Subgroups — #13047456

Order 1296.

460 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	3	yes	maximal, complemented
3	C₄×C₉⋊D₉	C₂	1	3	yes	maximal, complemented
4	#88746	—	3	4	no	maximal
5	C₂×C₉⋊D₉	C₂²	1	3	yes	—
6	C₄×C₉²	C₂²	1	3	yes	—
7	C₉⋊Dic₉	C₂²	1	1	yes	characteristic
8	C₃×Q₈×C₉	D₃	1	4	yes	—
9	C₄⋊(C₃⋊D₉)	—	3	12	no	—
10	C₄×C₃⋊D₉	—	3	12	no	—
11	C₂×C₉²	C₂³	1	1	yes	characteristic
12	C₉⋊D₉	—	2	3	no	—
13	#3832	—	9	1	no	—
14	#3701	—	9	12	no	—
15	C₂×C₃⋊D₉	—	3	12	no	—
16	C₃×C₄×C₉	C₂×D₃	1	12	yes	—
17	C₃⋊Dic₉	—	3	4	no	—
18	C₉²	#41	1	1	yes	characteristic, complemented, Sylow
19	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
20	C₄⋊(C₃⋊D₃)	—	9	3	no	—
21	C₄×C₃⋊D₃	—	9	3	no	—
22	Q₈×C₉	D₉	1	12	yes	complemented
23	D₃₆	—	9	36	no	—
24	C₄×D₉	—	9	36	no	—
25	C₂×C₃×C₉	C₂²×D₃	1	4	yes	—
26	C₃⋊D₉	—	6	12	no	—
27	#239	—	27	4	no	—
28	C₂×C₃⋊D₃	—	9	3	no	—
29	C₃²×C₄	C₂×C₃⋊D₃	1	3	yes	—
30	C₃⋊Dic₃	—	9	1	no	—
31	C₂×D₉	—	9	36	no	—
32	C₄×C₉	C₂×D₉	1	36	yes	—
33	Dic₉	—	9	12	no	—
34	C₃×C₉	#239	1	4	yes	—
35	C₃×Q₈	C₃⋊D₉	1	4	yes	—
36	D₁₂	—	27	12	no	—
37	C₄×D₃	—	27	12	no	—
38	C₂×C₃²	C₂²×C₃⋊D₃	1	1	yes	characteristic
39	C₃⋊D₃	—	18	3	no	—
40	C₂×C₉	C₂²×D₉	1	12	yes	—
41	D₉	—	18	36	no	—
42	#41	—	81	1	no	Sylow
43	C₂×D₃	—	27	12	no	—
44	C₃×C₄	C₂×C₃⋊D₉	1	12	yes	—
45	Dic₃	—	27	4	no	—
46	C₃²	#3832	1	1	yes	characteristic
47	C₉	#3701	1	12	yes	complemented
48	Q₈	C₉⋊D₉	1	1	yes	characteristic, complemented
49	D₄	—	81	3	no	—
50	C₂×C₄	—	81	3	no	—
51	C₂×C₃	C₂²×C₃⋊D₉	1	4	yes	—
52	D₃	—	54	12	no	—
53	C₂²	—	81	3	no	—
54	C₄	—	81	1	no	—
55	C₄	C₂×C₉⋊D₉	1	3	yes	—
56	C₃	#88746	1	4	yes	—
57	C₂	—	162	3	no	—
58	C₂	C₂²×C₉⋊D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×C₉⋊D₉
commutator, cosocle	C₂×C₉²	C₂³
Frattini, socle	C₂×C₃²	C₂²×C₃⋊D₃
Fitting	Q₈×C₉²	C₂
hypercenter	Q₈	C₉⋊D₉
hypocenter	C₉²	#41
LCS	G ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×C₉²	C₂	semidirect
2. C₄⋊(C₉⋊D₉)	C₂	semidirect
3. C₄×C₉⋊D₉	C₂	semidirect
18. C₉²	#41	semidirect
22. Q₈×C₉	D₉	semidirect
47. C₉	#3701	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
C₄×C₉⋊D₉	2
#88746	3