Group #23161 — subgroups

Subgroups — #23161

Order 256.

397 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 183 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	C₂×#500	C₂	1	1	yes	maximal, characteristic
2	#2365	C₂	1	2	yes	maximal
3	#1988	C₂	1	2	yes	maximal
4	#1943	C₂	1	2	yes	maximal
5	#1921	C₂	1	4	yes	maximal
6	#1776	C₂	1	2	yes	maximal
7	C₄×#106	C₂	1	1	yes	maximal, characteristic
8	#1770	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈	—	2	2	no	—
11	#500	C₄	1	8	yes	complemented
12	C₂×#107	C₂²	1	2	yes	—
13	C₂×#107	C₂²	1	2	yes	—
14	C₂×#106	C₂²	1	1	yes	characteristic
15	C₂×#106	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	#391	—	2	2	no	—
20	#389	C₂²	1	4	yes	—
21	#387	—	2	4	no	—
22	C₄×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	#340	C₂²	1	2	yes	—
26	#340	C₂²	1	4	yes	—
27	#340	C₂²	1	4	yes	—
28	#336	C₂²	1	2	yes	—
29	C₂²×Q₈	—	2	2	no	—
30	C₂²×C₈	C₂³	1	2	yes	—
31	C₂²×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈	—	2	4	no	—
34	C₄⋊Q₈	C₂×C₄	1	8	yes	—
35	C₄×Q₈	—	4	8	no	—
36	C₂×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄⋊C₄	—	2	4	no	—
38	C₂×C₄⋊C₄	D₄	1	2	yes	—
39	C₂×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₄²	—	2	4	no	—
43	C₂×C₄²	—	2	2	no	—
44	C₂×C₄²	C₂³	1	2	yes	—
45	C₂×C₄²	C₂³	1	1	yes	characteristic
46	#107	C₂×C₄	1	8	yes	—
47	#107	C₂×C₄	1	8	yes	—
48	#106	—	2	2	no	—
49	#106	Q₈	1	4	yes	complemented
50	C₄⋊C₈	—	2	2	no	—
51	#97	—	2	4	no	—
52	C₄×C₈	C₂×C₄	1	4	yes	—
53	#95	—	2	2	no	—
54	#95	—	2	4	no	—
55	#95	—	2	4	no	—
56	#95	D₄	1	2	yes	—
57	C₂×Q₈	—	2	8	no	—
58	C₂×Q₈	—	4	4	no	—
59	C₂×Q₈	—	4	4	no	—
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	2	4	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×D₄	1	4	yes	—
64	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄	C₂⁴	1	1	yes	characteristic
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₈	C₂×Q₈	1	2	yes	—
75	C₂×C₈	#41	1	2	yes	—
76	C₄⋊C₄	—	2	8	no	—
77	C₄⋊C₄	—	2	4	no	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	4	8	no	—
83	C₄⋊C₄	C₂²×C₄	1	8	yes	—
84	C₄⋊C₄	C₂×Q₈	1	4	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	4	8	no	—
88	C₄²	—	4	4	no	—
89	C₄²	C₂²×C₄	1	2	yes	—
90	C₄²	#41	1	2	yes	—
91	C₂³	C₂²×D₄	1	1	yes	characteristic
92	Q₈	—	4	16	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	C₄×D₄	1	2	yes	—
107	C₂×C₄	C₄×D₄	1	2	yes	—
108	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
109	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄	C₂×#41	1	1	yes	characteristic
112	C₂×C₄	C₂×#41	1	1	yes	characteristic
113	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	—	4	2	no	—
117	C₈	C₄×Q₈	1	4	yes	complemented
118	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²	#547	1	1	yes	characteristic
120	C₂²	D₄×Q₈	1	1	yes	characteristic
121	C₂²	C₂×AGL(1,ℤ/8)	1	2	yes	—
122	C₂²	#578	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	4	4	no	—
126	C₄	—	4	8	no	—
127	C₄	—	8	2	no	—
128	C₄	#442	1	4	yes	—
129	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄	#519	1	1	yes	characteristic
131	C₄	#556	1	1	yes	characteristic
132	C₄	#557	1	1	yes	characteristic
133	C₂	#2365	1	1	yes	characteristic
134	C₂	C₂×#442	1	1	yes	characteristic
135	C₂	#2950	1	1	yes	characteristic
136	C₂	#3353	1	2	yes	—
137	C₂	#3386	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

#2365

C₂

yes

maximal

#1988

C₂

yes

maximal

#1943

C₂

yes

maximal

#1921

C₂

yes

maximal

#1776

C₂

yes

maximal

C₄×#106

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈

—

#500

C₄

yes

complemented

C₂×#107

C₂²

yes

—

C₂×#107

C₂²

yes

—

C₂×#106

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#391

—

#389

C₂²

yes

—

#387

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

characteristic

#340

C₂²

yes

—

#340

C₂²

yes

—

#340

C₂²

yes

—

#336

C₂²

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#107

C₂×C₄

yes

—

#107

C₂×C₄

yes

—

#106

—

#106

Q₈

yes

complemented

C₄⋊C₈

—

#97

—

C₄×C₈

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₄⋊C₄

C₂×Q₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

—

C₄²

#41

yes

—

C₂³

C₂²×D₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₄×D₄

yes

—

107

C₂×C₄

C₄×D₄

yes

—

108

C₂×C₄

AGL(1,ℤ/8)

yes

—

109

C₂×C₄

C₂³×C₄

yes

characteristic

110

C₂×C₄

C₂²×Q₈

yes

characteristic

111

C₂×C₄

C₂×#41

yes

characteristic

112

C₂×C₄

C₂×#41

yes

characteristic

113

C₂×C₄

2₋¹⁺⁴

yes

characteristic

114

C₂×C₄

2₋¹⁺⁴

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

C₄×Q₈

yes

complemented

118

C₂²

C₂×C₄×D₄

yes

characteristic

119

C₂²

#547

yes

characteristic

120

C₂²

D₄×Q₈

yes

characteristic

121

C₂²

C₂×AGL(1,ℤ/8)

yes

—

122

C₂²

#578

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#442

yes

—

129

C₄

C₂×C₄×Q₈

yes

characteristic

130

C₄

#519

yes

characteristic

131

C₄

#556

yes

characteristic

132

C₄

#557

yes

characteristic

133

C₂

#2365

yes

characteristic

134

C₂

C₂×#442

yes

characteristic

135

C₂

#2950

yes

characteristic

136

C₂

#3353

yes

—

137

C₂

#3386

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
11. #500	C₄	semidirect
49. #106	Q₈	semidirect
117. C₈	C₄×Q₈	semidirect

H ≅ G/N

type

11. #500

C₄

semidirect

49. #106

Q₈

semidirect

117. C₈

C₄×Q₈

semidirect

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

subgroup	index
C₂×#500	2
#2365	2
#1988	2
#1943	2
#1921	2
#1776	2
C₄×#106	2
#1770	2

subgroup

index

Subgroups — #23161

Conjugacy classes of subgroups

Complements

Maximal subgroups