Subgroups — #13046820

Order 1296.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₉²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊(C₉⋊D₉)	C₂	1	1	yes	maximal, characteristic
3	C₉⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#88548	—	3	4	no	maximal
5	C₂×C₉⋊D₉	—	2	1	no	—
6	C₄×C₉²	—	2	1	no	—
7	C₄×C₉²	C₂²	1	1	yes	characteristic
8	C₃×Q₈×C₉	D₃	1	4	yes	—
9	C₄⋊(C₃⋊D₉)	—	3	4	no	—
10	C₃⋊(C₉⋊C₈)	—	3	4	no	—
11	C₂×C₉²	D₄	1	1	yes	characteristic
12	C₉⋊D₉	—	4	1	no	—
13	#3755	—	9	1	no	—
14	#3675	—	9	12	no	—
15	C₂×C₃⋊D₉	—	6	4	no	—
16	C₃×C₄×C₉	—	2	4	no	—
17	C₃×C₄×C₉	C₂×D₃	1	4	yes	—
18	C₉²	SD₁₆	1	1	yes	characteristic, complemented, Sylow
19	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
20	C₄⋊(C₃⋊D₃)	—	9	1	no	—
21	C₃⋊(C₃⋊C₈)	—	9	1	no	—
22	Q₈×C₉	D₉	1	12	yes	complemented
23	D₃₆	—	9	12	no	—
24	C₉⋊C₈	—	9	12	no	—
25	C₂×C₃×C₉	#67	1	4	yes	—
26	C₃⋊D₉	—	12	4	no	—
27	#215	—	27	4	no	—
28	C₂×C₃⋊D₃	—	18	1	no	—
29	C₃²×C₄	—	2	1	no	—
30	C₃²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
31	C₂×D₉	—	18	12	no	—
32	C₄×C₉	—	2	12	no	—
33	C₄×C₉	C₂×D₉	1	12	yes	—
34	C₃×C₉	#215	1	4	yes	—
35	C₃×Q₈	C₃⋊D₉	1	4	yes	—
36	D₁₂	—	27	4	no	—
37	C₃⋊C₈	—	27	4	no	—
38	C₂×C₃²	#638	1	1	yes	characteristic
39	C₃⋊D₃	—	36	1	no	—
40	C₂×C₉	#611	1	12	yes	—
41	D₉	—	36	12	no	—
42	SD₁₆	—	81	1	no	Sylow
43	C₂×D₃	—	54	4	no	—
44	C₃×C₄	—	2	4	no	—
45	C₃×C₄	C₂×C₃⋊D₉	1	4	yes	—
46	C₃²	#3755	1	1	yes	characteristic
47	C₉	#3675	1	12	yes	complemented
48	Q₈	C₉⋊D₉	1	1	yes	characteristic, complemented
49	D₄	—	81	1	no	—
50	C₈	—	81	1	no	—
51	C₂×C₃	#6233	1	4	yes	—
52	D₃	—	108	4	no	—
53	C₂²	—	162	1	no	—
54	C₄	—	2	1	no	—
55	C₄	C₂×C₉⋊D₉	1	1	yes	characteristic
56	C₃	#88548	1	4	yes	—
57	C₂	—	324	1	no	—
58	C₂	#10619673	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10619673
commutator, cosocle	C₄×C₉²	C₂²
Frattini	C₃²×C₄	C₂×C₃⋊D₃
Fitting	Q₈×C₉²	C₂
socle	C₂×C₃²	#638
prime radical	C₄⋊(C₉⋊D₉)	C₂
hypercenter	Q₈	C₉⋊D₉
hypocenter	C₉²	SD₁₆
LCS	G ▷ C₄×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×C₉²	C₂	semidirect
3. C₉⋊(C₉⋊C₈)	C₂	semidirect
18. C₉²	SD₁₆	semidirect
22. Q₈×C₉	D₉	semidirect
47. C₉	#3675	semidirect
48. Q₈	C₉⋊D₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₉²	2
C₄⋊(C₉⋊D₉)	2
C₉⋊(C₉⋊C₈)	2
#88548	3