Group #8595 — subgroups

Subgroups — #8595

Order 256.

299 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	#1797	C₂	1	2	yes	maximal
3	C₂×#334	C₂	1	2	yes	maximal, complemented
4	#1277	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₂²⋊C₈	—	2	2	no	—
10	C₂×C₄×C₈	—	2	4	no	—
11	C₂×C₄×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#336	—	2	2	no	—
16	#334	—	2	4	no	—
17	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄	D₄	1	1	yes	characteristic
19	C₂³×C₄	Q₈	1	1	yes	characteristic
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	—	2	2	no	—
24	C₂²×C₈	—	2	2	no	—
25	C₂²×C₈	—	2	4	no	—
26	C₂²×C₈	C₈	1	4	yes	complemented
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²	—	2	2	no	—
32	C₂×C₄²	—	2	2	no	—
33	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²	D₄	1	1	yes	characteristic
35	C₂×C₄²	Q₈	1	1	yes	characteristic
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊C₈	—	2	4	no	—
38	C₄⋊C₈	—	2	4	no	—
39	C₂²⋊C₈	—	4	4	no	—
40	C₄×C₈	—	2	2	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	—	2	2	no	—
43	C₄×C₈	D₄	1	4	yes	—
44	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄	—	2	2	no	—
46	C₂²×C₄	—	2	2	no	—
47	C₂²×C₄	—	2	1	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	—	2	2	no	—
50	C₂²×C₄	—	2	2	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	—	2	2	no	—
54	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄	C₂×C₈	1	2	yes	—
58	C₂²×C₄	M₁₆	1	2	yes	—
59	C₂²×C₄	SD₁₆	1	4	yes	—
60	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈	—	2	8	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	8	no	—
71	C₂×C₈	—	4	4	no	—
72	C₂×C₈	—	4	4	no	—
73	C₂×C₈	—	4	4	no	—
74	C₂×C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈	C₂×C₈	1	2	yes	—
77	C₂×C₈	C₂×C₈	1	2	yes	—
78	C₂×C₈	M₁₆	1	2	yes	—
79	C₂×C₈	M₁₆	1	2	yes	—
80	C₄²	—	2	1	no	—
81	C₄²	—	2	1	no	—
82	C₄²	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	C₂×D₄	1	2	yes	—
85	C₄²	#41	1	2	yes	—
86	C₂³	—	2	1	no	—
87	C₂³	—	2	1	no	—
88	C₂³	—	2	1	no	—
89	C₂³	—	2	1	no	—
90	C₂³	C₄⋊C₈	1	2	yes	—
91	C₂³	#106	1	2	yes	—
92	C₂³	#108	1	2	yes	—
93	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	C₄⋊C₈	1	2	yes	—
109	C₂×C₄	#106	1	2	yes	—
110	C₂×C₄	#108	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄	#117	1	1	yes	characteristic
113	C₂×C₄	#121	1	2	yes	—
114	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄	#135	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	—	2	2	no	—
123	C₈	—	2	2	no	—
124	C₈	—	8	4	no	—
125	C₈	C₂²⋊C₈	1	4	yes	complemented
126	C₂²	—	2	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#334	1	4	yes	complemented
133	C₂²	#380	1	1	yes	characteristic
134	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₂²	#423	1	1	yes	characteristic
136	C₂²	C₂×#106	1	1	yes	characteristic
137	C₂²	#427	1	1	yes	characteristic
138	C₂²	C₂×#108	1	1	yes	characteristic
139	C₂²	C₂×#108	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄	#407	1	1	yes	characteristic
146	C₄	#431	1	1	yes	characteristic
147	C₄	#432	1	1	yes	characteristic
148	C₄	#465	1	4	yes	—
149	C₂	—	2	4	no	—
150	C₂	C₂×#334	1	1	yes	characteristic
151	C₂	#1564	1	1	yes	characteristic
152	C₂	#1566	1	1	yes	characteristic
153	C₂	#1568	1	1	yes	characteristic
154	C₂	#1797	1	1	yes	characteristic
155	C₂	#1807	1	1	yes	characteristic
156	C₂	#1810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×C₈

C₂

yes

maximal, characteristic

#1797

C₂

yes

maximal

C₂×#334

C₂

yes

maximal, complemented

#1277

C₂

yes

maximal, complemented

C₂³×C₈

C₄

yes

characteristic

C₂³×C₈

C₄

yes

characteristic

C₂²×C₄²

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#336

—

#334

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂³×C₄

D₄

yes

characteristic

C₂³×C₄

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₈

yes

complemented

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

Q₈

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₂⁴

C₄⋊C₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

SD₁₆

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

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

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄⋊C₈

yes

—

C₂³

#106

yes

—

C₂³

#108

yes

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

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

—

107

C₂×C₄

—

108

C₂×C₄

C₄⋊C₈

yes

—

109

C₂×C₄

#106

yes

—

110

C₂×C₄

#108

yes

—

111

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

112

C₂×C₄

#117

yes

characteristic

113

C₂×C₄

#121

yes

—

114

C₂×C₄

C₂²⋊Q₈

yes

—

115

C₂×C₄

C₂²×C₈

yes

characteristic

116

C₂×C₄

C₂×M₁₆

yes

characteristic

117

C₂×C₄

C₂×M₁₆

yes

characteristic

118

C₂×C₄

C₂×M₁₆

yes

characteristic

119

C₂×C₄

C₂×SD₁₆

yes

—

120

C₂×C₄

#135

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂²⋊C₈

yes

complemented

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#334

yes

complemented

133

C₂²

#380

yes

characteristic

134

C₂²

C₂×C₄⋊C₈

yes

characteristic

135

C₂²

#423

yes

characteristic

136

C₂²

C₂×#106

yes

characteristic

137

C₂²

#427

yes

characteristic

138

C₂²

C₂×#108

yes

characteristic

139

C₂²

C₂×#108

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

C₂×C₂²⋊C₈

yes

characteristic

145

C₄

#407

yes

characteristic

146

C₄

#431

yes

characteristic

147

C₄

#432

yes

characteristic

148

C₄

#465

yes

—

149

C₂

—

150

C₂

C₂×#334

yes

characteristic

151

C₂

#1564

yes

characteristic

152

C₂

#1566

yes

characteristic

153

C₂

#1568

yes

characteristic

154

C₂

#1797

yes

characteristic

155

C₂

#1807

yes

characteristic

156

C₂

#1810

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂³	C₂×C₄⋊C₄
prime radical	C₂⁴	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂³

C₂×C₄⋊C₄

prime radical

C₂⁴

C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. C₂×#334	C₂	semidirect
4. #1277	C₂	semidirect
26. C₂²×C₈	C₈	semidirect
125. C₈	C₂²⋊C₈	semidirect
132. C₂²	#334	semidirect

H ≅ G/N

type

3. C₂×#334

C₂

semidirect

4. #1277

C₂

semidirect

26. C₂²×C₈

C₈

semidirect

125. C₈

C₂²⋊C₈

semidirect

132. C₂²

#334

semidirect

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

subgroup	index
C₂²×C₄×C₈	2
#1797	2
C₂×#334	2
#1277	2

subgroup

index

C₂²×C₄×C₈

#1797

C₂×#334

#1277

Subgroups — #8595

Conjugacy classes of subgroups

Complements

Maximal subgroups