Subgroups — #13047781

Order 1296.

139 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊(C₄²⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#88455	C₃	1	2	yes	maximal, complemented
4	C₉⋊(C₄²⋊C₃)	C₃	1	6	yes	maximal, complemented
5	C₉×C₄²⋊C₃	C₃	1	3	yes	maximal, complemented
6	#66436	—	4	1	no	maximal
7	C₂×C₃×C₄×C₉	—	3	1	no	—
8	C₃²×C₄²	C₃²	1	1	yes	characteristic
9	C₃×C₄²⋊C₃	C₃²	1	3	yes	—
10	C₄²×C₉	C₃²	1	1	yes	characteristic
11	C₄²×C₉	C₃²	1	2	yes	—
12	C₄²⋊C₉	C₃²	1	6	yes	—
13	C₂²×C₃×C₉	A₄	1	1	yes	characteristic
14	C₂²⋊H(3,𝔽₃)	—	4	1	no	—
15	#1094	—	4	2	no	—
16	C₉⋊A₄	—	4	6	no	—
17	C₉×A₄	—	4	3	no	—
18	C₃×C₄×C₉	—	3	2	no	—
19	#737	—	16	1	no	Sylow
20	C₂×C₃²×C₄	—	3	1	no	—
21	C₂×C₄×C₉	—	3	1	no	—
22	C₂×C₄×C₉	—	3	2	no	—
23	C₂×C₃×C₉	—	3	1	no	—
24	C₃×C₄²	—	3	1	no	—
25	C₃×C₄²	C₃³	1	1	yes	characteristic
26	C₄²⋊C₃	—	3	3	no	—
27	C₂²×C₃²	C₃×A₄	1	1	yes	characteristic
28	C₃×A₄	—	4	3	no	—
29	C₃²×C₄	—	3	2	no	—
30	C₂²×C₉	C₃×A₄	1	1	yes	characteristic
31	C₂²×C₉	C₃×A₄	1	2	yes	—
32	C₂²⋊C₉	—	4	6	no	—
33	C₄×C₉	—	3	4	no	—
34	C₄×C₉	—	3	2	no	—
35	3₋¹⁺²	—	16	2	no	—
36	3₋¹⁺²	—	16	6	no	—
37	H(3,𝔽₃)	—	16	1	no	—
38	C₃×C₉	—	16	3	no	—
39	C₃×C₉	C₄²⋊C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×C₄	—	3	3	no	—
41	C₂×C₃×C₄	—	3	1	no	—
42	C₂×C₃²	—	3	1	no	—
43	C₂×C₉	—	3	2	no	—
44	C₂×C₉	—	3	1	no	—
45	C₄²	#737	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₃	—	3	1	no	—
47	C₂²×C₃	C₃²×A₄	1	1	yes	characteristic
48	A₄	—	12	3	no	—
49	C₃×C₄	—	3	2	no	—
50	C₃×C₄	—	3	6	no	—
51	C₃²	—	16	3	no	—
52	C₃²	C₃×C₄²⋊C₃	1	1	yes	characteristic
53	C₉	—	16	6	no	—
54	C₉	C₃×C₄²⋊C₃	1	2	yes	—
55	C₉	C₃×C₄²⋊C₃	1	1	yes	characteristic
56	C₂×C₄	—	3	1	no	—
57	C₂×C₃	—	3	3	no	—
58	C₂×C₃	—	3	1	no	—
59	C₂²	#66436	1	1	yes	characteristic
60	C₄	—	3	2	no	—
61	C₃	—	3	1	no	—
62	C₃	—	48	3	no	—
63	C₃	C₃²×C₄²⋊C₃	1	1	yes	characteristic
64	C₂	—	3	1	no	—

	subgroup	G/H
center	C₉	C₃×C₄²⋊C₃
commutator, cosocle	C₃×C₄²	C₃³
Frattini, socle	C₂²×C₃	C₃²×A₄
Fitting	C₃×C₄²×C₉	C₃
prime radical	C₃²⋊(C₄²⋊C₃)	C₃
hypercenter	C₃×C₉	C₄²⋊C₃
hypocenter	C₄²	#737
LCS	G ▷ C₃×C₄² ▷ C₄²
UCS	C₁ ◁ C₉ ◁ C₃×C₉

Complements

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

N	H ≅ G/N	type
1. C₃×C₄²×C₉	C₃	semidirect
3. #88455	C₃	semidirect
4. C₉⋊(C₄²⋊C₃)	C₃	semidirect
5. C₉×C₄²⋊C₃	C₃	semidirect
39. C₃×C₉	C₄²⋊C₃	semidirect
45. C₄²	#737	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄²×C₉	3
C₃²⋊(C₄²⋊C₃)	3
#88455	3
C₉⋊(C₄²⋊C₃)	3
C₉×C₄²⋊C₃	3
#66436	4