Group #19661 — subgroups

Subgroups — #19661

Order 256.

495 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 139 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	#3205	C₂	1	4	yes	maximal, complemented
2	#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	#1721	C₂	1	4	yes	maximal, complemented
5	#1712	C₂	1	2	yes	maximal, complemented
6	#1709	C₂	1	2	yes	maximal, complemented
7	#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	—	2	2	no	—
9	C₂²×D₈	—	2	2	no	—
10	#532	C₂²	1	2	yes	—
11	C₂×#127	C₂²	1	2	yes	—
12	#495	—	2	2	no	—
13	#494	C₂²	1	1	yes	characteristic, complemented
14	#493	—	2	2	no	—
15	#493	C₂²	1	2	yes	—
16	#493	C₂²	1	2	yes	—
17	#493	C₂²	1	1	yes	characteristic, complemented
18	#493	C₂²	1	8	yes	complemented
19	#492	C₂²	1	2	yes	complemented
20	#486	C₂²	1	8	yes	complemented
21	#459	—	2	8	no	—
22	#447	—	2	8	no	—
23	#431	C₂²	1	2	yes	—
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	#322	C₂²	1	2	yes	complemented
26	C₈²	C₂²	1	2	yes	complemented
27	C₂×#41	—	2	2	no	—
28	C₂²×D₄	—	2	2	no	—
29	C₂²×D₄	—	2	4	no	—
30	#135	—	4	8	no	—
31	C₂×Q₁₆	—	2	2	no	—
32	C₂×SD₁₆	—	2	4	no	—
33	C₂×D₈	—	2	4	no	—
34	C₂×D₈	—	2	4	no	—
35	C₂×D₈	—	2	2	no	—
36	C₂×D₈	—	2	16	no	—
37	C₂×D₈	—	4	8	no	—
38	C₂²×C₈	D₄	1	2	yes	—
39	C₄⋊Q₈	C₂³	1	2	yes	—
40	#127	—	2	4	no	—
41	#127	—	2	2	no	—
42	#127	C₂³	1	4	yes	—
43	#127	C₂³	1	2	yes	—
44	#124	—	2	2	no	—
45	#121	—	2	4	no	—
46	C₄×D₄	—	2	4	no	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	C₄⋊C₈	D₄	1	4	yes	—
49	#102	—	2	16	no	—
50	C₂²⋊C₈	D₄	1	4	yes	—
51	C₄×C₈	D₄	1	2	yes	—
52	C₄×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₈	C₂³	1	4	yes	—
54	C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂⁴	—	4	2	no	—
56	#41	—	4	8	no	—
57	C₂×Q₈	—	2	2	no	—
58	C₂×D₄	—	2	8	no	—
59	C₂×D₄	—	2	4	no	—
60	C₂×D₄	—	2	2	no	—
61	C₂×D₄	—	2	4	no	—
62	C₂×D₄	—	4	2	no	—
63	C₂×D₄	—	4	4	no	—
64	C₂×D₄	—	4	2	no	—
65	C₂×D₄	—	4	8	no	—
66	C₂×D₄	—	8	8	no	—
67	C₂²×C₄	—	4	2	no	—
68	C₂²×C₄	C₂×D₄	1	2	yes	—
69	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
70	Q₁₆	—	4	4	no	—
71	SD₁₆	—	4	8	no	—
72	D₈	—	4	8	no	—
73	D₈	—	4	8	no	—
74	D₈	—	4	4	no	—
75	D₈	—	8	16	no	—
76	C₂×C₈	D₈	1	8	yes	complemented
77	C₂×C₈	C₂×D₄	1	8	yes	—
78	C₂×C₈	C₂×D₄	1	2	yes	—
79	C₂×C₈	C₂×D₄	1	2	yes	—
80	C₄⋊C₄	—	2	4	no	—
81	C₂²⋊C₄	—	4	4	no	—
82	C₄²	C₂×D₄	1	1	yes	characteristic
83	C₄²	C₂×D₄	1	2	yes	—
84	C₄²	C₂⁴	1	1	yes	characteristic
85	C₂³	—	4	2	no	—
86	C₂³	—	4	4	no	—
87	C₂³	—	8	8	no	—
88	C₂³	—	8	4	no	—
89	C₂³	#127	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	4	8	no	—
92	D₄	—	4	4	no	—
93	D₄	—	4	8	no	—
94	D₄	—	4	16	no	—
95	D₄	—	8	8	no	—
96	D₄	—	8	4	no	—
97	D₄	—	8	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	8	4	no	—
101	C₂×C₄	#127	1	2	yes	—
102	C₂×C₄	#127	1	1	yes	characteristic
103	C₂×C₄	C₂×D₈	1	4	yes	—
104	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄	C₂²×D₄	1	2	yes	—
106	C₈	—	2	8	no	—
107	C₈	C₂×D₈	1	4	yes	—
108	C₈	C₂×D₈	1	4	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	—	8	2	no	—
111	C₂²	—	8	4	no	—
112	C₂²	—	8	8	no	—
113	C₂²	—	8	4	no	—
114	C₂²	—	16	8	no	—
115	C₂²	#493	1	2	yes	—
116	C₂²	C₂×#127	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	8	2	no	—
119	C₄	#493	1	2	yes	—
120	C₄	C₂²×D₈	1	2	yes	—
121	C₄	#576	1	4	yes	—
122	C₂	—	2	2	no	—
123	C₂	—	16	2	no	—
124	C₂	—	16	4	no	—
125	C₂	C₂×#493	1	1	yes	characteristic
126	C₂	#3155	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3205

C₂

yes

maximal, complemented

#3147

C₂

yes

maximal, characteristic, complemented

C₂×#493

C₂

yes

maximal, characteristic, complemented

#1721

C₂

yes

maximal, complemented

#1712

C₂

yes

maximal, complemented

#1709

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

C₂²×D₈

—

#532

C₂²

yes

—

C₂×#127

C₂²

yes

—

#495

—

#494

C₂²

yes

characteristic, complemented

#493

—

#493

C₂²

yes

—

#493

C₂²

yes

—

#493

C₂²

yes

characteristic, complemented

#493

C₂²

yes

complemented

#492

C₂²

yes

complemented

#486

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

complemented

C₈²

C₂²

yes

complemented

C₂×#41

—

C₂²×D₄

—

C₂²×D₄

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂³

yes

—

#127

—

#127

—

#127

C₂³

yes

—

#127

C₂³

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

D₄

yes

—

#102

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

C₂×C₈

D₈

yes

complemented

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#127

yes

—

102

C₂×C₄

#127

yes

characteristic

103

C₂×C₄

C₂×D₈

yes

—

104

C₂×C₄

C₂²×D₄

yes

characteristic

105

C₂×C₄

C₂²×D₄

yes

—

106

C₈

—

107

C₈

C₂×D₈

yes

—

108

C₈

C₂×D₈

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#493

yes

—

116

C₂²

C₂×#127

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#493

yes

—

120

C₄

C₂²×D₈

yes

—

121

C₄

#576

yes

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

C₂×#493

yes

characteristic

126

C₂

#3155

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3205	C₂	semidirect
2. #3147	C₂	semidirect
3. C₂×#493	C₂	semidirect
4. #1721	C₂	semidirect
5. #1712	C₂	semidirect
6. #1709	C₂	semidirect
7. #1450	C₂	semidirect
13. #494	C₂²	semidirect
17. #493	C₂²	semidirect
18. #493	C₂²	semidirect
19. #492	C₂²	semidirect
20. #486	C₂²	semidirect
25. #322	C₂²	semidirect
26. C₈²	C₂²	semidirect
76. C₂×C₈	D₈	semidirect

H ≅ G/N

type

1. #3205

C₂

semidirect

2. #3147

C₂

semidirect

3. C₂×#493

C₂

semidirect

4. #1721

C₂

semidirect

5. #1712

C₂

semidirect

6. #1709

C₂

semidirect

7. #1450

C₂

semidirect

13. #494

C₂²

semidirect

17. #493

C₂²

semidirect

18. #493

C₂²

semidirect

19. #492

C₂²

semidirect

20. #486

C₂²

semidirect

25. #322

C₂²

semidirect

26. C₈²

C₂²

semidirect

76. C₂×C₈

D₈

semidirect

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

subgroup	index
#3205	2
#3147	2
C₂×#493	2
#1721	2
#1712	2
#1709	2
#1450	2

subgroup

index

Subgroups — #19661

Conjugacy classes of subgroups

Complements

Maximal subgroups