Group #93063 — subgroups

Subgroups — #93063

Order 512.

192 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₂×#1370	C₂	1	1	yes	maximal, characteristic
2	C₂×#1367	C₂	1	1	yes	maximal, characteristic
3	#8814	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊C₁₆	—	2	1	no	—
6	#1845	C₄	1	2	yes	—
7	#1844	—	2	1	no	—
8	#1834	—	2	1	no	—
9	C₂×C₈²	C₂²	1	1	yes	characteristic
10	#1370	—	2	1	no	—
11	#1370	C₄	1	2	yes	complemented
12	#1367	—	2	1	no	—
13	#1367	C₄	1	2	yes	complemented
14	C₂²×C₁₆	—	4	1	no	—
15	C₂²×C₁₆	—	4	1	no	—
16	C₂×#107	—	4	2	no	—
17	C₂×#106	—	4	1	no	—
18	C₂×#106	—	4	1	no	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	#384	—	2	2	no	—
23	C₄⋊C₁₆	—	2	2	no	—
24	C₄⋊C₁₆	—	2	2	no	—
25	C₄⋊C₁₆	—	4	1	no	—
26	C₄⋊C₁₆	—	4	1	no	—
27	#340	—	4	1	no	—
28	#340	—	4	1	no	—
29	C₈²	C₂×C₄	1	1	yes	characteristic
30	C₈²	C₂×C₄	1	1	yes	characteristic
31	C₈²	D₄	1	1	yes	characteristic
32	C₈²	Q₈	1	1	yes	characteristic
33	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	D₈	1	1	yes	characteristic
35	C₂²×C₈	D₈	1	1	yes	characteristic
36	C₂²×C₈	SD₁₆	1	1	yes	characteristic
37	C₂²×C₈	SD₁₆	1	1	yes	characteristic
38	C₂×C₄⋊C₄	—	4	2	no	—
39	C₂×C₄⋊C₄	—	4	2	no	—
40	C₂×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₁₆	—	4	2	no	—
43	C₂×C₁₆	—	4	2	no	—
44	C₂×C₁₆	—	8	2	no	—
45	C₂×C₁₆	—	8	1	no	—
46	C₂×C₁₆	—	8	1	no	—
47	#107	—	8	4	no	—
48	#106	—	8	2	no	—
49	#106	—	8	1	no	—
50	#106	—	8	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	C₄²	1	1	yes	characteristic
53	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₈	SD₁₆	1	1	yes	characteristic
59	C₄×C₈	SD₁₆	1	1	yes	characteristic
60	C₄×C₈	Q₁₆	1	1	yes	characteristic
61	C₄×C₈	Q₁₆	1	1	yes	characteristic
62	C₂²×C₄	—	8	2	no	—
63	C₂²×C₄	#100	1	1	yes	characteristic
64	C₂²×C₄	#102	1	1	yes	characteristic
65	C₂²×C₄	#102	1	1	yes	characteristic
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	#102	1	1	yes	characteristic
74	C₂×C₈	#102	1	1	yes	characteristic
75	C₂×C₈	#103	1	1	yes	characteristic
76	C₂×C₈	#103	1	1	yes	characteristic
77	C₂×C₈	#106	1	1	yes	characteristic
78	C₂×C₈	#106	1	1	yes	characteristic
79	C₂×C₈	#107	1	1	yes	characteristic
80	C₂×C₈	#107	1	1	yes	characteristic
81	C₂×C₈	D₁₆	1	1	yes	characteristic
82	C₂×C₈	SD₃₂	1	1	yes	characteristic
83	C₂×C₈	SD₃₂	1	1	yes	characteristic
84	C₂×C₈	Q₃₂	1	1	yes	characteristic
85	C₄⋊C₄	—	8	4	no	—
86	C₄⋊C₄	—	8	2	no	—
87	C₄⋊C₄	—	8	2	no	—
88	C₄⋊C₄	—	8	2	no	—
89	C₄⋊C₄	—	8	2	no	—
90	C₄²	#95	1	1	yes	characteristic
91	C₄²	#101	1	1	yes	characteristic
92	C₄²	#103	1	1	yes	characteristic
93	C₄²	#103	1	1	yes	characteristic
94	C₁₆	—	8	2	no	—
95	C₁₆	—	8	2	no	—
96	C₂³	#331	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	16	2	no	—
99	C₂×C₄	—	16	2	no	—
100	C₂×C₄	—	16	2	no	—
101	C₂×C₄	#332	1	1	yes	characteristic
102	C₂×C₄	#332	1	1	yes	characteristic
103	C₂×C₄	#333	1	1	yes	characteristic
104	C₂×C₄	#340	1	1	yes	characteristic
105	C₂×C₄	#340	1	1	yes	characteristic
106	C₂×C₄	#343	1	1	yes	characteristic
107	C₂×C₄	#357	1	1	yes	characteristic
108	C₂×C₄	#358	1	1	yes	characteristic
109	C₂×C₄	#359	1	2	yes	—
110	C₂×C₄	#360	1	2	yes	—
111	C₂×C₄	#361	1	1	yes	characteristic
112	C₂×C₄	#362	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	1	no	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	—	2	1	no	—
121	C₈	#357	1	1	yes	characteristic
122	C₈	#358	1	1	yes	characteristic
123	C₈	#366	1	1	yes	characteristic
124	C₈	#367	1	1	yes	characteristic
125	C₂²	#1296	1	1	yes	characteristic
126	C₂²	#1359	1	2	yes	—
127	C₂²	#1360	1	2	yes	—
128	C₂²	#1361	1	1	yes	characteristic
129	C₂²	#1362	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	32	2	no	—
133	C₄	#1363	1	2	yes	—
134	C₄	#1364	1	1	yes	characteristic
135	C₄	#1365	1	1	yes	characteristic
136	C₄	#1380	1	1	yes	characteristic
137	C₄	#1382	1	1	yes	characteristic
138	C₄	#1386	1	1	yes	characteristic
139	C₄	#1387	1	1	yes	characteristic
140	C₂	#7094	1	2	yes	—
141	C₂	#7095	1	2	yes	—
142	C₂	#7268	1	1	yes	characteristic
143	C₂	#7274	1	1	yes	characteristic
144	C₂	#7275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1370

C₂

yes

maximal, characteristic

C₂×#1367

C₂

yes

maximal, characteristic

#8814

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

—

#1845

C₄

yes

—

#1844

—

#1834

—

C₂×C₈²

C₂²

yes

characteristic

#1370

—

#1370

C₄

yes

complemented

#1367

—

#1367

C₄

yes

complemented

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

#384

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#340

—

#340

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

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

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

D₁₆

yes

characteristic

C₂×C₈

SD₃₂

yes

characteristic

C₂×C₈

SD₃₂

yes

characteristic

C₂×C₈

Q₃₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂³

#331

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#332

yes

characteristic

102

C₂×C₄

#332

yes

characteristic

103

C₂×C₄

#333

yes

characteristic

104

C₂×C₄

#340

yes

characteristic

105

C₂×C₄

#340

yes

characteristic

106

C₂×C₄

#343

yes

characteristic

107

C₂×C₄

#357

yes

characteristic

108

C₂×C₄

#358

yes

characteristic

109

C₂×C₄

#359

yes

—

110

C₂×C₄

#360

yes

—

111

C₂×C₄

#361

yes

characteristic

112

C₂×C₄

#362

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

#357

yes

characteristic

122

C₈

#358

yes

characteristic

123

C₈

#366

yes

characteristic

124

C₈

#367

yes

characteristic

125

C₂²

#1296

yes

characteristic

126

C₂²

#1359

yes

—

127

C₂²

#1360

yes

—

128

C₂²

#1361

yes

characteristic

129

C₂²

#1362

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#1363

yes

—

134

C₄

#1364

yes

characteristic

135

C₄

#1365

yes

characteristic

136

C₄

#1380

yes

characteristic

137

C₄

#1382

yes

characteristic

138

C₄

#1386

yes

characteristic

139

C₄

#1387

yes

characteristic

140

C₂

#7094

yes

—

141

C₂

#7095

yes

—

142

C₂

#7268

yes

characteristic

143

C₂

#7274

yes

characteristic

144

C₂

#7275

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#331
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#331

commutator

C₄×C₈

C₄²

Frattini, cosocle

C₂×C₈²

C₂²

LCS

G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
11. #1370	C₄	semidirect
13. #1367	C₄	semidirect

H ≅ G/N

type

11. #1370

C₄

semidirect

13. #1367

C₄

semidirect

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

subgroup	index
C₂×#1370	2
C₂×#1367	2
#8814	2

subgroup

index

C₂×#1370

C₂×#1367

#8814

Subgroups — #93063

Conjugacy classes of subgroups

Complements

Maximal subgroups