Group #94346 — subgroups

Subgroups — #94346

Order 512.

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

normal subgroups only

This group has 220 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	#13553	C₂	1	1	yes	maximal, characteristic
2	#12837	C₂	1	1	yes	maximal, characteristic
3	C₂×#1381	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₁₆	C₂²	1	1	yes	characteristic
5	C₂²×#108	—	2	1	no	—
6	C₂×M₆₄	—	2	1	no	—
7	C₂×#368	C₄	1	2	yes	—
8	#1810	—	2	1	no	—
9	C₂²⋊C₃₂	—	2	1	no	—
10	C₂²⋊C₃₂	C₄	1	2	yes	—
11	#1381	—	2	1	no	—
12	#1381	—	2	1	no	—
13	#1381	C₄	1	2	yes	—
14	C₂²×M₁₆	—	4	1	no	—
15	C₂³×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	—	4	2	no	—
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	Q₈	1	1	yes	characteristic
23	C₂×#108	—	2	4	no	—
24	C₂×#108	—	2	2	no	—
25	C₂×#108	—	2	2	no	—
26	C₂×#108	—	4	2	no	—
27	C₂×#108	—	4	1	no	—
28	#407	—	4	1	no	—
29	M₆₄	—	4	2	no	—
30	C₂×C₃₂	—	2	2	no	—
31	#368	—	4	2	no	—
32	#341	—	4	1	no	—
33	C₂³×C₄	D₈	1	1	yes	characteristic
34	C₂×M₁₆	—	4	4	no	—
35	C₂×M₁₆	—	4	2	no	—
36	C₂×M₁₆	—	4	2	no	—
37	C₂×M₁₆	—	8	2	no	—
38	C₂×M₁₆	—	8	1	no	—
39	C₂²×C₈	—	2	2	no	—
40	C₂²×C₈	—	2	2	no	—
41	C₂²×C₈	—	8	1	no	—
42	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₈	SD₁₆	1	1	yes	characteristic
46	C₂²×C₈	SD₁₆	1	1	yes	characteristic
47	C₂²×C₈	Q₁₆	1	1	yes	characteristic
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	—	4	1	no	—
51	C₂×C₁₆	—	4	1	no	—
52	C₂×C₁₆	—	4	1	no	—
53	C₂×C₁₆	—	4	1	no	—
54	C₂×C₁₆	C₄²	1	1	yes	characteristic
55	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
58	#108	—	2	4	no	—
59	#108	—	4	4	no	—
60	#108	—	4	2	no	—
61	#108	—	4	2	no	—
62	#108	—	4	2	no	—
63	C₂²⋊C₈	—	8	1	no	—
64	C₂²⋊C₈	—	8	1	no	—
65	C₃₂	—	4	2	no	—
66	C₂⁴	Q₃₂	1	1	yes	characteristic
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	#102	1	1	yes	characteristic
70	C₂²×C₄	#102	1	1	yes	characteristic
71	C₂²×C₄	#107	1	1	yes	characteristic
72	C₂²×C₄	D₁₆	1	1	yes	characteristic
73	C₂²×C₄	SD₃₂	1	1	yes	characteristic
74	C₂²×C₄	SD₃₂	1	1	yes	characteristic
75	M₁₆	—	4	4	no	—
76	M₁₆	—	8	4	no	—
77	M₁₆	—	8	2	no	—
78	M₁₆	—	8	2	no	—
79	M₁₆	—	8	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	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	8	4	no	—
88	C₂×C₈	—	8	2	no	—
89	C₂×C₈	—	8	2	no	—
90	C₂×C₈	#95	1	1	yes	characteristic
91	C₂×C₈	#99	1	2	yes	—
92	C₂×C₈	#103	1	1	yes	characteristic
93	C₂×C₈	#103	1	1	yes	characteristic
94	C₂×C₈	#104	1	2	yes	—
95	C₂×C₈	#106	1	1	yes	characteristic
96	C₁₆	—	2	1	no	—
97	C₁₆	—	2	1	no	—
98	C₁₆	—	2	1	no	—
99	C₁₆	—	2	1	no	—
100	C₂³	—	2	2	no	—
101	C₂³	—	4	1	no	—
102	C₂³	—	4	1	no	—
103	C₂³	#358	1	1	yes	characteristic
104	C₂³	#358	1	1	yes	characteristic
105	C₂³	#366	1	1	yes	characteristic
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	#327	1	2	yes	—
114	C₂×C₄	#340	1	1	yes	characteristic
115	C₂×C₄	#357	1	1	yes	characteristic
116	C₂×C₄	#357	1	1	yes	characteristic
117	C₂×C₄	#360	1	2	yes	—
118	C₂×C₄	#367	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	2	1	no	—
121	C₈	—	8	4	no	—
122	C₈	—	16	2	no	—
123	C₈	#328	1	2	yes	—
124	C₈	#339	1	1	yes	characteristic
125	C₈	#342	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	4	4	no	—
135	C₂²	#1348	1	2	yes	—
136	C₂²	#1380	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	#1294	1	1	yes	characteristic
140	C₄	#1347	1	2	yes	—
141	C₄	#1386	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	—	4	1	no	—
144	C₂	—	4	1	no	—
145	C₂	#7258	1	1	yes	characteristic
146	C₂	#7476	1	1	yes	characteristic
147	C₂	#7479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13553

C₂

yes

maximal, characteristic

#12837

C₂

yes

maximal, characteristic

C₂×#1381

C₂

yes

maximal, characteristic

C₂³×C₁₆

C₂²

yes

characteristic

C₂²×#108

—

C₂×M₆₄

—

C₂×#368

C₄

yes

—

#1810

—

C₂²⋊C₃₂

—

C₂²⋊C₃₂

C₄

yes

—

#1381

—

#1381

—

#1381

C₄

yes

—

C₂²×M₁₆

—

C₂³×C₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

Q₈

yes

—

—

—

—

—

—

M₆₄

—

C₂×C₃₂

—

#368

—

#341

—

C₂³×C₄

D₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

—

—

—

—

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₃₂

—

C₂⁴

Q₃₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#95

yes

characteristic

C₂×C₈

#99

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#106

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

#358

yes

characteristic

104

C₂³

#358

yes

characteristic

105

C₂³

#366

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#327

yes

—

114

C₂×C₄

#340

yes

characteristic

115

C₂×C₄

#357

yes

characteristic

116

C₂×C₄

#357

yes

characteristic

117

C₂×C₄

#360

yes

—

118

C₂×C₄

#367

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

#328

yes

—

124

C₈

#339

yes

characteristic

125

C₈

#342

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#1348

yes

—

136

C₂²

#1380

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#1294

yes

characteristic

140

C₄

#1347

yes

—

141

C₄

#1386

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

#7258

yes

characteristic

146

C₂

#7476

yes

characteristic

147

C₂

#7479

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂³×C₁₆	C₂²
prime radical	C₂⁴	Q₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1380

commutator

C₂×C₁₆

C₄²

Frattini, cosocle

C₂³×C₁₆

C₂²

prime radical

C₂⁴

Q₃₂

LCS

G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G

LECS

G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13553	2
#12837	2
C₂×#1381	2

subgroup

index

#13553

#12837

C₂×#1381

Subgroups — #94346

Conjugacy classes of subgroups

Maximal subgroups