Group #99280 — subgroups

Subgroups — #99280

Order 512.

276 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 138 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	#8807	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	#7841	C₂	1	1	yes	maximal, characteristic
4	#7018	C₂	1	4	yes	maximal
5	#1831	C₂²	1	2	yes	—
6	#1831	C₂²	1	2	yes	—
7	#1744	C₂²	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#335	—	2	1	no	—
10	C₂×#335	—	2	1	no	—
11	C₂×#334	—	2	1	no	—
12	C₂×#334	—	2	1	no	—
13	C₂×#322	C₄	1	1	yes	characteristic
14	C₂×#322	C₄	1	1	yes	characteristic
15	C₂×C₈²	C₄	1	2	yes	—
16	C₂×C₄⋊C₈	—	2	2	no	—
17	C₂×C₄⋊C₈	—	2	2	no	—
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
25	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
26	C₄³	C₂³	1	1	yes	characteristic
27	#336	—	2	2	no	—
28	#336	—	2	2	no	—
29	#336	—	2	4	no	—
30	#336	—	2	4	no	—
31	#335	—	4	2	no	—
32	#335	—	4	2	no	—
33	#334	—	4	2	no	—
34	#334	—	4	2	no	—
35	#322	—	2	2	no	—
36	#322	C₈	1	4	yes	complemented
37	C₈²	—	2	4	no	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	—	2	2	no	—
40	C₂²×C₈	—	4	2	no	—
41	C₂²×C₈	—	4	2	no	—
42	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄²	#41	1	2	yes	—
50	C₂×C₄²	#41	1	2	yes	—
51	C₄⋊C₈	—	4	2	no	—
52	C₄⋊C₈	—	4	2	no	—
53	C₄⋊C₈	—	4	2	no	—
54	C₄⋊C₈	—	4	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	C₂×C₈	1	4	yes	—
61	C₄×C₈	C₂×C₈	1	2	yes	—
62	C₄×C₈	M₁₆	1	4	yes	—
63	C₄×C₈	M₁₆	1	2	yes	—
64	C₄×C₈	D₈	1	2	yes	—
65	C₄×C₈	SD₁₆	1	2	yes	—
66	C₄×C₈	SD₁₆	1	2	yes	—
67	C₄×C₈	Q₁₆	1	2	yes	—
68	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄	#117	1	2	yes	—
70	C₂²×C₄	#124	1	1	yes	characteristic
71	C₂²×C₄	#124	1	1	yes	characteristic
72	C₂²×C₄	#125	1	2	yes	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	—	4	2	no	—
83	C₂×C₈	—	8	4	no	—
84	C₂×C₈	—	8	4	no	—
85	C₂×C₈	—	8	2	no	—
86	C₂×C₈	—	8	2	no	—
87	C₂×C₈	C₄⋊C₈	1	2	yes	—
88	C₂×C₈	C₄⋊C₈	1	2	yes	—
89	C₂×C₈	#106	1	2	yes	—
90	C₂×C₈	#106	1	2	yes	—
91	C₂×C₈	#107	1	2	yes	—
92	C₂×C₈	#107	1	2	yes	—
93	C₂×C₈	#108	1	2	yes	—
94	C₂×C₈	#108	1	2	yes	—
95	C₄²	—	2	1	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	4	2	no	—
98	C₄²	—	4	2	no	—
99	C₄²	C₂²×C₈	1	1	yes	characteristic
100	C₄²	C₂×M₁₆	1	1	yes	characteristic
101	C₄²	C₂×M₁₆	1	2	yes	—
102	C₄²	C₂×D₈	1	1	yes	characteristic
103	C₄²	C₂×SD₁₆	1	1	yes	characteristic
104	C₄²	C₂×SD₁₆	1	1	yes	characteristic
105	C₄²	C₂×Q₁₆	1	1	yes	characteristic
106	C₂³	#382	1	1	yes	characteristic
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₄	#423	1	1	yes	characteristic
116	C₂×C₄	C₂×#106	1	1	yes	characteristic
117	C₂×C₄	C₂×#107	1	1	yes	characteristic
118	C₂×C₄	C₂×#108	1	1	yes	characteristic
119	C₂×C₄	C₂×#108	1	1	yes	characteristic
120	C₂×C₄	#431	1	2	yes	—
121	C₂×C₄	#432	1	2	yes	—
122	C₂×C₄	#486	1	1	yes	characteristic
123	C₂×C₄	#486	1	1	yes	characteristic
124	C₂×C₄	#487	1	1	yes	characteristic
125	C₂×C₄	#487	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	—	4	4	no	—
129	C₈	—	4	4	no	—
130	C₈	—	16	2	no	—
131	C₈	—	16	2	no	—
132	C₈	#334	1	4	yes	—
133	C₈	#335	1	4	yes	—
134	C₂²	#1831	1	1	yes	characteristic
135	C₂²	#1833	1	2	yes	—
136	C₂²	#1833	1	2	yes	—
137	C₂²	#1834	1	1	yes	characteristic
138	C₂²	#1837	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	2	no	—
146	C₄	C₂×#334	1	1	yes	characteristic
147	C₄	C₂×#335	1	1	yes	characteristic
148	C₄	#1566	1	1	yes	characteristic
149	C₄	#1567	1	1	yes	characteristic
150	C₂	#8781	1	2	yes	—
151	C₂	#8784	1	1	yes	characteristic
152	C₂	#8790	1	1	yes	characteristic
153	C₂	#8793	1	1	yes	characteristic
154	C₂	#8797	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#8807

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#7841

C₂

yes

maximal, characteristic

#7018

C₂

yes

maximal

#1831

C₂²

yes

—

#1831

C₂²

yes

—

#1744

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

—

—

—

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×C₈²

C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

C₂³

yes

—

—

—

—

—

—

—

—

—

C₈

yes

complemented

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

#41

yes

—

C₂×C₄²

#41

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#117

yes

—

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₈

yes

characteristic

100

C₄²

C₂×M₁₆

yes

characteristic

101

C₄²

C₂×M₁₆

yes

—

102

C₄²

C₂×D₈

yes

characteristic

103

C₄²

C₂×SD₁₆

yes

characteristic

104

C₄²

C₂×SD₁₆

yes

characteristic

105

C₄²

C₂×Q₁₆

yes

characteristic

106

C₂³

#382

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

115

C₂×C₄

#423

yes

characteristic

116

C₂×C₄

C₂×#106

yes

characteristic

117

C₂×C₄

C₂×#107

yes

characteristic

118

C₂×C₄

C₂×#108

yes

characteristic

119

C₂×C₄

C₂×#108

yes

characteristic

120

C₂×C₄

#431

yes

—

121

C₂×C₄

#432

yes

—

122

C₂×C₄

#486

yes

characteristic

123

C₂×C₄

#486

yes

characteristic

124

C₂×C₄

#487

yes

characteristic

125

C₂×C₄

#487

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

#334

yes

—

133

C₈

#335

yes

—

134

C₂²

#1831

yes

characteristic

135

C₂²

#1833

yes

—

136

C₂²

#1833

yes

—

137

C₂²

#1834

yes

characteristic

138

C₂²

#1837

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

C₂×#334

yes

characteristic

147

C₄

C₂×#335

yes

characteristic

148

C₄

#1566

yes

characteristic

149

C₄

#1567

yes

characteristic

150

C₂

#8781

yes

—

151

C₂

#8784

yes

characteristic

152

C₂

#8790

yes

characteristic

153

C₂

#8793

yes

characteristic

154

C₂

#8797

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#382

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₄³

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
36. #322	C₈	semidirect

H ≅ G/N

type

36. #322

C₈

semidirect

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

subgroup	index
#8807	2
#7841	2
#7018	2

subgroup

index

#8807

#7841

#7018

Subgroups — #99280

Conjugacy classes of subgroups

Complements

Maximal subgroups