Group #13046830 — subgroups

Subgroups — #13046830

Order 1296.

160 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 35 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	1	yes	maximal, characteristic
3	#10619511	C₂	1	1	yes	maximal, characteristic, complemented
4	#3755⋊C₃	—	3	3	no	maximal
5	C₃×#3675	—	3	1	no	maximal
6	C₂×#4185	—	2	1	no	—
7	C₄×#732	—	2	1	no	—
8	C₄×#732	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	3	no	—
12	C₃×D₃₆	—	3	1	no	—
13	(C₃⋊(C₃⋊C₈))⋊C₃	—	3	3	no	—
14	C₃×C₉⋊C₈	—	3	1	no	—
15	C₂×#732	D₄	1	1	yes	characteristic
16	#4185	—	4	1	no	—
17	#3755	—	3	3	no	—
18	C₃×#215	—	9	1	no	—
19	#3675	—	3	1	no	—
20	C₂×(C₃⋊D₃)⋊C₃	—	6	3	no	—
21	C₂×C₃×D₉	—	6	1	no	—
22	C₄×H(3,𝔽₃)	—	2	3	no	—
23	C₄×H(3,𝔽₃)	C₂×D₃	1	3	yes	—
24	C₃×C₄×C₉	—	2	1	no	—
25	C₃×C₄×C₉	C₂×D₃	1	1	yes	characteristic
26	#732	SD₁₆	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	3	no	—
30	C₃×D₁₂	—	9	1	no	—
31	C₃⋊(C₃⋊C₈)	—	3	3	no	—
32	C₃×C₃⋊C₈	—	9	1	no	—
33	Q₈×C₉	—	3	1	no	—
34	D₃₆	—	3	1	no	—
35	C₉⋊C₈	—	3	1	no	—
36	C₂×H(3,𝔽₃)	#67	1	3	yes	—
37	C₂×C₃×C₉	#67	1	1	yes	characteristic
38	(C₃⋊D₃)⋊C₃	—	12	3	no	—
39	C₃×D₉	—	12	1	no	—
40	C₃×SD₁₆	—	27	1	no	—
41	#215	—	9	1	no	—
42	#215	—	27	3	no	—
43	C₂×C₃⋊D₃	—	6	3	no	—
44	C₂×C₃×D₃	—	18	1	no	—
45	C₃²×C₄	—	2	1	no	—
46	C₃²×C₄	—	3	3	no	—
47	C₃²×C₄	—	6	3	no	—
48	C₃²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
49	C₂×D₉	—	6	1	no	—
50	C₄×C₉	—	3	1	no	—
51	C₄×C₉	—	6	1	no	—
52	H(3,𝔽₃)	#215	1	3	yes	complemented
53	C₃×C₉	#215	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	1	no	—
58	D₁₂	—	9	1	no	—
59	D₁₂	—	27	3	no	—
60	C₃×C₈	—	27	1	no	—
61	C₃⋊C₈	—	9	1	no	—
62	C₃⋊C₈	—	27	3	no	—
63	C₂×C₃²	—	3	3	no	—
64	C₂×C₃²	#638	1	1	yes	characteristic
65	C₃⋊D₃	—	12	3	no	—
66	C₃×D₃	—	36	1	no	—
67	C₂×C₉	—	3	1	no	—
68	D₉	—	12	1	no	—
69	SD₁₆	—	27	1	no	Sylow
70	C₂²×C₃	—	54	1	no	—
71	C₂×D₃	—	18	1	no	—
72	C₂×D₃	—	54	3	no	—
73	C₃×C₄	—	2	1	no	—
74	C₃×C₄	—	3	1	no	—
75	C₃×C₄	—	6	1	no	—
76	C₃×C₄	—	9	3	no	—
77	C₃×C₄	—	18	3	no	—
78	C₃×C₄	C₂×#272	1	1	yes	characteristic
79	C₃²	—	3	3	no	—
80	C₃²	#3755	1	1	yes	characteristic
81	C₉	—	3	1	no	—
82	Q₈	#4185	1	1	yes	characteristic, complemented
83	D₄	—	27	1	no	—
84	C₈	—	27	1	no	—
85	C₂×C₃	—	3	1	no	—
86	C₂×C₃	—	9	3	no	—
87	C₂×C₃	—	108	1	no	—
88	C₂×C₃	#6235	1	1	yes	characteristic
89	D₃	—	36	1	no	—
90	D₃	—	108	3	no	—
91	C₂²	—	54	1	no	—
92	C₄	—	2	1	no	—
93	C₄	C₂×#4185	1	1	yes	characteristic
94	C₃	—	3	1	no	—
95	C₃	—	9	3	no	—
96	C₃	#88549	1	1	yes	characteristic
97	C₂	—	108	1	no	—
98	C₂	#10619683	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#732

C₂

yes

maximal, characteristic, complemented

C₄⋊#4185

C₂

yes

maximal, characteristic

#10619511

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

Q₈×H(3,𝔽₃)

D₃

yes

complemented

C₃×Q₈×C₉

D₃

yes

characteristic, complemented

C₄⋊((C₃⋊D₃)⋊C₃)

—

C₃×D₃₆

—

(C₃⋊(C₃⋊C₈))⋊C₃

—

C₃×C₉⋊C₈

—

C₂×#732

D₄

yes

—

—

—

—

C₂×(C₃⋊D₃)⋊C₃

—

C₂×C₃×D₉

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×D₃

yes

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×D₃

yes

characteristic

#732

SD₁₆

yes

characteristic, complemented, Sylow

C₃²×Q₈

—

C₃²×Q₈

C₃⋊D₃

yes

characteristic

C₄⋊(C₃⋊D₃)

—

C₃×D₁₂

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

D₃₆

—

C₉⋊C₈

—

C₂×H(3,𝔽₃)

#67

yes

—

C₂×C₃×C₉

#67

yes

characteristic

(C₃⋊D₃)⋊C₃

—

C₃×D₉

—

C₃×SD₁₆

—

#215

—

#215

—

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×C₃⋊D₃

yes

characteristic

C₂×D₉

—

C₄×C₉

—

C₄×C₉

—

yes

C₃×C₉

yes

characteristic, complemented

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

#272

yes

characteristic

C₃×D₄

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#638

yes

characteristic

C₃⋊D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

SD₁₆

—

Sylow

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×#272

yes

characteristic

C₃²

—

C₃²

#3755

yes

characteristic

C₉

—

Q₈

#4185

yes

characteristic, complemented

D₄

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

108

—

C₂×C₃

#6235

yes

characteristic

D₃

—

D₃

—

108

—

C₂²

—

C₄

—

C₄

C₂×#4185

yes

characteristic

C₃

—

C₃

—

C₃

#88549

yes

characteristic

C₂

—

108

—

C₂

#10619683

yes

characteristic

	subgroup	G/H
center	C₂	#10619683
commutator, cosocle	C₄×#732	C₂²
Frattini	C₃²×C₄	C₂×C₃⋊D₃
Fitting	Q₈×#732	C₂
socle	C₂×C₃	#6235
prime radical	C₄⋊#4185	C₂
hypercenter	Q₈	#4185
hypocenter	#732	SD₁₆
LCS	G ▷ C₄×#732 ▷ C₂×#732 ▷ #732
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

subgroup

G/H

C₂

C₂²

C₃²×C₄

C₂×C₃⋊D₃

C₂

C₂×C₃

C₂

Q₈

SD₁₆

G ▷ C₄×#732 ▷ C₂×#732 ▷ #732

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₈

N	H ≅ G/N	type
1. Q₈×#732	C₂	semidirect
3. #10619511	C₂	semidirect
9. Q₈×H(3,𝔽₃)	D₃	semidirect
10. C₃×Q₈×C₉	D₃	semidirect
26. #732	SD₁₆	semidirect
52. H(3,𝔽₃)	#215	semidirect
53. C₃×C₉	#215	semidirect
82. Q₈	#4185	semidirect

H ≅ G/N

type

1. Q₈×#732

C₂

semidirect

3. #10619511

C₂

semidirect

9. Q₈×H(3,𝔽₃)

D₃

semidirect

10. C₃×Q₈×C₉

D₃

semidirect

26. #732

SD₁₆

semidirect

52. H(3,𝔽₃)

#215

semidirect

53. C₃×C₉

#215

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
#10619511	2
#3755⋊C₃	3
C₃×#3675	3

subgroup

index

Subgroups — #13046830

Conjugacy classes of subgroups

Complements

Maximal subgroups