Group #21458673 — subgroups

Subgroups — Q₈×D₄₈₈ #21458673

Order 7808.

404 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 149 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×D₂₄₄	C₂	1	2	yes	maximal, complemented
2	#16840167	C₂	1	3	yes	maximal, complemented
3	#16840043	C₂	1	6	yes	maximal, complemented
4	C₄×D₄₈₈	C₂	1	3	yes	maximal
5	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₆₁	—	2	2	no	—
7	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
8	#14519777	C₂²	1	6	yes	—
9	#14519757	—	2	6	no	—
10	C₄×D₂₄₄	C₂²	1	6	yes	—
11	#14519711	C₂²	1	6	yes	—
12	#14519711	C₂²	1	3	yes	—
13	#14519699	C₂²	1	6	yes	—
14	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₁	C₂²	1	3	yes	—
16	C₄×C₈×C₆₁	C₂²	1	3	yes	—
17	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
18	Q₈×D₆₁	—	4	8	no	—
19	C₂×D₂₄₄	C₂³	1	2	yes	—
20	C₂×C₄×D₆₁	—	2	6	no	—
21	C₂×Dic₁₂₂	—	2	6	no	—
22	C₂×C₈×C₆₁	C₂³	1	3	yes	—
23	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₆₁	D₄	1	3	yes	—
25	C₄²×C₆₁	C₂³	1	3	yes	—
26	#11758198	—	2	6	no	—
27	C₄⋊Dic₆₁	—	2	6	no	—
28	C₄⋊Dic₆₁	C₂³	1	6	yes	—
29	D₄₈₈	Q₈	1	4	yes	direct product factor
30	C₂²×D₆₁	—	2	2	no	—
31	Q₈×C₆₁	D₈	1	4	yes	complemented
32	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₆₁	—	2	6	no	—
36	D₂₄₄	—	2	2	no	—
37	D₂₄₄	C₂×Q₈	1	4	yes	complemented
38	C₄×D₆₁	—	4	12	no	—
39	Dic₁₂₂	—	4	12	no	—
40	C₈×C₆₁	—	2	3	no	—
41	C₈×C₆₁	C₂×Q₈	1	2	yes	complemented
42	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
43	C₂×D₆₁	—	2	4	no	—
44	C₂×D₆₁	—	4	2	no	—
45	C₂×D₆₁	—	4	2	no	—
46	C₄×C₆₁	—	2	3	no	—
47	C₄×C₆₁	C₂×D₈	1	6	yes	—
48	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
49	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
50	Dic₆₁	—	4	6	no	—
51	Q₈×D₈	—	61	1	no	maximal, Sylow
52	C₂×C₆₁	D₄×Q₈	1	1	yes	characteristic
53	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
54	C₂×C₆₁	#578	1	1	yes	characteristic
55	D₆₁	—	4	4	no	—
56	D₄×Q₈	—	61	2	no	—
57	#500	—	61	3	no	—
58	#474	—	61	6	no	—
59	C₈×Q₈	D₆₁	1	1	yes	characteristic, complemented
60	C₄×D₈	—	61	3	no	—
61	C₆₁	Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
62	C₂²×Q₈	—	122	2	no	—
63	C₂×D₈	—	61	1	no	—
64	C₄⋊Q₈	—	61	6	no	—
65	C₂²⋊Q₈	—	122	6	no	—
66	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
67	C₄×D₄	—	61	6	no	—
68	#107	—	61	3	no	—
69	#107	—	61	6	no	—
70	C₄⋊C₈	C₂×D₆₁	1	3	yes	—
71	#102	—	61	6	no	—
72	C₄×C₈	C₂×D₆₁	1	3	yes	—
73	C₂×Q₈	—	122	6	no	—
74	C₂×Q₈	—	244	8	no	—
75	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
76	C₂×D₄	—	61	2	no	—
77	C₂²×C₄	—	122	6	no	—
78	D₈	—	61	4	no	—
79	C₂×C₈	C₂²×D₆₁	1	3	yes	—
80	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
81	C₄⋊C₄	—	61	6	no	—
82	C₄⋊C₄	—	122	6	no	—
83	C₄⋊C₄	D₂₄₄	1	3	yes	—
84	C₂²⋊C₄	—	122	6	no	—
85	C₄²	C₂²×D₆₁	1	3	yes	—
86	C₂³	—	122	2	no	—
87	Q₈	—	244	12	no	—
88	Q₈	D₄₈₈	1	4	yes	direct product factor
89	D₄	—	61	4	no	—
90	D₄	—	122	2	no	—
91	C₂×C₄	—	122	6	no	—
92	C₂×C₄	—	244	12	no	—
93	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
94	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
95	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
96	C₈	—	2	3	no	—
97	C₈	Q₈×D₆₁	1	2	yes	complemented
98	C₂²	—	122	4	no	—
99	C₂²	—	244	2	no	—
100	C₂²	—	244	2	no	—
101	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
102	C₄	—	2	3	no	—
103	C₄	—	244	6	no	—
104	C₄	C₂×D₄₈₈	1	6	yes	—
105	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
106	C₄	#14519824	1	1	yes	characteristic
107	C₂	—	244	4	no	—
108	C₂	Q₈×D₂₄₄	1	1	yes	characteristic
109	C₂	C₂²×D₄₈₈	1	1	yes	characteristic
110	C₂	#16840579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₄₄

C₂

yes

maximal, complemented

#16840167

C₂

yes

maximal, complemented

#16840043

C₂

yes

maximal, complemented

C₄×D₄₈₈

C₂

yes

maximal

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₁

—

C₂×D₄₈₈

C₂²

yes

characteristic

#14519777

C₂²

yes

—

#14519757

—

C₄×D₂₄₄

C₂²

yes

—

#14519711

C₂²

yes

—

#14519711

C₂²

yes

—

#14519699

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

Q₈×D₆₁

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

D₄₈₈

Q₈

yes

direct product factor

C₂²×D₆₁

—

Q₈×C₆₁

D₈

yes

complemented

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

C₂×Q₈

yes

complemented

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Q₈×D₈

—

maximal, Sylow

C₂×C₆₁

D₄×Q₈

yes

characteristic

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

#578

yes

characteristic

D₆₁

—

D₄×Q₈

—

#500

—

#474

—

C₈×Q₈

D₆₁

yes

characteristic, complemented

C₄×D₈

—

C₆₁

Q₈×D₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂×D₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

C₂×D₆₁

yes

—

#102

—

C₄×C₈

C₂×D₆₁

yes

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂×Q₈

D₂₄₄

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

122

—

D₈

—

C₂×C₈

C₂²×D₆₁

yes

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

D₂₄₄

yes

—

C₂²⋊C₄

—

122

—

C₄²

C₂²×D₆₁

yes

—

C₂³

—

122

—

Q₈

—

244

—

Q₈

D₄₈₈

yes

direct product factor

D₄

—

D₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂×D₂₄₄

yes

—

C₂×C₄

C₂×D₂₄₄

yes

—

C₂×C₄

C₂³×D₆₁

yes

characteristic

C₈

—

C₈

Q₈×D₆₁

yes

complemented

C₂²

—

122

—

C₂²

—

244

—

100

C₂²

—

244

—

101

C₂²

C₂²×D₂₄₄

yes

characteristic

102

C₄

—

103

C₄

—

244

—

104

C₄

C₂×D₄₈₈

yes

—

105

C₄

C₂×Q₈×D₆₁

yes

characteristic

106

C₄

#14519824

yes

characteristic

107

C₂

—

244

—

108

C₂

Q₈×D₂₄₄

yes

characteristic

109

C₂

C₂²×D₄₈₈

yes

characteristic

110

C₂

#16840579

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	Q₈×D₈
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

Q₈×D₈

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×D₂₄₄	C₂	semidirect
2. #16840167	C₂	semidirect
3. #16840043	C₂	semidirect
5. C₈×Q₈×C₆₁	C₂	semidirect
29. D₄₈₈	Q₈	direct (N × H)
31. Q₈×C₆₁	D₈	semidirect
37. D₂₄₄	C₂×Q₈	semidirect
41. C₈×C₆₁	C₂×Q₈	semidirect
59. C₈×Q₈	D₆₁	semidirect
61. C₆₁	Q₈×D₈	semidirect
88. Q₈	D₄₈₈	direct (N × H)
97. C₈	Q₈×D₆₁	semidirect

H ≅ G/N

type

1. Q₈×D₂₄₄

C₂

semidirect

2. #16840167

C₂

semidirect

3. #16840043

C₂

semidirect

5. C₈×Q₈×C₆₁

C₂

semidirect

29. D₄₈₈

Q₈

direct (N × H)

31. Q₈×C₆₁

D₈

semidirect

37. D₂₄₄

C₂×Q₈

semidirect

41. C₈×C₆₁

C₂×Q₈

semidirect

59. C₈×Q₈

D₆₁

semidirect

61. C₆₁

Q₈×D₈

semidirect

88. Q₈

D₄₈₈

direct (N × H)

97. C₈

Q₈×D₆₁

semidirect

Subgroups — Q₈×D₄₈₈ #21458673

Conjugacy classes of subgroups

Complements

Subgroups — Q8×D488 #21458673

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×D₄₈₈ #21458673