Subgroups — #14516347

Order 1944.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6900	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(3₋¹⁺²⋊C₈)	—	3	4	no	maximal
3	C₉⋊(C₉⋊C₈)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×#6900	C₄	1	1	yes	characteristic
5	C₃×C₄×3₋¹⁺²	D₃	1	4	yes	—
6	C₄×C₉⋊C₉	—	2	4	no	—
7	C₄×C₉²	C₂×C₃	1	1	yes	characteristic
8	#6900	C₈	1	1	yes	characteristic, complemented, Sylow
9	C₃×C₃⋊(C₃⋊C₈)	—	9	1	no	—
10	C₃⋊(C₉⋊C₈)	—	3	4	no	—
11	3₋¹⁺²⋊C₈	—	9	12	no	—
12	C₂×C₃×3₋¹⁺²	Dic₃	1	4	yes	—
13	C₂×C₉⋊C₉	—	2	4	no	—
14	C₂×C₉²	C₃×C₄	1	1	yes	characteristic
15	C₃³×C₄	C₃⋊D₃	1	1	yes	characteristic
16	C₄×3₋¹⁺²	—	3	12	no	—
17	C₃×C₄×C₉	—	2	4	no	—
18	C₃×C₄×C₉	C₃×D₃	1	4	yes	—
19	C₃×3₋¹⁺²	C₃⋊C₈	1	4	yes	—
20	C₉⋊C₉	—	2	4	no	—
21	C₉²	C₃×C₈	1	1	yes	characteristic, complemented
22	C₃⋊(C₃⋊C₈)	—	9	1	no	—
23	C₃×C₃⋊C₈	—	27	4	no	—
24	C₉⋊C₈	—	9	12	no	—
25	C₂×C₃³	C₃⋊Dic₃	1	1	yes	characteristic
26	C₂×3₋¹⁺²	—	3	12	no	—
27	C₂×C₃×C₉	—	2	4	no	—
28	C₂×C₃×C₉	C₃×Dic₃	1	4	yes	—
29	C₃²×C₄	—	3	4	no	—
30	C₃²×C₄	C₃×C₃⋊D₃	1	1	yes	characteristic
31	C₄×C₉	—	6	4	no	—
32	C₄×C₉	AGL(1,ℤ/9)	1	12	yes	—
33	C₃³	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
34	3₋¹⁺²	—	3	12	no	—
35	C₃×C₉	—	2	4	no	—
36	C₃×C₉	C₃×C₃⋊C₈	1	4	yes	—
37	C₃×C₈	—	81	1	no	—
38	C₃⋊C₈	—	27	4	no	—
39	C₂×C₃²	—	3	4	no	—
40	C₂×C₃²	C₃×C₃⋊Dic₃	1	1	yes	characteristic
41	C₂×C₉	—	6	4	no	—
42	C₂×C₉	3₋¹⁺²⋊C₄	1	12	yes	—
43	C₃×C₄	—	9	1	no	—
44	C₃×C₄	C₃⋊AGL(1,ℤ/9)	1	4	yes	—
45	C₃²	—	3	4	no	—
46	C₃²	C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
47	C₉	—	6	4	no	—
48	C₉	3₋¹⁺²⋊C₈	1	12	yes	complemented
49	C₈	—	81	1	no	Sylow
50	C₂×C₃	—	9	1	no	—
51	C₂×C₃	C₃⋊(3₋¹⁺²⋊C₄)	1	4	yes	—
52	C₄	#92314	1	1	yes	characteristic
53	C₃	—	9	1	no	—
54	C₃	C₃⋊(3₋¹⁺²⋊C₈)	1	4	yes	—
55	C₂	#11757618	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	#92314
commutator, hypocenter	C₉²	C₃×C₈
Frattini	C₃²×C₄	C₃×C₃⋊D₃
Fitting	C₄×#6900	C₂
socle	C₂×C₃²	C₃×C₃⋊Dic₃
prime radical	C₂×C₃³	C₃⋊Dic₃
cosocle	C₄×C₉²	C₂×C₃
LCS	G ▷ C₉²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
3. C₉⋊(C₉⋊C₈)	C₃	semidirect
8. #6900	C₈	semidirect
21. C₉²	C₃×C₈	semidirect
48. C₉	3₋¹⁺²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6900	2
C₃⋊(3₋¹⁺²⋊C₈)	3
C₉⋊(C₉⋊C₈)	3