	yes	—
69	C₃	#66363	1	1	yes	characteristic
70	C₃	#66368	1	1	yes	characteristic
71	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃³	C₃²
Frattini	C₃³	C₃×A₄
Fitting	C₂²×C₃×H(3,𝔽₃)	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	#6856
LCS	G ▷ C₂²×C₃³ ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×H(3,𝔽₃)	C₃	semidirect
2. C₃×C₂²⋊H(3,𝔽₃)	C₃	semidirect
3. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
13. C₃×H(3,𝔽₃)	A₄	semidirect
64. C₂²	#6856	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×H(3,𝔽₃)	3
C₃×C₂²⋊H(3,𝔽₃)	3
C₃²⋊(C₂²⋊C₉)	3
#6856	4