Group #93609 — subgroups

Subgroups — #93609

Order 512.

194 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 70 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	#12894	C₂	1	1	yes	maximal, characteristic
2	#8785	C₂	1	1	yes	maximal, characteristic
3	#8274	C₂	1	1	yes	maximal, characteristic
4	#1831	—	2	1	no	—
5	#1767	—	2	1	no	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₂×#335	—	2	1	no	—
8	#1374	—	2	1	no	—
9	#1374	—	2	1	no	—
10	#1374	C₄	1	2	yes	—
11	#1277	—	2	1	no	—
12	#1277	C₄	1	2	yes	—
13	#1277	C₄	1	2	yes	—
14	C₂²×C₁₆	—	2	2	no	—
15	C₂×C₄⋊C₈	—	2	2	no	—
16	C₂×C₄⋊C₈	—	2	2	no	—
17	C₂×#97	—	4	1	no	—
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
24	C₄³	D₄	1	1	yes	characteristic
25	#336	—	4	1	no	—
26	#336	—	4	1	no	—
27	#336	—	4	1	no	—
28	#335	—	2	4	no	—
29	C₂²×C₈	—	4	2	no	—
30	C₂²×C₈	—	4	2	no	—
31	C₂²×C₈	C₄²	1	1	yes	characteristic
32	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄²	D₈	1	1	yes	characteristic
39	C₂×C₄²	SD₁₆	1	1	yes	characteristic
40	C₂×C₄²	SD₁₆	1	1	yes	characteristic
41	C₂×C₄²	Q₁₆	1	1	yes	characteristic
42	C₂×C₁₆	—	4	2	no	—
43	C₂×C₁₆	—	4	2	no	—
44	C₂×C₁₆	—	4	2	no	—
45	C₄⋊C₈	—	2	4	no	—
46	C₄⋊C₈	—	4	2	no	—
47	C₄⋊C₈	—	4	2	no	—
48	C₄⋊C₈	—	4	2	no	—
49	#97	—	8	2	no	—
50	C₄×C₈	—	2	2	no	—
51	C₄×C₈	—	2	2	no	—
52	C₄×C₈	—	4	1	no	—
53	C₄×C₈	—	4	1	no	—
54	C₄×C₈	—	4	1	no	—
55	C₄×C₈	—	4	1	no	—
56	C₂²×C₄	#95	1	1	yes	characteristic
57	C₂²×C₄	#102	1	1	yes	characteristic
58	C₂²×C₄	#102	1	1	yes	characteristic
59	C₂²×C₄	#103	1	1	yes	characteristic
60	C₂²×C₄	#103	1	1	yes	characteristic
61	C₂²×C₄	#106	1	1	yes	characteristic
62	C₂²×C₄	#107	1	1	yes	characteristic
63	C₂×C₈	—	2	1	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	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	4	4	no	—
74	C₂×C₈	—	8	2	no	—
75	C₂×C₈	—	8	2	no	—
76	C₂×C₈	—	8	2	no	—
77	C₂×C₈	—	8	2	no	—
78	C₂×C₈	—	8	2	no	—
79	C₂×C₈	#104	1	2	yes	—
80	C₂×C₈	#108	1	2	yes	—
81	C₄²	—	2	1	no	—
82	C₄²	—	2	1	no	—
83	C₄²	—	2	1	no	—
84	C₄²	—	2	1	no	—
85	C₄²	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₄²	—	2	2	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	2	1	no	—
90	C₄²	—	4	1	no	—
91	C₄²	—	4	1	no	—
92	C₁₆	—	8	2	no	—
93	C₂³	#340	1	1	yes	characteristic
94	C₂×C₄	—	2	1	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	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
111	C₂×C₄	#339	1	1	yes	characteristic
112	C₂×C₄	#341	1	1	yes	characteristic
113	C₂×C₄	#344	1	1	yes	characteristic
114	C₂×C₄	#344	1	1	yes	characteristic
115	C₂×C₄	#344	1	1	yes	characteristic
116	C₂×C₄	#359	1	2	yes	—
117	C₂×C₄	#361	1	1	yes	characteristic
118	C₂×C₄	#362	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	2	2	no	—
121	C₈	—	4	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	8	4	no	—
124	C₈	—	16	2	no	—
125	C₂²	#1286	1	1	yes	characteristic
126	C₂²	#1301	1	1	yes	characteristic
127	C₂²	#1303	1	1	yes	characteristic
128	C₂²	#1382	1	1	yes	characteristic
129	C₂²	#1384	1	1	yes	characteristic
130	C₂²	#1387	1	1	yes	characteristic
131	C₂²	#1388	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	4	1	no	—
137	C₄	—	4	1	no	—
138	C₄	—	4	1	no	—
139	C₄	#1287	1	1	yes	characteristic
140	C₄	#1288	1	1	yes	characteristic
141	C₄	#1302	1	1	yes	characteristic
142	C₄	#1308	1	1	yes	characteristic
143	C₂	#7041	1	1	yes	characteristic
144	C₂	#7218	1	1	yes	characteristic
145	C₂	#7223	1	1	yes	characteristic
146	C₂	#7291	1	1	yes	characteristic
147	C₂	#7293	1	1	yes	characteristic
148	C₂	#7297	1	1	yes	characteristic
149	C₂	#7301	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12894

C₂

yes

maximal, characteristic

#8785

C₂

yes

maximal, characteristic

#8274

C₂

yes

maximal, characteristic

#1831

—

#1767

—

C₄²×C₈

C₂²

yes

—

—

—

C₄

yes

—

#1277

—

#1277

C₄

yes

—

#1277

C₄

yes

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

D₄

yes

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

Q₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

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

#104

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₂³

#340

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

—

109

C₂×C₄

—

110

C₂×C₄

H(3,ℤ/4)

yes

characteristic

111

C₂×C₄

#339

yes

characteristic

112

C₂×C₄

#341

yes

characteristic

113

C₂×C₄

#344

yes

characteristic

114

C₂×C₄

#344

yes

characteristic

115

C₂×C₄

#344

yes

characteristic

116

C₂×C₄

#359

yes

—

117

C₂×C₄

#361

yes

characteristic

118

C₂×C₄

#362

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

#1286

yes

characteristic

126

C₂²

#1301

yes

characteristic

127

C₂²

#1303

yes

characteristic

128

C₂²

#1382

yes

characteristic

129

C₂²

#1384

yes

characteristic

130

C₂²

#1387

yes

characteristic

131

C₂²

#1388

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#1287

yes

characteristic

140

C₄

#1288

yes

characteristic

141

C₄

#1302

yes

characteristic

142

C₄

#1308

yes

characteristic

143

C₂

#7041

yes

characteristic

144

C₂

#7218

yes

characteristic

145

C₂

#7223

yes

characteristic

146

C₂

#7291

yes

characteristic

147

C₂

#7293

yes

characteristic

148

C₂

#7297

yes

characteristic

149

C₂

#7301

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#340
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₄²×C₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#340

commutator

C₂²×C₈

C₄²

Frattini, cosocle

C₄²×C₈

C₂²

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12894	2
#8785	2
#8274	2

subgroup

index

#12894

#8785

#8274

Subgroups — #93609

Conjugacy classes of subgroups

Maximal subgroups