Group #93024 — subgroups

Subgroups — #93024

Order 512.

182 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 54 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	#13264	C₂	1	1	yes	maximal, characteristic
2	#8623	C₂	1	1	yes	maximal, characteristic
3	#8617	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×#346	C₄	1	1	yes	characteristic
7	#1801	—	2	1	no	—
8	#1568	—	2	1	no	—
9	#1567	—	2	1	no	—
10	#1467	—	2	1	no	—
11	#1374	—	2	1	no	—
12	#1374	—	2	1	no	—
13	#1292	C₄	1	2	yes	—
14	#1290	C₄	1	2	yes	—
15	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	—	2	1	no	—
17	C₂²×C₁₆	—	2	1	no	—
18	C₈×Q₈	—	4	1	no	—
19	C₈×Q₈	—	4	1	no	—
20	#432	—	2	2	no	—
21	#423	—	2	2	no	—
22	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊C₈	Q₈	1	1	yes	characteristic
25	C₂×#101	—	4	1	no	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₄⋊C₁₆	—	4	1	no	—
30	#346	—	4	1	no	—
31	#343	—	4	1	no	—
32	#335	—	4	1	no	—
33	#335	—	4	1	no	—
34	#335	—	4	1	no	—
35	#334	—	4	1	no	—
36	#324	—	4	1	no	—
37	#324	—	4	1	no	—
38	C₂²×Q₈	SD₁₆	1	1	yes	characteristic
39	C₂×M₁₆	—	4	2	no	—
40	C₂²×C₈	C₄²	1	1	yes	characteristic
41	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Q₈	—	4	1	no	—
45	C₄×Q₈	—	4	1	no	—
46	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
47	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
48	C₂×C₄⋊C₄	Q₁₆	1	1	yes	characteristic
49	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	4	1	no	—
54	C₂×C₁₆	—	4	1	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	—	4	1	no	—
57	C₄⋊C₈	—	4	1	no	—
58	C₄⋊C₈	—	4	1	no	—
59	C₄⋊C₈	—	4	1	no	—
60	C₄⋊C₈	—	4	2	no	—
61	C₄⋊C₈	—	4	2	no	—
62	C₄⋊C₈	—	4	2	no	—
63	#101	—	8	2	no	—
64	C₂²⋊C₈	—	4	2	no	—
65	#97	—	4	2	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂×Q₈	—	2	1	no	—
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×Q₈	—	4	1	no	—
76	C₂²×C₄	#95	1	1	yes	characteristic
77	C₂²×C₄	#102	1	1	yes	characteristic
78	C₂²×C₄	#102	1	1	yes	characteristic
79	C₂²×C₄	#103	1	1	yes	characteristic
80	C₂²×C₄	#103	1	1	yes	characteristic
81	C₂²×C₄	#106	1	1	yes	characteristic
82	C₂²×C₄	#107	1	1	yes	characteristic
83	M₁₆	—	8	2	no	—
84	M₁₆	—	8	2	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	—	8	2	no	—
94	C₂×C₈	—	8	2	no	—
95	C₂×C₈	#99	1	1	yes	characteristic
96	C₂×C₈	#99	1	1	yes	characteristic
97	C₄⋊C₄	—	4	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₁₆	—	4	2	no	—
107	C₁₆	—	8	1	no	—
108	C₂³	#340	1	1	yes	characteristic
109	Q₈	—	4	2	no	—
110	Q₈	—	4	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
128	C₂×C₄	#342	1	1	yes	characteristic
129	C₂×C₄	#361	1	1	yes	characteristic
130	C₂×C₄	#362	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	16	2	no	—
136	C₈	—	16	2	no	—
137	C₈	#355	1	1	yes	characteristic
138	C₈	#356	1	1	yes	characteristic
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#1289	1	1	yes	characteristic
142	C₂²	#1387	1	1	yes	characteristic
143	C₂²	#1389	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	1	no	—
152	C₄	#1392	1	1	yes	characteristic
153	C₄	#1394	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#7227	1	1	yes	characteristic
157	C₂	#7235	1	1	yes	characteristic
158	C₂	#7238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13264

C₂

yes

maximal, characteristic

#8623

C₂

yes

maximal, characteristic

#8617

C₂

yes

maximal, characteristic

C₂×C₈×Q₈

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₄

yes

characteristic

C₂×#346

C₄

yes

—

—

—

—

—

—

C₄

yes

—

#1290

C₄

yes

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×Q₈

—

C₈×Q₈

—

#432

—

#423

—

C₂×C₄⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄⋊C₈

Q₈

yes

characteristic

C₂×#101

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄⋊C₁₆

—

—

—

—

—

—

—

—

—

C₂²×Q₈

SD₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

D₈

yes

characteristic

C₂×C₄⋊C₄

SD₁₆

yes

characteristic

C₂×C₄⋊C₄

Q₁₆

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₂³

#340

yes

characteristic

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

H(3,ℤ/4)

yes

characteristic

128

C₂×C₄

#342

yes

characteristic

129

C₂×C₄

#361

yes

characteristic

130

C₂×C₄

#362

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

#355

yes

characteristic

138

C₈

#356

yes

characteristic

139

C₂²

—

140

C₂²

—

141

C₂²

#1289

yes

characteristic

142

C₂²

#1387

yes

characteristic

143

C₂²

#1389

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#1392

yes

characteristic

153

C₄

#1394

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#7227

yes

characteristic

157

C₂

#7235

yes

characteristic

158

C₂

#7238

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1289
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×C₈×Q₈	C₂²
prime radical	C₂³	#340
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#1289

commutator

C₂²×C₈

C₄²

Frattini, cosocle

C₂×C₈×Q₈

C₂²

prime radical

C₂³

#340

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

subgroup	index
#13264	2
#8623	2
#8617	2

subgroup

index

#13264

#8623

#8617

Subgroups — #93024

Conjugacy classes of subgroups

Maximal subgroups