Subgroups — #11757413

Order 972.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
2	C₂²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
4	#6856	—	4	1	no	maximal, Sylow
5	C₂×C₃²⋊C₉	—	3	1	no	—
6	C₂²×C₃³	C₃²	1	1	yes	characteristic
7	C₃²×A₄	—	3	2	no	—
8	C₂²×C₃×C₉	—	3	1	no	—
9	C₂²⋊H(3,𝔽₃)	—	3	6	no	—
10	C₃×C₂²⋊C₉	—	3	1	no	—
11	C₃×H(3,𝔽₃)	—	4	2	no	—
12	C₃²⋊C₉	—	4	1	no	—
13	C₃²⋊C₉	A₄	1	1	yes	characteristic, complemented
14	C₂×C₃³	—	3	1	no	—
15	C₂×C₃×C₉	—	3	3	no	—
16	C₂²×C₃²	—	3	2	no	—
17	C₂²×C₃²	—	3	1	no	—
18	C₂²×C₃²	—	3	1	no	—
19	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃×A₄	—	3	6	no	—
21	C₃×A₄	—	9	2	no	—
22	C₃×A₄	—	9	2	no	—
23	C₃×A₄	—	9	2	no	—
24	C₂²×C₉	—	9	1	no	—
25	C₂²⋊C₉	—	9	1	no	—
26	C₃³	—	12	2	no	—
27	C₃³	C₃×A₄	1	1	yes	characteristic
28	H(3,𝔽₃)	—	12	6	no	—
29	C₃×C₉	—	3	1	no	—
30	C₃×C₉	—	12	1	no	—
31	C₂×C₃²	—	3	1	no	—
32	C₂×C₃²	—	3	3	no	—
33	C₂×C₃²	—	9	1	no	—
34	C₂×C₃²	—	9	2	no	—
35	C₂×C₉	—	9	3	no	—
36	C₂²×C₃	—	9	1	no	—
37	C₂²×C₃	AΣL(1,𝔽₂₇)	1	2	yes	—
38	C₂²×C₃	#732	1	1	yes	characteristic
39	C₂²×C₃	#732	1	1	yes	characteristic
40	A₄	—	9	6	no	—
41	C₃²	—	3	1	no	—
42	C₃²	—	3	1	no	—
43	C₃²	—	3	2	no	—
44	C₃²	—	12	6	no	—
45	C₃²	—	36	2	no	—
46	C₃²	—	36	2	no	—
47	C₃²	—	36	2	no	—
48	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₉	—	9	1	no	—
50	C₉	—	36	1	no	—
51	C₂×C₃	—	3	1	no	—
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	3	2	no	—
54	C₂×C₃	—	9	3	no	—
55	C₂²	#6856	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	9	1	no	—
57	C₃	—	36	6	no	—
58	C₃	#66358	1	1	yes	characteristic
59	C₃	#66362	1	1	yes	characteristic
60	C₃	#66366	1	2	yes	—
61	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃³	C₃²
Frattini	C₃³	C₃×A₄
Fitting	C₂²×C₃²⋊C₉	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
hypercenter	C₃²⋊C₉	A₄
hypocenter	C₂²	#6856
LCS	G ▷ C₂²×C₃³ ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

Complements

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

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

Maximal subgroups

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

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