Subgroups — #13046764

Order 1296.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₉²	C₂	1	1	yes	maximal, characteristic
2	C₉⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
3	#88532	—	3	4	no	maximal
4	C₂²×C₉²	C₄	1	1	yes	characteristic
5	C₄×C₉²	C₄	1	1	yes	characteristic
6	C₄×C₉²	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₉	D₃	1	4	yes	—
8	C₃⋊(C₉⋊C₈)	—	3	8	no	—
9	C₂×C₉²	—	2	1	no	—
10	C₂×C₉²	C₂×C₄	1	1	yes	characteristic
11	#3748	—	9	1	no	—
12	#3667	—	9	12	no	—
13	C₂²×C₃×C₉	Dic₃	1	4	yes	—
14	C₃×C₄×C₉	Dic₃	1	4	yes	—
15	C₃×C₄×C₉	C₂×D₃	1	4	yes	—
16	C₉²	M₁₆	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃²×C₄	C₃⋊D₃	1	1	yes	characteristic
18	C₃⋊(C₃⋊C₈)	—	9	2	no	—
19	C₂×C₄×C₉	D₉	1	12	yes	—
20	C₉⋊C₈	—	9	24	no	—
21	C₂×C₃×C₉	—	2	4	no	—
22	C₂×C₃×C₉	C₂×Dic₃	1	4	yes	—
23	#208	—	27	4	no	—
24	C₂²×C₃²	C₃⋊Dic₃	1	1	yes	characteristic
25	C₃²×C₄	C₃⋊Dic₃	1	1	yes	characteristic
26	C₃²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
27	C₂²×C₉	Dic₉	1	12	yes	—
28	C₄×C₉	Dic₉	1	12	yes	—
29	C₄×C₉	C₂×D₉	1	12	yes	—
30	C₃×C₉	#208	1	4	yes	—
31	C₂×C₃×C₄	C₃⋊D₉	1	4	yes	—
32	C₃⋊C₈	—	27	8	no	—
33	C₂×C₃²	—	2	1	no	—
34	C₂×C₃²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
35	C₂×C₉	—	2	12	no	—
36	C₂×C₉	C₂×Dic₉	1	12	yes	—
37	M₁₆	—	81	1	no	Sylow
38	C₂²×C₃	C₃⋊Dic₉	1	4	yes	—
39	C₃×C₄	C₃⋊Dic₉	1	4	yes	—
40	C₃×C₄	C₂×C₃⋊D₉	1	4	yes	—
41	C₃²	#3748	1	1	yes	characteristic
42	C₉	#3667	1	12	yes	complemented
43	C₂×C₄	C₉⋊D₉	1	1	yes	characteristic
44	C₈	—	81	2	no	—
45	C₂×C₃	—	2	4	no	—
46	C₂×C₃	C₂×C₃⋊Dic₉	1	4	yes	—
47	C₂²	C₉⋊Dic₉	1	1	yes	characteristic
48	C₄	C₉⋊Dic₉	1	1	yes	characteristic
49	C₄	C₂×C₉⋊D₉	1	1	yes	characteristic
50	C₃	#88532	1	4	yes	—
51	C₂	—	2	1	no	—
52	C₂	C₂×C₉⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₉⋊D₉
commutator	C₂×C₉²	C₂×C₄
Frattini	C₃²×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₄×C₉²	C₂
socle	C₂×C₃²	C₂×C₃⋊Dic₃
prime radical	C₂²×C₃²	C₃⋊Dic₃
cosocle	C₄×C₉²	C₂²
hypercenter	C₂×C₄	C₉⋊D₉
hypocenter	C₉²	M₁₆
LCS	G ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₉⋊(C₉⋊C₈)	C₂	semidirect
16. C₉²	M₁₆	semidirect
42. C₉	#3667	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₉²	2
C₉⋊(C₉⋊C₈)	2
#88532	3