Group #93447 — subgroups

Subgroups — #93447

Order 512.

210 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 98 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×#364	C₂	1	1	yes	maximal, characteristic
2	C₄×#364	C₂	1	1	yes	maximal, characteristic
3	#8807	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	—	2	1	no	—
5	C₂×#364	C₄	1	2	yes	—
6	C₄×M₃₂	—	2	1	no	—
7	C₄×M₃₂	—	2	1	no	—
8	#1831	C₄	1	2	yes	—
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×#335	—	2	1	no	—
11	C₂×#334	—	2	1	no	—
12	#1376	—	2	1	no	—
13	#1376	C₄	1	2	yes	—
14	C₂×M₃₂	—	2	2	no	—
15	C₂×M₃₂	—	2	2	no	—
16	C₂×C₄⋊C₈	—	2	2	no	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
23	C₄³	C₂×C₄	1	1	yes	characteristic
24	#364	—	2	4	no	—
25	#364	—	2	2	no	—
26	#364	C₈	1	4	yes	complemented
27	#346	—	4	1	no	—
28	#346	—	4	1	no	—
29	C₄×C₁₆	—	4	1	no	—
30	C₄×C₁₆	—	4	1	no	—
31	#336	—	4	2	no	—
32	#335	—	4	1	no	—
33	#335	—	4	1	no	—
34	#334	—	4	1	no	—
35	#334	—	4	1	no	—
36	C₂²×C₈	—	4	2	no	—
37	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄²	—	2	1	no	—
42	C₂×C₄²	—	2	1	no	—
43	C₂×C₄²	C₄²	1	1	yes	characteristic
44	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
46	M₃₂	—	2	4	no	—
47	M₃₂	—	2	4	no	—
48	M₃₂	—	4	2	no	—
49	M₃₂	—	4	2	no	—
50	C₂×C₁₆	—	4	2	no	—
51	C₂×C₁₆	—	4	2	no	—
52	C₄⋊C₈	—	4	2	no	—
53	C₄⋊C₈	—	4	2	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	4	1	no	—
57	C₄×C₈	C₂×C₈	1	2	yes	—
58	C₄×C₈	M₁₆	1	2	yes	—
59	C₄×C₈	D₈	1	1	yes	characteristic
60	C₄×C₈	D₈	1	1	yes	characteristic
61	C₄×C₈	SD₁₆	1	1	yes	characteristic
62	C₄×C₈	SD₁₆	1	1	yes	characteristic
63	C₄×C₈	SD₁₆	1	1	yes	characteristic
64	C₄×C₈	SD₁₆	1	1	yes	characteristic
65	C₄×C₈	Q₁₆	1	1	yes	characteristic
66	C₄×C₈	Q₁₆	1	1	yes	characteristic
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	#95	1	1	yes	characteristic
70	C₂²×C₄	#100	1	1	yes	characteristic
71	C₂²×C₄	#101	1	1	yes	characteristic
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	8	4	no	—
77	C₂×C₈	—	8	2	no	—
78	C₂×C₈	C₂²⋊C₈	1	2	yes	—
79	C₂×C₈	#102	1	1	yes	characteristic
80	C₂×C₈	#102	1	1	yes	characteristic
81	C₂×C₈	#103	1	1	yes	characteristic
82	C₂×C₈	#103	1	1	yes	characteristic
83	C₂×C₈	#104	1	2	yes	—
84	C₂×C₈	C₄⋊C₈	1	2	yes	—
85	C₂×C₈	#106	1	1	yes	characteristic
86	C₂×C₈	#106	1	1	yes	characteristic
87	C₂×C₈	#107	1	1	yes	characteristic
88	C₂×C₈	#107	1	1	yes	characteristic
89	C₂×C₈	#108	1	2	yes	—
90	C₄²	—	2	1	no	—
91	C₄²	—	2	1	no	—
92	C₄²	—	2	1	no	—
93	C₄²	—	4	2	no	—
94	C₄²	—	4	2	no	—
95	C₄²	C₄×C₈	1	1	yes	characteristic
96	C₄²	#97	1	1	yes	characteristic
97	C₄²	#102	1	1	yes	characteristic
98	C₄²	#102	1	1	yes	characteristic
99	C₄²	#103	1	1	yes	characteristic
100	C₄²	#103	1	1	yes	characteristic
101	C₁₆	—	4	4	no	—
102	C₁₆	—	4	4	no	—
103	C₂³	#343	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	#331	1	1	yes	characteristic
114	C₂×C₄	#332	1	1	yes	characteristic
115	C₂×C₄	#332	1	1	yes	characteristic
116	C₂×C₄	#333	1	1	yes	characteristic
117	C₂×C₄	#336	1	1	yes	characteristic
118	C₂×C₄	#339	1	1	yes	characteristic
119	C₂×C₄	#340	1	1	yes	characteristic
120	C₂×C₄	#340	1	1	yes	characteristic
121	C₂×C₄	#341	1	1	yes	characteristic
122	C₂×C₄	#344	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	16	2	no	—
128	C₈	D₄⋊C₈	1	2	yes	—
129	C₈	Q₈⋊C₈	1	2	yes	—
130	C₈	#334	1	2	yes	—
131	C₈	#335	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#1283	1	1	yes	characteristic
135	C₂²	#1296	1	1	yes	characteristic
136	C₂²	#1299	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	4	1	no	—
142	C₄	—	4	2	no	—
143	C₄	#1277	1	1	yes	characteristic
144	C₄	#1279	1	1	yes	characteristic
145	C₄	#1301	1	1	yes	characteristic
146	C₄	#1303	1	1	yes	characteristic
147	C₄	#1366	1	2	yes	—
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	#7031	1	1	yes	characteristic
151	C₂	#7271	1	1	yes	characteristic
152	C₂	#7273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#364

C₂

yes

maximal, characteristic

C₄×#364

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

C₂×#364

—

C₂×#364

C₄

yes

—

C₄×M₃₂

—

C₄×M₃₂

—

#1831

C₄

yes

—

C₄²×C₈

C₂²

yes

—

—

—

C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄⋊C₈

—

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

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

C₂×C₄

yes

—

—

C₈

yes

complemented

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×C₈

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

100

C₄²

#103

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₂³

#343

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#331

yes

characteristic

114

C₂×C₄

#332

yes

characteristic

115

C₂×C₄

#332

yes

characteristic

116

C₂×C₄

#333

yes

characteristic

117

C₂×C₄

#336

yes

characteristic

118

C₂×C₄

#339

yes

characteristic

119

C₂×C₄

#340

yes

characteristic

120

C₂×C₄

#340

yes

characteristic

121

C₂×C₄

#341

yes

characteristic

122

C₂×C₄

#344

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

D₄⋊C₈

yes

—

129

C₈

Q₈⋊C₈

yes

—

130

C₈

#334

yes

—

131

C₈

#335

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

#1283

yes

characteristic

135

C₂²

#1296

yes

characteristic

136

C₂²

#1299

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#1277

yes

characteristic

144

C₄

#1279

yes

characteristic

145

C₄

#1301

yes

characteristic

146

C₄

#1303

yes

characteristic

147

C₄

#1366

yes

—

148

C₂

—

149

C₂

—

150

C₂

#7031

yes

characteristic

151

C₂

#7271

yes

characteristic

152

C₂

#7273

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#333
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1296
prime radical	C₂³	#343
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#333

commutator

C₄²

C₄×C₈

Frattini, cosocle

C₄²×C₈

C₂²

C₂²

C₂³

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
26. #364	C₈	semidirect

H ≅ G/N

type

26. #364

C₈

semidirect

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

subgroup	index
C₄×#364	2
#8807	2

subgroup

index

C₄×#364

#8807

Subgroups — #93447

Conjugacy classes of subgroups

Complements

Maximal subgroups