Group #141809 — subgroups

Subgroups — #141809

Order 512.

293 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 74 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	#33748	C₂	1	1	yes	maximal, characteristic
2	#12593	C₂	1	1	yes	maximal, characteristic, complemented
3	#12021	C₂	1	1	yes	maximal, characteristic, complemented
4	#11675	C₂	1	2	yes	maximal, complemented
5	#9762	C₂	1	2	yes	maximal, complemented
6	#3169	C₂²	1	2	yes	—
7	#2254	C₄	1	1	yes	characteristic, complemented
8	#2246	C₄	1	1	yes	characteristic, complemented
9	#2240	C₄	1	2	yes	complemented
10	#2218	—	2	2	no	—
11	#2177	—	2	2	no	—
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₄×M₃₂	C₂²	1	1	yes	characteristic
14	#1726	C₂²	1	1	yes	characteristic
15	#1568	C₂²	1	2	yes	—
16	#1530	—	2	2	no	—
17	#1505	—	2	2	no	—
18	#1375	—	2	2	no	—
19	#1368	—	2	2	no	—
20	#1331	—	2	4	no	—
21	#532	—	2	2	no	—
22	C₂×D₁₆	—	2	2	no	—
23	C₂×M₃₂	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	#500	C₂×C₄	1	2	yes	—
26	#493	C₂×C₄	1	2	yes	—
27	#486	—	2	2	no	—
28	#466	—	2	2	no	—
29	#466	—	2	2	no	—
30	C₄×D₈	—	2	2	no	—
31	C₄×D₈	—	2	2	no	—
32	#432	—	2	2	no	—
33	C₂×#97	—	2	1	no	—
34	C₂×#97	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	—	2	1	no	—
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	#376	D₄	1	1	yes	characteristic
38	#367	—	2	2	no	—
39	#357	—	2	4	no	—
40	#346	—	2	1	no	—
41	#346	D₄	1	2	yes	—
42	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₄×C₁₆	D₄	1	2	yes	—
45	#335	—	2	2	no	—
46	#334	—	2	2	no	—
47	#329	—	4	4	no	—
48	#327	—	4	4	no	—
49	C₂×#41	—	4	2	no	—
50	C₂×D₈	—	2	2	no	—
51	C₂×D₈	—	2	2	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₈	#41	1	1	yes	characteristic
55	C₄⋊Q₈	—	2	2	no	—
56	#127	—	2	2	no	—
57	#124	—	4	2	no	—
58	#121	—	4	2	no	—
59	#121	—	4	2	no	—
60	C₄×D₄	—	4	2	no	—
61	C₄×D₄	—	4	2	no	—
62	C₂×C₄²	—	2	1	no	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	D₁₆	—	4	4	no	—
65	M₃₂	—	4	2	no	—
66	M₃₂	—	4	2	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
69	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
70	#107	—	2	2	no	—
71	#107	—	2	2	no	—
72	C₄⋊C₈	—	4	2	no	—
73	#102	—	4	2	no	—
74	#102	—	4	2	no	—
75	C₂²⋊C₈	—	4	2	no	—
76	#97	—	2	2	no	—
77	#97	—	4	2	no	—
78	#97	—	4	2	no	—
79	C₄×C₈	—	2	4	no	—
80	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₄×C₈	C₂×D₄	1	1	yes	characteristic
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	#95	—	2	2	no	—
85	#41	—	16	4	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂×D₄	—	4	2	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	—	8	2	no	—
91	C₂²×C₄	—	4	1	no	—
92	C₂²×C₄	—	8	2	no	—
93	C₂²×C₄	#121	1	1	yes	characteristic
94	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
95	D₈	—	4	4	no	—
96	D₈	—	4	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	8	2	no	—
103	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈	C₄×D₄	1	1	yes	characteristic
105	C₂×C₈	#121	1	1	yes	characteristic
106	C₂×C₈	#124	1	1	yes	characteristic
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₂²⋊C₄	—	8	2	no	—
110	C₂²⋊C₄	—	8	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	4	1	no	—
113	C₄²	C₄×D₄	1	1	yes	characteristic
114	C₄²	#127	1	1	yes	characteristic
115	C₄²	AGL(1,ℤ/8)	1	2	yes	—
116	C₁₆	—	2	2	no	—
117	C₁₆	—	2	2	no	—
118	C₁₆	—	4	2	no	—
119	C₂³	—	8	2	no	—
120	C₂³	#469	1	1	yes	characteristic
121	Q₈	—	16	2	no	—
122	D₄	—	8	2	no	—
123	D₄	—	8	2	no	—
124	D₄	—	16	2	no	—
125	D₄	—	16	2	no	—
126	D₄	—	16	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	—	16	4	no	—
134	C₂×C₄	#390	1	1	yes	characteristic
135	C₂×C₄	#442	1	2	yes	—
136	C₂×C₄	#469	1	1	yes	characteristic
137	C₂×C₄	#496	1	2	yes	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	2	no	—
141	C₈	—	16	2	no	—
142	C₈	#421	1	2	yes	—
143	C₈	#443	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	16	2	no	—
147	C₂²	—	16	2	no	—
148	C₂²	—	16	2	no	—
149	C₂²	#1972	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	8	2	no	—
154	C₄	—	16	2	no	—
155	C₄	#1963	1	1	yes	characteristic
156	C₄	#1974	1	1	yes	characteristic
157	C₂	—	4	1	no	—
158	C₂	—	32	2	no	—
159	C₂	#9852	1	1	yes	characteristic
160	C₂	#13247	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33748

C₂

yes

maximal, characteristic

#12593

C₂

yes

maximal, characteristic, complemented

#12021

C₂

yes

maximal, characteristic, complemented

#11675

C₂

yes

maximal, complemented

#9762

C₂

yes

maximal, complemented

#3169

C₂²

yes

—

#2254

C₄

yes

characteristic, complemented

#2246

C₄

yes

characteristic, complemented

#2240

C₄

yes

complemented

#2218

—

#2177

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

#1726

C₂²

yes

characteristic

#1568

C₂²

yes

—

—

—

—

—

—

—

C₂×D₁₆

—

C₂×M₃₂

—

C₂×M₃₂

D₄

yes

—

#500

C₂×C₄

yes

—

#493

C₂×C₄

yes

—

#486

—

#466

—

#466

—

C₄×D₈

—

C₄×D₈

—

#432

—

C₂×#97

—

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

#376

D₄

yes

—

—

—

D₄

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

—

C₂×#41

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

#107

—

#107

—

C₄⋊C₈

—

#102

—

#102

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₈

C₄×D₄

yes

characteristic

105

C₂×C₈

#121

yes

characteristic

106

C₂×C₈

#124

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₄×D₄

yes

characteristic

114

C₄²

#127

yes

characteristic

115

C₄²

AGL(1,ℤ/8)

yes

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₂³

—

120

C₂³

#469

yes

characteristic

121

Q₈

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#390

yes

characteristic

135

C₂×C₄

#442

yes

—

136

C₂×C₄

#469

yes

characteristic

137

C₂×C₄

#496

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

#421

yes

—

143

C₈

#443

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#1972

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#1963

yes

characteristic

156

C₄

#1974

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

#9852

yes

characteristic

160

C₂

#13247

yes

—

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33748	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1972

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

#33748

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #12593	C₂	semidirect
3. #12021	C₂	semidirect
4. #11675	C₂	semidirect
5. #9762	C₂	semidirect
7. #2254	C₄	semidirect
8. #2246	C₄	semidirect
9. #2240	C₄	semidirect

H ≅ G/N

type

2. #12593

C₂

semidirect

3. #12021

C₂

semidirect

4. #11675

C₂

semidirect

5. #9762

C₂

semidirect

7. #2254

C₄

semidirect

8. #2246

C₄

semidirect

9. #2240

C₄

semidirect

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

subgroup	index
#33748	2
#12593	2
#12021	2
#11675	2
#9762	2

subgroup

index

Subgroups — #141809

Conjugacy classes of subgroups

Complements

Maximal subgroups