Group #13047466 — subgroups

Subgroups — #13047466

Order 1296.

320 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#732	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#4185	C₂	1	3	yes	maximal, complemented
3	C₄×#4185	C₂	1	3	yes	maximal, complemented
4	#88722	—	3	3	no	maximal
5	C₃×#3701	—	3	1	no	maximal
6	C₂×#4185	C₂²	1	3	yes	—
7	C₄×#732	C₂²	1	3	yes	—
8	#66332	C₂²	1	1	yes	characteristic
9	Q₈×H(3,𝔽₃)	D₃	1	3	yes	complemented
10	C₃×Q₈×C₉	D₃	1	1	yes	characteristic, complemented
11	C₄⋊((C₃⋊D₃)⋊C₃)	—	3	9	no	—
12	C₄×(C₃⋊D₃)⋊C₃	—	3	9	no	—
13	C₃×D₃₆	—	3	3	no	—
14	C₃×C₄×D₉	—	3	3	no	—
15	C₂×#732	C₂³	1	1	yes	characteristic
16	#4185	—	2	3	no	—
17	#3832	—	3	3	no	—
18	C₃×#239	—	9	1	no	—
19	#3701	—	3	1	no	—
20	C₂×(C₃⋊D₃)⋊C₃	—	3	9	no	—
21	C₂×C₃×D₉	—	3	3	no	—
22	C₄×H(3,𝔽₃)	C₂×D₃	1	9	yes	—
23	C₃×C₄×C₉	C₂×D₃	1	3	yes	—
24	(C₃⋊Dic₃)⋊C₃	—	3	3	no	—
25	C₃×Dic₉	—	3	1	no	—
26	#732	#41	1	1	yes	characteristic, complemented, Sylow
27	C₃²×Q₈	—	3	3	no	—
28	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
29	C₄⋊(C₃⋊D₃)	—	3	9	no	—
30	C₄×C₃⋊D₃	—	3	9	no	—
31	C₃×D₁₂	—	9	3	no	—
32	C₃×C₄×D₃	—	9	3	no	—
33	Q₈×C₉	—	3	1	no	—
34	D₃₆	—	3	3	no	—
35	C₄×D₉	—	3	3	no	—
36	C₂×H(3,𝔽₃)	C₂²×D₃	1	3	yes	—
37	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
38	(C₃⋊D₃)⋊C₃	—	6	9	no	—
39	C₃×D₉	—	6	3	no	—
40	C₃×#41	—	27	1	no	—
41	#239	—	9	1	no	—
42	#239	—	27	3	no	—
43	C₂×C₃⋊D₃	—	3	9	no	—
44	C₂×C₃×D₃	—	9	3	no	—
45	C₃²×C₄	—	3	9	no	—
46	C₃²×C₄	C₂×C₃⋊D₃	1	3	yes	—
47	C₃⋊Dic₃	—	3	3	no	—
48	C₃×Dic₃	—	9	1	no	—
49	C₂×D₉	—	3	3	no	—
50	C₄×C₉	—	3	3	no	—
51	Dic₉	—	3	1	no	—
52	H(3,𝔽₃)	#239	1	3	yes	complemented
53	C₃×C₉	#239	1	1	yes	characteristic, complemented
54	C₃×Q₈	—	3	1	no	—
55	C₃×Q₈	—	9	3	no	—
56	C₃×Q₈	#272	1	1	yes	characteristic
57	C₃×D₄	—	27	3	no	—
58	C₂×C₃×C₄	—	27	3	no	—
59	D₁₂	—	9	3	no	—
60	D₁₂	—	27	9	no	—
61	C₄×D₃	—	9	3	no	—
62	C₄×D₃	—	27	9	no	—
63	C₂×C₃²	—	3	3	no	—
64	C₂×C₃²	C₂²×C₃⋊D₃	1	1	yes	characteristic
65	C₃⋊D₃	—	6	9	no	—
66	C₃×D₃	—	18	3	no	—
67	C₂×C₉	—	3	1	no	—
68	D₉	—	6	3	no	—
69	#41	—	27	1	no	Sylow
70	C₂²×C₃	—	27	3	no	—
71	C₂×D₃	—	9	3	no	—
72	C₂×D₃	—	27	9	no	—
73	C₃×C₄	—	3	3	no	—
74	C₃×C₄	—	9	9	no	—
75	C₃×C₄	—	27	1	no	—
76	C₃×C₄	C₂×#272	1	3	yes	—
77	Dic₃	—	9	1	no	—
78	Dic₃	—	27	3	no	—
79	C₃²	—	3	3	no	—
80	C₃²	#3832	1	1	yes	characteristic
81	C₉	—	3	1	no	—
82	Q₈	#4185	1	1	yes	characteristic, complemented
83	D₄	—	27	3	no	—
84	C₂×C₄	—	27	3	no	—
85	C₂×C₃	—	3	1	no	—
86	C₂×C₃	—	9	3	no	—
87	C₂×C₃	—	54	3	no	—
88	C₂×C₃	C₂²×#272	1	1	yes	characteristic
89	D₃	—	18	3	no	—
90	D₃	—	54	9	no	—
91	C₂²	—	27	3	no	—
92	C₄	—	27	1	no	—
93	C₄	C₂×#4185	1	3	yes	—
94	C₃	—	3	1	no	—
95	C₃	—	9	3	no	—
96	C₃	#88748	1	1	yes	characteristic
97	C₂	—	54	3	no	—
98	C₂	C₂²×#4185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#732

C₂

yes

maximal, characteristic, complemented

C₄⋊#4185

C₂

yes

maximal, complemented

C₄×#4185

C₂

yes

maximal, complemented

—

—

C₂²

yes

—

C₄×#732

C₂²

yes

—

#66332

C₂²

yes

characteristic

Q₈×H(3,𝔽₃)

D₃

yes

complemented

C₃×Q₈×C₉

D₃

yes

characteristic, complemented

C₄⋊((C₃⋊D₃)⋊C₃)

—

C₄×(C₃⋊D₃)⋊C₃

—

C₃×D₃₆

—

C₃×C₄×D₉

—

C₂×#732

C₂³

yes

—

—

—

—

C₂×(C₃⋊D₃)⋊C₃

—

C₂×C₃×D₉

—

C₄×H(3,𝔽₃)

C₂×D₃

yes

—

C₃×C₄×C₉

C₂×D₃

yes

—

(C₃⋊Dic₃)⋊C₃

—

C₃×Dic₉

—

#732

#41

yes

characteristic, complemented, Sylow

C₃²×Q₈

—

C₃²×Q₈

C₃⋊D₃

yes

characteristic

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

—

C₃×D₁₂

—

C₃×C₄×D₃

—

Q₈×C₉

—

D₃₆

—

C₄×D₉

—

C₂×H(3,𝔽₃)

C₂²×D₃

yes

—

C₂×C₃×C₉

C₂²×D₃

yes

characteristic

(C₃⋊D₃)⋊C₃

—

C₃×D₉

—

C₃×#41

—

#239

—

#239

—

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

C₂×C₃⋊D₃

yes

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₂×D₉

—

C₄×C₉

—

Dic₉

—

yes

C₃×C₉

yes

characteristic, complemented

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

#272

yes

characteristic

C₃×D₄

—

C₂×C₃×C₄

—

D₁₂

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

—

C₂×C₃²

C₂²×C₃⋊D₃

yes

characteristic

C₃⋊D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

#41

—

Sylow

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×#272

yes

—

Dic₃

—

Dic₃

—

C₃²

—

C₃²

#3832

yes

characteristic

C₉

—

Q₈

#4185

yes

characteristic, complemented

D₄

—

C₂×C₄

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂²×#272

yes

characteristic

D₃

—

D₃

—

C₂²

—

C₄

—

C₄

C₂×#4185

yes

—

C₃

—

C₃

—

C₃

#88748

yes

characteristic

C₂

—

C₂

C₂²×#4185

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#4185
commutator, cosocle	C₂×#732	C₂³
Frattini	C₂×C₃²	C₂²×C₃⋊D₃
Fitting	Q₈×#732	C₂
socle	C₂×C₃	C₂²×#272
hypercenter	Q₈	#4185
hypocenter	#732	#41
LCS	G ▷ C₂×#732 ▷ #732
UCS	C₁ ◁ C₂ ◁ Q₈

subgroup

G/H

center

C₂

C₂²×#4185

commutator, cosocle

C₂×#732

C₂³

Frattini

C₂×C₃²

C₂²×C₃⋊D₃

Fitting

Q₈×#732

C₂

socle

C₂×C₃

C₂²×#272

Q₈

G ▷ C₂×#732 ▷ #732

C₁ ◁ C₂ ◁ Q₈

N	H ≅ G/N	type
1. Q₈×#732	C₂	semidirect
2. C₄⋊#4185	C₂	semidirect
3. C₄×#4185	C₂	semidirect
9. Q₈×H(3,𝔽₃)	D₃	semidirect
10. C₃×Q₈×C₉	D₃	semidirect
26. #732	#41	semidirect
52. H(3,𝔽₃)	#239	semidirect
53. C₃×C₉	#239	semidirect
82. Q₈	#4185	semidirect

H ≅ G/N

type

1. Q₈×#732

C₂

semidirect

2. C₄⋊#4185

C₂

semidirect

3. C₄×#4185

C₂

semidirect

9. Q₈×H(3,𝔽₃)

D₃

semidirect

10. C₃×Q₈×C₉

D₃

semidirect

26. #732

#41

semidirect

52. H(3,𝔽₃)

#239

semidirect

53. C₃×C₉

#239

semidirect

82. Q₈

#4185

semidirect

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

subgroup	index
Q₈×#732	2
C₄⋊#4185	2
C₄×#4185	2
#88722	3
C₃×#3701	3

subgroup

index

Subgroups — #13047466

Conjugacy classes of subgroups

Complements

Maximal subgroups