Group #96900 — subgroups

Subgroups — #96900

Order 512.

374 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 182 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×#334	C₂	1	2	yes	maximal
2	#8261	C₂	1	2	yes	maximal
3	C₄×#322	C₂	1	1	yes	maximal, characteristic
4	#7015	C₂	1	1	yes	maximal, characteristic
5	#7015	C₂	1	1	yes	maximal, characteristic
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×#334	C₄	1	4	yes	—
12	C₂×#322	—	2	2	no	—
13	C₂×#322	C₄	1	2	yes	—
14	C₂×#322	C₄	1	2	yes	—
15	#1277	C₄	1	4	yes	—
16	#1273	—	2	2	no	—
17	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₈	—	2	2	no	—
20	C₂×C₄×C₈	—	2	2	no	—
21	C₂×C₄×C₈	—	2	2	no	—
22	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	Q₈	1	2	yes	—
31	C₄³	C₂³	1	1	yes	characteristic
32	#336	—	2	2	no	—
33	#336	—	2	2	no	—
34	#334	—	2	8	no	—
35	#322	—	2	4	no	—
36	#322	—	2	4	no	—
37	#322	—	4	4	no	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	—	2	4	no	—
40	C₂²×C₈	—	2	4	no	—
41	C₂²×C₈	C₄²	1	1	yes	characteristic
42	C₂²×C₈	C₄²	1	1	yes	characteristic
43	C₂²×C₈	C₄²	1	2	yes	—
44	C₂²×C₈	C₄⋊C₄	1	2	yes	—
45	C₂²×C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄²	C₂²×C₄	1	2	yes	—
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄²	#41	1	2	yes	—
51	C₄⋊C₈	—	2	4	no	—
52	C₄⋊C₈	—	2	4	no	—
53	C₄⋊C₈	—	2	4	no	—
54	C₄⋊C₈	M₁₆	1	8	yes	—
55	C₄×C₈	—	2	4	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	4	no	—
61	C₄×C₈	—	2	4	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	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	M₁₆	1	4	yes	—
70	C₄×C₈	M₁₆	1	4	yes	—
71	C₄×C₈	SD₁₆	1	4	yes	—
72	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
73	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄	#117	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₄	1	2	yes	—
76	C₂²×C₄	C₄×Q₈	1	2	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	2	8	no	—
89	C₂×C₈	—	4	4	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	#97	1	4	yes	—
96	C₂×C₈	#106	1	4	yes	—
97	C₂×C₈	#108	1	4	yes	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	4	no	—
100	C₄²	C₂×M₁₆	1	2	yes	—
101	C₄²	C₂×M₁₆	1	2	yes	—
102	C₄²	C₂×M₁₆	1	4	yes	—
103	C₄²	#131	1	4	yes	—
104	C₄²	C₂×SD₁₆	1	2	yes	—
105	C₄²	#135	1	1	yes	characteristic
106	C₄²	#135	1	1	yes	characteristic
107	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	C₂×#97	1	1	yes	characteristic
112	C₂×C₄	C₄×M₁₆	1	2	yes	—
113	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
114	C₂×C₄	#405	1	2	yes	—
115	C₂×C₄	#423	1	2	yes	—
116	C₂×C₄	C₂×#106	1	1	yes	characteristic
117	C₂×C₄	#427	1	1	yes	characteristic
118	C₂×C₄	C₂×#108	1	2	yes	—
119	C₂×C₄	#432	1	2	yes	—
120	C₂×C₄	#435	1	4	yes	—
121	C₂×C₄	C₄×SD₁₆	1	2	yes	—
122	C₂×C₄	C₄×SD₁₆	1	2	yes	—
123	C₂×C₄	#444	1	4	yes	—
124	C₂×C₄	#446	1	4	yes	—
125	C₈	—	2	4	no	—
126	C₈	—	2	4	no	—
127	C₈	—	2	4	no	—
128	C₈	—	4	4	no	—
129	C₈	—	4	8	no	—
130	C₈	—	4	1	no	—
131	C₈	—	4	1	no	—
132	C₈	—	8	4	no	—
133	C₂²	#1767	1	1	yes	characteristic
134	C₂²	#1771	1	2	yes	—
135	C₂²	#1773	1	2	yes	—
136	C₂²	C₄×#106	1	1	yes	characteristic
137	C₂²	C₄×#108	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	4	no	—
140	C₄	#1451	1	2	yes	—
141	C₄	#1453	1	1	yes	characteristic
142	C₄	#1455	1	1	yes	characteristic
143	C₄	#1566	1	2	yes	—
144	C₄	#1568	1	2	yes	—
145	C₄	#1582	1	4	yes	—
146	C₄	#1583	1	4	yes	—
147	C₂	#8263	1	1	yes	characteristic
148	C₂	#8269	1	1	yes	characteristic
149	C₂	#8273	1	1	yes	characteristic
150	C₂	#8290	1	2	yes	—
151	C₂	#8297	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#334

C₂

yes

maximal

#8261

C₂

yes

maximal

C₄×#322

C₂

yes

maximal, characteristic

#7015

C₂

yes

maximal, characteristic

#7015

C₂

yes

maximal, characteristic

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×#334

C₄

yes

—

C₂×#322

—

C₂×#322

C₄

yes

—

C₂×#322

C₄

yes

—

#1277

C₄

yes

—

#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

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

—

—

—

—

—

—

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

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

SD₁₆

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

—

C₂×C₈

—

C₂×C₈

#97

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

100

C₄²

C₂×M₁₆

yes

—

101

C₄²

C₂×M₁₆

yes

—

102

C₄²

C₂×M₁₆

yes

—

103

C₄²

#131

yes

—

104

C₄²

C₂×SD₁₆

yes

—

105

C₄²

#135

yes

characteristic

106

C₄²

#135

yes

characteristic

107

C₂³

C₄×C₄⋊C₄

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×#97

yes

characteristic

112

C₂×C₄

C₄×M₁₆

yes

—

113

C₂×C₄

C₄×M₁₆

yes

characteristic

114

C₂×C₄

#405

yes

—

115

C₂×C₄

#423

yes

—

116

C₂×C₄

C₂×#106

yes

characteristic

117

C₂×C₄

#427

yes

characteristic

118

C₂×C₄

C₂×#108

yes

—

119

C₂×C₄

#432

yes

—

120

C₂×C₄

#435

yes

—

121

C₂×C₄

C₄×SD₁₆

yes

—

122

C₂×C₄

C₄×SD₁₆

yes

—

123

C₂×C₄

#444

yes

—

124

C₂×C₄

#446

yes

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₂²

#1767

yes

characteristic

134

C₂²

#1771

yes

—

135

C₂²

#1773

yes

—

136

C₂²

C₄×#106

yes

characteristic

137

C₂²

C₄×#108

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

#1451

yes

—

141

C₄

#1453

yes

characteristic

142

C₄

#1455

yes

characteristic

143

C₄

#1566

yes

—

144

C₄

#1568

yes

—

145

C₄

#1582

yes

—

146

C₄

#1583

yes

—

147

C₂

#8263

yes

characteristic

148

C₂

#8269

yes

characteristic

149

C₂

#8273

yes

characteristic

150

C₂

#8290

yes

—

151

C₂

#8297

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂×C₄²

C₂×D₄

commutator

C₂²×C₄

C₂×C₄²

Frattini, cosocle

C₄³

C₂³

socle, prime radical

C₂³

C₄×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

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

subgroup	index
C₄×#334	2
#8261	2
C₄×#322	2
#7015	2

subgroup

index

Subgroups — #96900

Conjugacy classes of subgroups

Maximal subgroups