Group #96899 — subgroups

Subgroups — #96899

Order 512.

390 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 214 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	C₄×#335	C₂	1	2	yes	maximal
2	#8260	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×#322	C₂	1	1	yes	maximal, characteristic
5	#7015	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	C₄	1	4	yes	—
12	C₂×#322	—	2	1	no	—
13	C₂×#322	C₄	1	2	yes	—
14	C₂×#322	C₄	1	2	yes	—
15	C₂×C₈²	—	2	1	no	—
16	#1277	C₄	1	4	yes	—
17	#1273	—	2	1	no	—
18	#1273	—	2	1	no	—
19	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
20	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
21	C₂×C₄×C₈	—	2	2	no	—
22	C₂×C₄×C₈	—	2	2	no	—
23	C₂×C₄×C₈	—	2	2	no	—
24	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	Q₈	1	2	yes	—
33	C₄³	C₂³	1	1	yes	characteristic
34	#336	—	2	2	no	—
35	#336	—	2	2	no	—
36	#335	—	2	8	no	—
37	#322	—	2	4	no	—
38	#322	—	4	2	no	—
39	#322	C₈	1	8	yes	complemented
40	C₈²	—	4	2	no	—
41	C₂²×C₈	—	2	2	no	—
42	C₂²×C₈	—	2	4	no	—
43	C₂²×C₈	—	2	4	no	—
44	C₂²×C₈	C₄²	1	1	yes	characteristic
45	C₂²×C₈	C₄²	1	1	yes	characteristic
46	C₂²×C₈	C₄²	1	2	yes	—
47	C₂²×C₈	C₄⋊C₄	1	2	yes	—
48	C₂²×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄²	C₂²×C₄	1	2	yes	—
51	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄²	#41	1	2	yes	—
54	C₄⋊C₈	—	2	4	no	—
55	C₄⋊C₈	—	2	4	no	—
56	C₄⋊C₈	—	2	4	no	—
57	C₄⋊C₈	M₁₆	1	8	yes	—
58	C₄×C₈	—	2	4	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	4	no	—
62	C₄×C₈	—	2	4	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	4	4	no	—
65	C₄×C₈	—	4	2	no	—
66	C₄×C₈	—	4	2	no	—
67	C₄×C₈	—	4	2	no	—
68	C₄×C₈	—	4	2	no	—
69	C₄×C₈	C₂×C₈	1	4	yes	—
70	C₄×C₈	C₂×C₈	1	4	yes	—
71	C₄×C₈	C₂×C₈	1	4	yes	—
72	C₄×C₈	M₁₆	1	4	yes	—
73	C₄×C₈	D₈	1	2	yes	—
74	C₄×C₈	Q₁₆	1	2	yes	—
75	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	#117	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₄	1	2	yes	—
79	C₂²×C₄	C₄×Q₈	1	2	yes	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	8	no	—
90	C₂×C₈	—	4	4	no	—
91	C₂×C₈	—	4	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	—	4	4	no	—
94	C₂×C₈	—	4	4	no	—
95	C₂×C₈	—	4	4	no	—
96	C₂×C₈	C₄×C₈	1	4	yes	—
97	C₂×C₈	C₄⋊C₈	1	4	yes	—
98	C₂×C₈	C₄⋊C₈	1	4	yes	—
99	C₂×C₈	#107	1	4	yes	—
100	C₂×C₈	#108	1	4	yes	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	4	no	—
103	C₄²	C₂²×C₈	1	2	yes	—
104	C₄²	C₂×M₁₆	1	2	yes	—
105	C₄²	C₂×M₁₆	1	4	yes	—
106	C₄²	#131	1	4	yes	—
107	C₄²	C₂×D₈	1	1	yes	characteristic
108	C₄²	C₂×Q₁₆	1	1	yes	characteristic
109	C₄²	#135	1	2	yes	—
110	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂×C₄	C₄×M₁₆	1	2	yes	—
116	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
117	C₂×C₄	#405	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
119	C₂×C₄	C₂×#107	1	1	yes	characteristic
120	C₂×C₄	#427	1	1	yes	characteristic
121	C₂×C₄	C₂×#108	1	2	yes	—
122	C₂×C₄	#431	1	2	yes	—
123	C₂×C₄	#435	1	4	yes	—
124	C₂×C₄	C₄×D₈	1	2	yes	—
125	C₂×C₄	C₄×Q₁₆	1	2	yes	—
126	C₂×C₄	#444	1	4	yes	—
127	C₂×C₄	#446	1	4	yes	—
128	C₈	—	2	4	no	—
129	C₈	—	2	4	no	—
130	C₈	—	4	4	no	—
131	C₈	—	4	8	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	8	4	no	—
135	C₈	#335	1	8	yes	complemented
136	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
137	C₂²	#1771	1	2	yes	—
138	C₂²	#1773	1	2	yes	—
139	C₂²	C₄×#107	1	1	yes	characteristic
140	C₂²	C₄×#108	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	4	no	—
143	C₄	C₈×M₁₆	1	2	yes	—
144	C₄	#1453	1	1	yes	characteristic
145	C₄	#1455	1	1	yes	characteristic
146	C₄	C₂×#335	1	2	yes	—
147	C₄	#1564	1	2	yes	—
148	C₄	#1581	1	4	yes	—
149	C₄	#1584	1	4	yes	—
150	C₂	#8262	1	1	yes	characteristic
151	C₂	#8268	1	1	yes	characteristic
152	C₂	C₄×#335	1	1	yes	characteristic
153	C₂	#8291	1	2	yes	—
154	C₂	#8297	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#335

C₂

yes

maximal

#8260

C₂

yes

maximal, characteristic

C₈×C₄⋊C₈

C₂

yes

maximal, characteristic

C₄×#322

C₂

yes

maximal, characteristic

#7015

C₂

yes

maximal

C₄×C₄⋊C₈

C₂²

yes

—

C₄×C₄⋊C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₄

yes

—

C₂×#322

—

C₂×#322

C₄

yes

—

C₂×#322

C₄

yes

—

C₂×C₈²

—

#1277

C₄

yes

—

#1273

—

#1273

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

—

—

—

—

—

C₈

yes

complemented

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₄²

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

#41

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

M₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

yes

—

C₂²×C₄

C₂×C₄²

yes

characteristic

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#117

yes

characteristic

C₂²×C₄

C₄×D₄

yes

—

C₂²×C₄

C₄×Q₈

yes

—

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

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#107

yes

—

100

C₂×C₈

#108

yes

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂²×C₈

yes

—

104

C₄²

C₂×M₁₆

yes

—

105

C₄²

C₂×M₁₆

yes

—

106

C₄²

#131

yes

—

107

C₄²

C₂×D₈

yes

characteristic

108

C₄²

C₂×Q₁₆

yes

characteristic

109

C₄²

#135

yes

—

110

C₂³

C₄×C₄⋊C₄

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄×C₈

yes

characteristic

115

C₂×C₄

C₄×M₁₆

yes

—

116

C₂×C₄

C₄×M₁₆

yes

characteristic

117

C₂×C₄

#405

yes

—

118

C₂×C₄

C₂×C₄⋊C₈

yes

—

119

C₂×C₄

C₂×#107

yes

characteristic

120

C₂×C₄

#427

yes

characteristic

121

C₂×C₄

C₂×#108

yes

—

122

C₂×C₄

#431

yes

—

123

C₂×C₄

#435

yes

—

124

C₂×C₄

C₄×D₈

yes

—

125

C₂×C₄

C₄×Q₁₆

yes

—

126

C₂×C₄

#444

yes

—

127

C₂×C₄

#446

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#335

yes

complemented

136

C₂²

C₄×C₄⋊C₈

yes

characteristic

137

C₂²

#1771

yes

—

138

C₂²

#1773

yes

—

139

C₂²

C₄×#107

yes

characteristic

140

C₂²

C₄×#108

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

C₈×M₁₆

yes

—

144

C₄

#1453

yes

characteristic

145

C₄

#1455

yes

characteristic

146

C₄

C₂×#335

yes

—

147

C₄

#1564

yes

—

148

C₄

#1581

yes

—

149

C₄

#1584

yes

—

150

C₂

#8262

yes

characteristic

151

C₂

#8268

yes

characteristic

152

C₂

C₄×#335

yes

characteristic

153

C₂

#8291

yes

—

154

C₂

#8297

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂×C₄²

C₂×D₄

commutator

C₂×C₄

C₂×C₄×C₈

Frattini, cosocle

C₄³

C₂³

socle, prime radical

C₂³

C₄×C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
39. #322	C₈	semidirect
135. C₈	#335	semidirect

H ≅ G/N

type

39. #322

C₈

semidirect

135. C₈

#335

semidirect

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

subgroup	index
C₄×#335	2
#8260	2
C₈×C₄⋊C₈	2
C₄×#322	2
#7015	2

subgroup

index

C₈×C₄⋊C₈

Subgroups — #96899

Conjugacy classes of subgroups

Complements

Maximal subgroups