Group #8205 — subgroups

Subgroups — #8205

Order 256.

241 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 111 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#1770	C₂	1	1	yes	maximal, characteristic
3	C₄×#103	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#322	C₂	1	1	yes	maximal, characteristic
6	#1279	C₂	1	1	yes	maximal, characteristic
7	#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
9	C₈×Q₈	—	2	2	no	—
10	C₈×Q₈	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈	—	2	1	no	—
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#103	C₄	1	2	yes	—
15	C₂×#97	—	2	1	no	—
16	C₂×C₄×C₈	—	2	1	no	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#340	C₄	1	2	yes	—
22	#336	—	2	1	no	—
23	Q₈⋊C₈	C₄	1	4	yes	complemented
24	#322	—	2	2	no	—
25	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	—	2	1	no	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	—	2	1	no	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	C₄×Q₈	—	2	1	no	—
33	C₄×Q₈	—	2	1	no	—
34	C₄×Q₈	C₂×C₄	1	2	yes	—
35	C₄×Q₈	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²	—	2	1	no	—
40	C₂×C₄²	—	2	1	no	—
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₄⋊C₈	—	2	4	no	—
43	C₄⋊C₈	—	2	1	no	—
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₄⋊C₈	C₂×C₄	1	2	yes	—
47	#103	—	2	4	no	—
48	#97	—	4	2	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	C₂×C₄	1	2	yes	—
54	C₄×C₈	C₂×C₄	1	2	yes	—
55	C₄×C₈	D₄	1	2	yes	—
56	#95	—	2	1	no	—
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	—	2	2	no	—
59	C₂×Q₈	C₄²	1	2	yes	—
60	C₂×Q₈	M₁₆	1	4	yes	—
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	—	2	2	no	—
64	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄	#41	1	1	yes	characteristic
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	4	1	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈	C₂²⋊C₄	1	2	yes	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	C₄²	1	2	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	—	4	1	no	—
93	C₄²	—	4	1	no	—
94	C₄²	C₂²×C₄	1	1	yes	characteristic
95	C₄²	C₂²×C₄	1	1	yes	characteristic
96	C₄²	C₂×D₄	1	1	yes	characteristic
97	C₄²	#41	1	1	yes	characteristic
98	C₂³	C₄×D₄	1	1	yes	characteristic
99	Q₈	—	2	2	no	—
100	Q₈	—	2	4	no	—
101	Q₈	#97	1	4	yes	complemented
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄	#117	1	1	yes	characteristic
113	C₂×C₄	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄	C₂×M₁₆	1	2	yes	—
117	C₂×C₄	#131	1	2	yes	—
118	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄	#137	1	1	yes	characteristic
120	C₂×C₄	#137	1	1	yes	characteristic
121	C₂×C₄	#137	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	2	no	—
126	C₈	—	4	2	no	—
127	C₈	—	4	2	no	—
128	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²	#435	1	2	yes	—
130	C₂²	#440	1	1	yes	characteristic
131	C₂²	#441	1	1	yes	characteristic
132	C₂²	#443	1	2	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	2	no	—
138	C₄	C₂×#97	1	1	yes	characteristic
139	C₄	#405	1	1	yes	characteristic
140	C₄	#417	1	1	yes	characteristic
141	C₄	#419	1	1	yes	characteristic
142	C₄	#435	1	2	yes	—
143	C₄	#440	1	1	yes	characteristic
144	C₄	#441	1	1	yes	characteristic
145	C₂	#1587	1	2	yes	—
146	C₂	#1588	1	2	yes	—
147	C₂	#1752	1	1	yes	characteristic
148	C₂	#1763	1	1	yes	characteristic
149	C₂	#1764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal, characteristic

C₄×#103

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

#1279

C₂

yes

maximal, characteristic

#1277

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

—

C₈×Q₈

C₄

yes

complemented

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

C₄

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

characteristic

#340

C₄

yes

—

#336

—

Q₈⋊C₈

C₄

yes

complemented

#322

—

C₂²×Q₈

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄²

yes

—

C₂×Q₈

M₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₄²

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

#97

yes

complemented

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×C₄²

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

112

C₂×C₄

#117

yes

characteristic

113

C₂×C₄

C₄×D₄

yes

characteristic

114

C₂×C₄

C₄×D₄

yes

characteristic

115

C₂×C₄

C₄×D₄

yes

characteristic

116

C₂×C₄

C₂×M₁₆

yes

—

117

C₂×C₄

#131

yes

—

118

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

119

C₂×C₄

#137

yes

characteristic

120

C₂×C₄

#137

yes

characteristic

121

C₂×C₄

#137

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂²

C₄×C₂²⋊C₄

yes

characteristic

129

C₂²

#435

yes

—

130

C₂²

#440

yes

characteristic

131

C₂²

#441

yes

characteristic

132

C₂²

#443

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₂×#97

yes

characteristic

139

C₄

#405

yes

characteristic

140

C₄

#417

yes

characteristic

141

C₄

#419

yes

characteristic

142

C₄

#435

yes

—

143

C₄

#440

yes

characteristic

144

C₄

#441

yes

characteristic

145

C₂

#1587

yes

—

146

C₂

#1588

yes

—

147

C₂

#1752

yes

characteristic

148

C₂

#1763

yes

characteristic

149

C₂

#1764

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
10. C₈×Q₈	C₄	semidirect
23. Q₈⋊C₈	C₄	semidirect
101. Q₈	#97	semidirect

H ≅ G/N

type

10. C₈×Q₈

C₄

semidirect

23. Q₈⋊C₈

C₄

semidirect

101. Q₈

#97

semidirect

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

subgroup	index
C₂×C₈×Q₈	2
#1770	2
C₄×#103	2
C₂×Q₈⋊C₈	2
C₂×#322	2
#1279	2
#1277	2

subgroup

index

C₂×C₈×Q₈

C₂×Q₈⋊C₈

Subgroups — #8205

Conjugacy classes of subgroups

Complements

Maximal subgroups