Group #102288 — subgroups

Subgroups — #102288

Order 512.

448 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 166 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	#33310	C₂	1	1	yes	maximal, characteristic
2	#9896	C₂	1	2	yes	maximal
3	#9691	C₂	1	2	yes	maximal
4	#8806	C₂	1	1	yes	maximal, characteristic
5	#8260	C₂	1	1	yes	maximal, characteristic
6	C₂×#498	C₄	1	4	yes	—
7	#2868	C₂²	1	2	yes	—
8	#1844	—	2	1	no	—
9	#1844	—	2	2	no	—
10	#1844	—	2	1	no	—
11	#1842	C₂²	1	1	yes	characteristic
12	#1842	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₈	C₂²	1	2	yes	—
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×#335	—	2	2	no	—
16	C₂×#333	—	2	4	no	—
17	C₂×#322	—	2	1	no	—
18	#1296	—	2	4	no	—
19	#1273	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×C₄⋊Q₈	C₂×C₄	1	4	yes	—
23	#498	—	4	8	no	—
24	C₂×#106	—	2	4	no	—
25	C₂×#106	—	2	4	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	—	2	1	no	—
30	C₂×C₄⋊C₈	D₄	1	4	yes	—
31	C₂×C₄×C₈	—	2	2	no	—
32	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	2	yes	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	#384	—	2	2	no	—
37	#384	—	2	2	no	—
38	#384	—	2	2	no	—
39	#384	—	2	2	no	—
40	C₄³	C₂³	1	1	yes	characteristic
41	#336	—	2	2	no	—
42	#335	—	4	4	no	—
43	#333	—	4	8	no	—
44	#322	—	4	2	no	—
45	C₂²×Q₈	—	2	4	no	—
46	C₂²×C₈	—	2	4	no	—
47	C₂²×C₈	—	4	1	no	—
48	C₂²×C₈	—	4	1	no	—
49	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
51	C₄⋊Q₈	—	4	8	no	—
52	C₄⋊Q₈	—	4	4	no	—
53	C₄⋊Q₈	—	4	4	no	—
54	C₄⋊Q₈	—	8	4	no	—
55	C₂×C₄⋊C₄	—	2	4	no	—
56	C₂×C₄⋊C₄	—	2	4	no	—
57	C₂×C₄⋊C₄	—	4	2	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₄	1	2	yes	—
65	C₂×C₄²	#41	1	2	yes	—
66	#106	—	4	8	no	—
67	#106	—	4	8	no	—
68	C₄⋊C₈	—	2	4	no	—
69	C₄⋊C₈	—	2	4	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	4	2	no	—
72	C₄⋊C₈	—	4	4	no	—
73	C₄⋊C₈	—	4	4	no	—
74	C₄×C₈	—	2	4	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	—	4	2	no	—
77	C₄×C₈	—	4	2	no	—
78	C₄×C₈	SD₁₆	1	2	yes	—
79	C₄×C₈	SD₁₆	1	4	yes	—
80	C₄×C₈	Q₁₆	1	2	yes	—
81	C₄×C₈	Q₁₆	1	4	yes	—
82	C₂×Q₈	—	4	8	no	—
83	C₂×Q₈	—	8	4	no	—
84	C₂×Q₈	—	8	4	no	—
85	C₂²×C₄	—	4	4	no	—
86	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₄	1	2	yes	—
88	C₂²×C₄	#121	1	2	yes	—
89	C₂²×C₄	#124	1	1	yes	characteristic
90	C₂²×C₄	#127	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	4	no	—
94	C₂×C₈	—	4	4	no	—
95	C₂×C₈	—	4	4	no	—
96	C₂×C₈	—	8	1	no	—
97	C₂×C₈	—	8	1	no	—
98	C₂×C₈	—	8	2	no	—
99	C₂×C₈	—	8	2	no	—
100	C₂×C₈	#100	1	4	yes	—
101	C₂×C₈	#103	1	4	yes	—
102	C₂×C₈	#103	1	4	yes	—
103	C₂×C₈	#103	1	4	yes	—
104	C₄⋊C₄	—	4	8	no	—
105	C₄⋊C₄	—	8	4	no	—
106	C₄⋊C₄	—	8	4	no	—
107	C₄⋊C₄	—	8	4	no	—
108	C₄⋊C₄	—	8	4	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₄⋊C₄	—	8	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	4	no	—
113	C₄²	C₂×SD₁₆	1	1	yes	characteristic
114	C₄²	C₂×SD₁₆	1	2	yes	—
115	C₄²	C₂×Q₁₆	1	1	yes	characteristic
116	C₄²	C₂×Q₁₆	1	2	yes	—
117	C₄²	AGL(1,ℤ/8)	1	4	yes	—
118	C₄²	#137	1	2	yes	—
119	C₄²	#137	1	4	yes	—
120	C₂³	#390	1	1	yes	characteristic
121	Q₈	—	16	8	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	8	4	no	—
126	C₂×C₄	—	8	4	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	C₂×#100	1	2	yes	—
129	C₂×C₄	C₂×#103	1	1	yes	characteristic
130	C₂×C₄	C₂×#103	1	2	yes	—
131	C₂×C₄	#419	1	1	yes	characteristic
132	C₂×C₄	#440	1	4	yes	—
133	C₂×C₄	#461	1	2	yes	—
134	C₂×C₄	#462	1	2	yes	—
135	C₂×C₄	#468	1	4	yes	—
136	C₂×C₄	#471	1	4	yes	—
137	C₂×C₄	#487	1	2	yes	—
138	C₂×C₄	#492	1	1	yes	characteristic
139	C₂×C₄	#494	1	1	yes	characteristic
140	C₂×C₄	#496	1	2	yes	—
141	C₂×C₄	#497	1	2	yes	—
142	C₈	—	2	4	no	—
143	C₈	—	8	4	no	—
144	C₈	—	16	1	no	—
145	C₈	—	16	1	no	—
146	C₈	#333	1	8	yes	complemented
147	C₂²	#1956	1	1	yes	characteristic
148	C₂²	#1965	1	1	yes	characteristic
149	C₂²	#1973	1	2	yes	—
150	C₂²	#1977	1	2	yes	—
151	C₂²	#1981	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	4	no	—
154	C₄	—	16	4	no	—
155	C₄	C₂×#333	1	2	yes	—
156	C₄	#1548	1	2	yes	—
157	C₄	#1691	1	4	yes	—
158	C₄	#1692	1	4	yes	—
159	C₄	#1715	1	1	yes	characteristic
160	C₄	#1716	1	1	yes	characteristic
161	C₄	#1723	1	2	yes	—
162	C₂	#9691	1	1	yes	characteristic
163	C₂	#9741	1	2	yes	—
164	C₂	#9767	1	1	yes	characteristic
165	C₂	#9805	1	2	yes	—
166	C₂	#9841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33310

C₂

yes

maximal, characteristic

#9896

C₂

yes

maximal

#9691

C₂

yes

maximal

#8806

C₂

yes

maximal, characteristic

#8260

C₂

yes

maximal, characteristic

C₂×#498

C₄

yes

—

#2868

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#1842

C₂²

yes

characteristic

C₄×C₄⋊C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂×C₄

yes

—

#498

—

C₂×#106

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

—

—

—

C₄³

C₂³

yes

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

#41

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄×D₄

yes

—

C₂²×C₄

#121

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

#100

yes

—

101

C₂×C₈

#103

yes

—

102

C₂×C₈

#103

yes

—

103

C₂×C₈

#103

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂×SD₁₆

yes

characteristic

114

C₄²

C₂×SD₁₆

yes

—

115

C₄²

C₂×Q₁₆

yes

characteristic

116

C₄²

C₂×Q₁₆

yes

—

117

C₄²

AGL(1,ℤ/8)

yes

—

118

C₄²

#137

yes

—

119

C₄²

#137

yes

—

120

C₂³

#390

yes

characteristic

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×#100

yes

—

129

C₂×C₄

C₂×#103

yes

characteristic

130

C₂×C₄

C₂×#103

yes

—

131

C₂×C₄

#419

yes

characteristic

132

C₂×C₄

#440

yes

—

133

C₂×C₄

#461

yes

—

134

C₂×C₄

#462

yes

—

135

C₂×C₄

#468

yes

—

136

C₂×C₄

#471

yes

—

137

C₂×C₄

#487

yes

—

138

C₂×C₄

#492

yes

characteristic

139

C₂×C₄

#494

yes

characteristic

140

C₂×C₄

#496

yes

—

141

C₂×C₄

#497

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#333

yes

complemented

147

C₂²

#1956

yes

characteristic

148

C₂²

#1965

yes

characteristic

149

C₂²

#1973

yes

—

150

C₂²

#1977

yes

—

151

C₂²

#1981

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

C₂×#333

yes

—

156

C₄

#1548

yes

—

157

C₄

#1691

yes

—

158

C₄

#1692

yes

—

159

C₄

#1715

yes

characteristic

160

C₄

#1716

yes

characteristic

161

C₄

#1723

yes

—

162

C₂

#9691

yes

characteristic

163

C₂

#9741

yes

—

164

C₂

#9767

yes

characteristic

165

C₂

#9805

yes

—

166

C₂

#9841

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#390

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
146. C₈	#333	semidirect

H ≅ G/N

type

146. C₈

#333

semidirect

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

subgroup	index
#33310	2
#9896	2
#9691	2
#8806	2
#8260	2

subgroup

index

Subgroups — #102288

Conjugacy classes of subgroups

Complements

Maximal subgroups