Group #145516 — subgroups

Subgroups — #145516

Order 512.

277 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 72 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	#33815	C₂	1	1	yes	maximal, characteristic
2	#12879	C₂	1	1	yes	maximal, characteristic
3	#11983	C₂	1	2	yes	maximal
4	#9845	C₂	1	2	yes	maximal
5	#7241	C₂	1	1	yes	maximal, characteristic
6	#3170	C₂²	1	2	yes	—
7	#2227	—	2	2	no	—
8	#2216	—	2	2	no	—
9	#2209	—	2	2	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	#1994	—	2	2	no	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	#1567	C₂²	1	2	yes	—
15	#1493	—	2	2	no	—
16	#1374	—	2	2	no	—
17	#1365	—	2	2	no	—
18	#1362	—	2	2	no	—
19	#533	—	2	2	no	—
20	C₂×Q₃₂	—	2	2	no	—
21	C₂²×C₁₆	—	2	1	no	—
22	C₂²×C₁₆	D₄	1	2	yes	—
23	#500	D₄	1	2	yes	—
24	#494	D₄	1	2	yes	—
25	#487	—	2	2	no	—
26	#467	—	2	2	no	—
27	#467	—	2	2	no	—
28	C₄×Q₁₆	—	2	2	no	—
29	C₄×Q₁₆	—	2	2	no	—
30	#423	—	2	2	no	—
31	C₂×C₄×C₈	—	2	1	no	—
32	C₂×C₄×C₈	—	2	1	no	—
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₄³	D₄	1	1	yes	characteristic
36	#366	—	2	2	no	—
37	C₄⋊C₁₆	—	2	2	no	—
38	C₄⋊C₁₆	—	2	2	no	—
39	#358	—	2	4	no	—
40	#339	—	4	2	no	—
41	#339	—	4	2	no	—
42	#335	—	2	2	no	—
43	#335	—	2	2	no	—
44	Q₈⋊C₈	—	4	4	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂×M₁₆	—	4	2	no	—
48	C₂²×C₈	—	2	1	no	—
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₈	#41	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	2	no	—
52	C₄⋊Q₈	—	2	2	no	—
53	#125	—	4	2	no	—
54	C₂²⋊Q₈	—	4	2	no	—
55	C₂²⋊Q₈	—	4	2	no	—
56	C₄×Q₈	—	4	2	no	—
57	C₄×Q₈	—	4	2	no	—
58	#117	—	4	2	no	—
59	C₂×C₄²	—	2	1	no	—
60	C₂×C₄²	—	2	1	no	—
61	C₂×C₄²	D₈	1	1	yes	characteristic
62	C₂×C₄²	D₈	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	Q₃₂	—	8	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	4	2	no	—
67	C₂×C₁₆	—	4	2	no	—
68	C₂×C₁₆	—	4	1	no	—
69	#107	—	2	2	no	—
70	#107	—	2	2	no	—
71	C₄⋊C₈	—	4	2	no	—
72	C₄⋊C₈	—	4	2	no	—
73	#103	—	4	2	no	—
74	#103	—	4	2	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	—	4	1	no	—
77	C₄×C₈	—	4	1	no	—
78	C₄×C₈	—	4	2	no	—
79	C₄×C₈	C₂×D₄	1	2	yes	—
80	C₄×C₈	#41	1	2	yes	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	4	2	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	#121	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
87	C₂²×C₄	#135	1	1	yes	characteristic
88	Q₁₆	—	4	2	no	—
89	Q₁₆	—	4	2	no	—
90	Q₁₆	—	8	2	no	—
91	M₁₆	—	8	2	no	—
92	M₁₆	—	8	2	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	8	2	no	—
100	C₂×C₈	#120	1	1	yes	characteristic
101	C₂×C₈	#123	1	2	yes	—
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	1	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	4	1	no	—
113	C₄²	—	4	1	no	—
114	C₄²	—	4	2	no	—
115	C₄²	—	8	2	no	—
116	C₄²	Q₃₂	1	2	yes	complemented
117	C₄²	#121	1	2	yes	—
118	C₄²	AGL(1,ℤ/8)	1	2	yes	—
119	C₁₆	—	4	2	no	—
120	C₁₆	—	8	1	no	—
121	C₂³	#466	1	1	yes	characteristic
122	Q₈	—	8	4	no	—
123	Q₈	—	8	2	no	—
124	Q₈	—	8	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	#394	1	1	yes	characteristic
137	C₂×C₄	#459	1	2	yes	—
138	C₂×C₄	#463	1	2	yes	—
139	C₂×C₄	#469	1	1	yes	characteristic
140	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
141	C₂×C₄	#508	1	1	yes	characteristic
142	C₂×C₄	#510	1	2	yes	—
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	2	no	—
146	C₈	—	16	2	no	—
147	C₈	#454	1	1	yes	characteristic
148	C₈	#455	1	1	yes	characteristic
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#2054	1	1	yes	characteristic
152	C₂²	#2216	1	1	yes	characteristic
153	C₂²	#2219	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	2	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	2	no	—
158	C₄	—	16	2	no	—
159	C₄	—	16	2	no	—
160	C₄	#2040	1	1	yes	characteristic
161	C₄	#2050	1	1	yes	characteristic
162	C₄	#2209	1	2	yes	—
163	C₄	#2211	1	2	yes	—
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#10519	1	1	yes	characteristic
167	C₂	#13444	1	1	yes	characteristic
168	C₂	#13450	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33815

C₂

yes

maximal, characteristic

#12879

C₂

yes

maximal, characteristic

#11983

C₂

yes

maximal

#9845

C₂

yes

maximal

#7241

C₂

yes

maximal, characteristic

#3170

C₂²

yes

—

#2227

—

#2216

—

#2209

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

#1994

—

C₄²×C₈

C₂²

yes

characteristic

#1567

C₂²

yes

—

—

—

—

—

—

C₂×Q₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

#500

D₄

yes

—

#494

D₄

yes

—

#487

—

#467

—

#467

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

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

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

#120

yes

characteristic

101

C₂×C₈

#123

yes

—

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

—

115

C₄²

—

116

C₄²

Q₃₂

yes

complemented

117

C₄²

#121

yes

—

118

C₄²

AGL(1,ℤ/8)

yes

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

#466

yes

characteristic

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#394

yes

characteristic

137

C₂×C₄

#459

yes

—

138

C₂×C₄

#463

yes

—

139

C₂×C₄

#469

yes

characteristic

140

C₂×C₄

C₂×Q₃₂

yes

characteristic

141

C₂×C₄

#508

yes

characteristic

142

C₂×C₄

#510

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

#454

yes

characteristic

148

C₈

#455

yes

characteristic

149

C₂²

—

150

C₂²

—

151

C₂²

#2054

yes

characteristic

152

C₂²

#2216

yes

characteristic

153

C₂²

#2219

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#2040

yes

characteristic

161

C₄

#2050

yes

characteristic

162

C₄

#2209

yes

—

163

C₄

#2211

yes

—

164

C₂

—

165

C₂

—

166

C₂

#10519

yes

characteristic

167

C₂

#13444

yes

characteristic

168

C₂

#13450

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2054

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#466

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
116. C₄²	Q₃₂	semidirect

H ≅ G/N

type

116. C₄²

Q₃₂

semidirect

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

subgroup	index
#33815	2
#12879	2
#11983	2
#9845	2
#7241	2

subgroup

index

Subgroups — #145516

Conjugacy classes of subgroups

Complements

Maximal subgroups