Group #141808 — subgroups

Subgroups — #141808

Order 512.

285 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 78 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	#33749	C₂	1	1	yes	maximal, characteristic
2	#12594	C₂	1	1	yes	maximal, characteristic
3	#12020	C₂	1	1	yes	maximal, characteristic
4	#11681	C₂	1	2	yes	maximal
5	#9850	C₂	1	2	yes	maximal
6	#3170	C₂²	1	2	yes	—
7	#2254	C₄	1	1	yes	characteristic, complemented
8	#2247	C₄	1	1	yes	characteristic, complemented
9	#2241	C₄	1	2	yes	complemented
10	#2219	—	2	2	no	—
11	#2176	—	2	2	no	—
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₄×M₃₂	C₂²	1	1	yes	characteristic
14	#1994	—	2	2	no	—
15	C₄×#97	C₂²	1	1	yes	characteristic
16	#1566	C₂²	1	2	yes	—
17	#1493	—	2	2	no	—
18	#1375	—	2	2	no	—
19	#1368	—	2	2	no	—
20	#1334	—	2	4	no	—
21	#533	—	2	2	no	—
22	C₂×Q₃₂	—	2	2	no	—
23	C₂×M₃₂	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	#500	C₂×C₄	1	2	yes	—
26	#494	C₂×C₄	1	2	yes	—
27	#487	—	2	2	no	—
28	#467	—	2	2	no	—
29	#467	—	2	2	no	—
30	C₄×Q₁₆	—	2	2	no	—
31	C₄×Q₁₆	—	2	2	no	—
32	#423	—	2	2	no	—
33	C₂×#97	—	2	1	no	—
34	C₂×#97	—	2	1	no	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	C₄³	D₄	1	1	yes	characteristic
38	#367	—	2	2	no	—
39	#358	—	2	4	no	—
40	#346	D₄	1	2	yes	—
41	#346	D₄	1	2	yes	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
45	#339	—	4	4	no	—
46	#334	—	2	2	no	—
47	#334	—	2	2	no	—
48	Q₈⋊C₈	—	4	4	no	—
49	C₂×Q₁₆	—	2	2	no	—
50	C₂×Q₁₆	—	2	2	no	—
51	C₂×M₁₆	—	4	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	C₄⋊Q₈	—	2	2	no	—
57	#125	—	4	2	no	—
58	C₂²⋊Q₈	—	4	2	no	—
59	C₂²⋊Q₈	—	4	2	no	—
60	C₄×Q₈	—	4	2	no	—
61	C₄×Q₈	—	4	2	no	—
62	#117	—	4	2	no	—
63	C₂×C₄²	—	2	1	no	—
64	C₂×C₄²	—	2	2	no	—
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	Q₃₂	—	4	4	no	—
67	M₃₂	—	4	2	no	—
68	M₃₂	—	4	2	no	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
71	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
72	#107	—	2	2	no	—
73	#107	—	2	2	no	—
74	C₄⋊C₈	—	4	2	no	—
75	C₄⋊C₈	—	4	2	no	—
76	#103	—	4	2	no	—
77	#103	—	4	2	no	—
78	#97	—	2	4	no	—
79	#97	—	4	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	C₂×D₄	1	1	yes	characteristic
84	C₄×C₈	#41	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	#121	1	1	yes	characteristic
89	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
90	Q₁₆	—	4	4	no	—
91	Q₁₆	—	4	4	no	—
92	M₁₆	—	8	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	8	2	no	—
99	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₈	C₄×D₄	1	1	yes	characteristic
101	C₂×C₈	#121	1	1	yes	characteristic
102	C₂×C₈	#124	1	1	yes	characteristic
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	8	2	no	—
108	C₄⋊C₄	—	8	2	no	—
109	C₂²⋊C₄	—	8	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	4	1	no	—
112	C₄²	—	4	4	no	—
113	C₄²	—	8	2	no	—
114	C₄²	C₄×D₄	1	1	yes	characteristic
115	C₄²	#127	1	1	yes	characteristic
116	C₄²	AGL(1,ℤ/8)	1	2	yes	—
117	C₁₆	—	2	2	no	—
118	C₁₆	—	2	2	no	—
119	C₁₆	—	4	2	no	—
120	C₂³	#469	1	1	yes	characteristic
121	Q₈	—	8	4	no	—
122	Q₈	—	8	2	no	—
123	Q₈	—	8	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	#390	1	1	yes	characteristic
132	C₂×C₄	#442	1	2	yes	—
133	C₂×C₄	#469	1	1	yes	characteristic
134	C₂×C₄	#496	1	2	yes	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	2	no	—
138	C₈	—	16	2	no	—
139	C₈	#421	1	2	yes	—
140	C₈	#443	1	2	yes	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	2	no	—
143	C₂²	#1972	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	2	no	—
146	C₄	—	8	2	no	—
147	C₄	—	16	2	no	—
148	C₄	—	16	2	no	—
149	C₄	#1963	1	1	yes	characteristic
150	C₄	#1974	1	1	yes	characteristic
151	C₄	#2180	1	4	yes	—
152	C₂	—	4	1	no	—
153	C₂	#9852	1	1	yes	characteristic
154	C₂	#13250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33749

C₂

yes

maximal, characteristic

#12594

C₂

yes

maximal, characteristic

#12020

C₂

yes

maximal, characteristic

#11681

C₂

yes

maximal

#9850

C₂

yes

maximal

#3170

C₂²

yes

—

#2254

C₄

yes

characteristic, complemented

#2247

C₄

yes

characteristic, complemented

#2241

C₄

yes

complemented

#2219

—

#2176

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

#1994

—

C₄×#97

C₂²

yes

characteristic

#1566

C₂²

yes

—

—

—

—

—

—

C₂×Q₃₂

—

C₂×M₃₂

—

C₂×M₃₂

D₄

yes

—

#500

C₂×C₄

yes

—

#494

C₂×C₄

yes

—

#487

—

#467

—

#467

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#423

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

—

—

D₄

yes

—

#346

D₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

—

—

—

Q₈⋊C₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₈

C₄×D₄

yes

characteristic

101

C₂×C₈

#121

yes

characteristic

102

C₂×C₈

#124

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₄×D₄

yes

characteristic

115

C₄²

#127

yes

characteristic

116

C₄²

AGL(1,ℤ/8)

yes

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₂³

#469

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#390

yes

characteristic

132

C₂×C₄

#442

yes

—

133

C₂×C₄

#469

yes

characteristic

134

C₂×C₄

#496

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

#421

yes

—

140

C₈

#443

yes

—

141

C₂²

—

142

C₂²

—

143

C₂²

#1972

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#1963

yes

characteristic

150

C₄

#1974

yes

characteristic

151

C₄

#2180

yes

—

152

C₂

—

153

C₂

#9852

yes

characteristic

154

C₂

#13250

yes

—

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ 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

C₂³

#469

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
7. #2254	C₄	semidirect
8. #2247	C₄	semidirect
9. #2241	C₄	semidirect

H ≅ G/N

type

7. #2254

C₄

semidirect

8. #2247

C₄

semidirect

9. #2241

C₄

semidirect

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

subgroup	index
#33749	2
#12594	2
#12020	2
#11681	2
#9850	2

subgroup

index

Subgroups — #141808

Conjugacy classes of subgroups

Complements

Maximal subgroups