Group #8682 — subgroups

Subgroups — #8682

Order 256.

215 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 81 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	#2931	C₂	1	1	yes	maximal, characteristic
2	#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	#1547	C₂	1	2	yes	maximal
5	#1292	C₂	1	2	yes	maximal, complemented
6	C₂×#131	—	2	1	no	—
7	#532	C₂²	1	1	yes	characteristic
8	#436	—	2	1	no	—
9	#436	C₄	1	2	yes	—
10	C₈×D₄	—	2	1	no	—
11	C₈×D₄	C₄	1	2	yes	—
12	C₂×#108	—	2	1	no	—
13	C₂×#108	—	2	1	no	—
14	#423	C₂²	1	2	yes	—
15	#423	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	2	yes	—
17	#408	—	2	1	no	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	#334	—	2	1	no	—
20	#334	—	2	1	no	—
21	#332	—	2	2	no	—
22	#331	—	2	2	no	—
23	C₂×#41	—	2	1	no	—
24	#131	—	4	2	no	—
25	#131	—	4	2	no	—
26	C₂×M₁₆	—	2	1	no	—
27	C₂×M₁₆	—	2	2	no	—
28	C₂×M₁₆	—	2	2	no	—
29	C₂²×C₈	—	2	2	no	—
30	C₂²×C₈	—	2	2	no	—
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₄⋊Q₈	D₄	1	1	yes	characteristic
34	#127	D₄	1	1	yes	characteristic
35	#124	—	2	1	no	—
36	#121	—	2	1	no	—
37	#121	Q₈	1	2	yes	—
38	C₄×D₄	—	2	1	no	—
39	C₄×D₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	#108	—	4	2	no	—
42	#108	—	4	2	no	—
43	C₄⋊C₈	—	2	1	no	—
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	—	2	2	no	—
46	C₄⋊C₈	—	2	2	no	—
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₂²⋊C₈	—	2	2	no	—
50	C₂²⋊C₈	—	2	2	no	—
51	C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈	D₄	1	1	yes	characteristic
54	C₄×C₈	D₄	1	1	yes	characteristic
55	#41	—	4	1	no	—
56	#41	—	4	1	no	—
57	#41	—	4	1	no	—
58	#41	—	4	1	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂×D₄	—	2	2	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄	—	2	2	no	—
65	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄	#41	1	1	yes	characteristic
68	M₁₆	—	4	1	no	—
69	M₁₆	—	4	1	no	—
70	M₁₆	—	4	4	no	—
71	M₁₆	—	4	2	no	—
72	M₁₆	—	4	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	8	2	no	—
80	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈	D₈	1	2	yes	—
85	C₂×C₈	SD₁₆	1	2	yes	—
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
88	C₂²⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₄²	C₂²×C₄	1	1	yes	characteristic
91	C₄²	C₂×D₄	1	1	yes	characteristic
92	C₄²	C₂×D₄	1	1	yes	characteristic
93	C₄²	#41	1	1	yes	characteristic
94	C₂³	—	2	2	no	—
95	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
96	Q₈	—	4	1	no	—
97	Q₈	—	4	1	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	1	no	—
102	D₄	—	4	1	no	—
103	D₄	—	4	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄	#117	1	1	yes	characteristic
112	C₂×C₄	#121	1	1	yes	characteristic
113	C₂×C₄	#121	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄	#137	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	2	1	no	—
121	C₈	—	4	2	no	—
122	C₈	—	8	2	no	—
123	C₈	—	8	2	no	—
124	C₈	#102	1	2	yes	—
125	C₈	#102	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	#380	1	1	yes	characteristic
132	C₂²	#474	1	1	yes	characteristic
133	C₂²	#476	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	2	no	—
138	C₄	C₂×#102	1	1	yes	characteristic
139	C₄	#417	1	1	yes	characteristic
140	C₄	#459	1	1	yes	characteristic
141	C₄	#461	1	1	yes	characteristic
142	C₄	#471	1	2	yes	—
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	8	2	no	—
146	C₂	#1802	1	1	yes	characteristic
147	C₂	#1813	1	1	yes	characteristic
148	C₂	#1815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2931

C₂

yes

maximal, characteristic

#1848

C₂

yes

maximal, characteristic, complemented

C₂×#334

C₂

yes

maximal, characteristic, complemented

#1547

C₂

yes

maximal

#1292

C₂

yes

maximal, complemented

C₂×#131

—

#532

C₂²

yes

characteristic

#436

—

#436

C₄

yes

—

C₈×D₄

—

C₈×D₄

C₄

yes

—

C₂×#108

—

C₂×#108

—

#423

C₂²

yes

—

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

#408

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

characteristic

#127

D₄

yes

—

—

Q₈

yes

—

C₄×D₄

—

C₄×D₄

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

—

C₂³

C₂²⋊Q₈

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

110

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

111

C₂×C₄

#117

yes

characteristic

112

C₂×C₄

#121

yes

characteristic

113

C₂×C₄

#121

yes

characteristic

114

C₂×C₄

C₂²⋊Q₈

yes

characteristic

115

C₂×C₄

C₂×D₈

yes

characteristic

116

C₂×C₄

C₂×SD₁₆

yes

characteristic

117

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

118

C₂×C₄

#137

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

#102

yes

—

125

C₈

#102

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#380

yes

characteristic

132

C₂²

#474

yes

characteristic

133

C₂²

#476

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₂×#102

yes

characteristic

139

C₄

#417

yes

characteristic

140

C₄

#459

yes

characteristic

141

C₄

#461

yes

characteristic

142

C₄

#471

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#1802

yes

characteristic

147

C₂

#1813

yes

characteristic

148

C₂

#1815

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#380
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#380

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂²

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1848	C₂	semidirect
3. C₂×#334	C₂	semidirect
5. #1292	C₂	semidirect

H ≅ G/N

type

2. #1848

C₂

semidirect

3. C₂×#334

C₂

semidirect

5. #1292

C₂

semidirect

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

subgroup	index
#2931	2
#1848	2
C₂×#334	2
#1547	2
#1292	2

subgroup

index

Subgroups — #8682

Conjugacy classes of subgroups

Complements

Maximal subgroups