Group #99308 — subgroups

Subgroups — #99308

Order 512.

368 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 198 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	#8806	C₂	1	1	yes	maximal, characteristic
2	#8806	C₂	1	2	yes	maximal
3	#8805	C₂	1	1	yes	maximal, characteristic
4	#8784	C₂	1	2	yes	maximal
5	C₄×#322	C₂	1	1	yes	maximal, characteristic
6	#1842	C₂²	1	1	yes	characteristic
7	#1842	C₂²	1	2	yes	—
8	#1842	C₂²	1	1	yes	characteristic
9	C₄²×C₈	C₂²	1	2	yes	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	—	2	4	no	—
12	C₂×#335	—	2	2	no	—
13	C₂×#334	—	2	2	no	—
14	C₂×#322	C₄	1	1	yes	characteristic
15	C₂×#322	C₄	1	2	yes	—
16	C₂×#322	C₄	1	1	yes	characteristic
17	#1277	—	2	4	no	—
18	C₂×C₄⋊C₈	—	2	1	no	—
19	C₂×C₄⋊C₈	—	2	1	no	—
20	C₂×C₄⋊C₈	—	2	2	no	—
21	C₂×C₄⋊C₈	—	2	2	no	—
22	C₂×C₄⋊C₈	—	2	2	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	—	2	2	no	—
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	4	yes	—
30	C₂×C₄×C₈	D₄	1	2	yes	—
31	C₂×C₄×C₈	Q₈	1	2	yes	—
32	C₂×C₄×C₈	Q₈	1	4	yes	—
33	C₄³	C₂³	1	1	yes	characteristic
34	#335	—	4	8	no	—
35	#335	—	4	4	no	—
36	#334	—	4	4	no	—
37	#322	—	2	4	no	—
38	#322	—	2	2	no	—
39	#322	C₈	1	4	yes	complemented
40	C₂²×C₈	—	4	1	no	—
41	C₂²×C₈	—	4	2	no	—
42	C₂²×C₈	—	4	1	no	—
43	C₂²×C₈	C₄⋊C₄	1	4	yes	—
44	C₂²×C₈	C₄⋊C₄	1	4	yes	—
45	C₂²×C₈	C₄⋊C₄	1	2	yes	—
46	C₂²×C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄²	C₂×D₄	1	2	yes	—
50	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄²	C₂×Q₈	1	2	yes	—
52	C₄⋊C₈	—	4	2	no	—
53	C₄⋊C₈	—	4	2	no	—
54	C₄⋊C₈	—	4	4	no	—
55	C₄⋊C₈	—	4	4	no	—
56	C₄⋊C₈	—	4	4	no	—
57	C₄⋊C₈	—	4	4	no	—
58	C₄⋊C₈	—	4	4	no	—
59	C₄×C₈	—	2	4	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	4	no	—
64	C₄×C₈	—	2	4	no	—
65	C₄×C₈	—	4	4	no	—
66	C₄×C₈	C₂×C₈	1	4	yes	—
67	C₄×C₈	C₂×C₈	1	2	yes	—
68	C₄×C₈	M₁₆	1	4	yes	—
69	C₄×C₈	M₁₆	1	2	yes	—
70	C₄×C₈	D₈	1	4	yes	—
71	C₄×C₈	Q₁₆	1	4	yes	—
72	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
74	C₂²×C₄	#127	1	1	yes	characteristic
75	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
76	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	8	1	no	—
85	C₂×C₈	—	8	2	no	—
86	C₂×C₈	—	8	1	no	—
87	C₂×C₈	—	8	2	no	—
88	C₂×C₈	—	8	2	no	—
89	C₂×C₈	—	8	2	no	—
90	C₂×C₈	—	8	2	no	—
91	C₂×C₈	C₄⋊C₈	1	8	yes	—
92	C₂×C₈	C₄⋊C₈	1	4	yes	—
93	C₂×C₈	#107	1	4	yes	—
94	C₂×C₈	#107	1	4	yes	—
95	C₂×C₈	#108	1	4	yes	—
96	C₄²	—	2	2	no	—
97	C₄²	—	2	4	no	—
98	C₄²	C₂²×C₈	1	1	yes	characteristic
99	C₄²	C₂×M₁₆	1	1	yes	characteristic
100	C₄²	C₂×M₁₆	1	2	yes	—
101	C₄²	C₂×D₈	1	2	yes	—
102	C₄²	C₂×Q₁₆	1	2	yes	—
103	C₄²	AGL(1,ℤ/8)	1	4	yes	—
104	C₄²	#137	1	4	yes	—
105	C₂³	#384	1	1	yes	characteristic
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
111	C₂×C₄	#423	1	1	yes	characteristic
112	C₂×C₄	#423	1	2	yes	—
113	C₂×C₄	C₂×#107	1	2	yes	—
114	C₂×C₄	#428	1	4	yes	—
115	C₂×C₄	C₂×#108	1	2	yes	—
116	C₂×C₄	#430	1	4	yes	—
117	C₂×C₄	#493	1	1	yes	characteristic
118	C₂×C₄	#494	1	1	yes	characteristic
119	C₂×C₄	#496	1	2	yes	—
120	C₂×C₄	#497	1	2	yes	—
121	C₂×C₄	#500	1	2	yes	—
122	C₂×C₄	#501	1	4	yes	—
123	C₈	—	2	8	no	—
124	C₈	—	2	4	no	—
125	C₈	—	4	4	no	—
126	C₈	—	16	1	no	—
127	C₈	—	16	2	no	—
128	C₈	—	16	1	no	—
129	C₈	#335	1	8	yes	complemented
130	C₂²	#1842	1	1	yes	characteristic
131	C₂²	#1845	1	1	yes	characteristic
132	C₂²	#1847	1	2	yes	—
133	C₂²	#1848	1	1	yes	characteristic
134	C₂²	#1849	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	—	2	4	no	—
137	C₄	C₂×#335	1	2	yes	—
138	C₄	#1565	1	4	yes	—
139	C₄	#1567	1	2	yes	—
140	C₄	#1569	1	4	yes	—
141	C₄	#1721	1	1	yes	characteristic
142	C₄	#1722	1	1	yes	characteristic
143	C₄	#1723	1	2	yes	—
144	C₂	#8806	1	1	yes	characteristic
145	C₂	#8808	1	2	yes	—
146	C₂	#8820	1	1	yes	characteristic
147	C₂	#8826	1	1	yes	characteristic
148	C₂	#8828	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#8806

C₂

yes

maximal, characteristic

#8806

C₂

yes

maximal

#8805

C₂

yes

maximal, characteristic

#8784

C₂

yes

maximal

C₄×#322

C₂

yes

maximal, characteristic

#1842

C₂²

yes

characteristic

#1842

C₂²

yes

—

#1842

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

—

—

C₄

yes

characteristic

C₂×#322

C₄

yes

—

C₂×#322

C₄

yes

characteristic

#1277

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

—

—

—

—

—

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

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

Q₁₆

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×C₄⋊C₄

yes

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂²×C₄

C₄⋊Q₈

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

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

100

C₄²

C₂×M₁₆

yes

—

101

C₄²

C₂×D₈

yes

—

102

C₄²

C₂×Q₁₆

yes

—

103

C₄²

AGL(1,ℤ/8)

yes

—

104

C₄²

#137

yes

—

105

C₂³

#384

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

110

C₂×C₄

C₂×C₄⋊C₈

yes

—

111

C₂×C₄

#423

yes

characteristic

112

C₂×C₄

#423

yes

—

113

C₂×C₄

C₂×#107

yes

—

114

C₂×C₄

#428

yes

—

115

C₂×C₄

C₂×#108

yes

—

116

C₂×C₄

#430

yes

—

117

C₂×C₄

#493

yes

characteristic

118

C₂×C₄

#494

yes

characteristic

119

C₂×C₄

#496

yes

—

120

C₂×C₄

#497

yes

—

121

C₂×C₄

#500

yes

—

122

C₂×C₄

#501

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#335

yes

complemented

130

C₂²

#1842

yes

characteristic

131

C₂²

#1845

yes

characteristic

132

C₂²

#1847

yes

—

133

C₂²

#1848

yes

characteristic

134

C₂²

#1849

yes

—

135

C₄

—

136

C₄

—

137

C₄

C₂×#335

yes

—

138

C₄

#1565

yes

—

139

C₄

#1567

yes

—

140

C₄

#1569

yes

—

141

C₄

#1721

yes

characteristic

142

C₄

#1722

yes

characteristic

143

C₄

#1723

yes

—

144

C₂

#8806

yes

characteristic

145

C₂

#8808

yes

—

146

C₂

#8820

yes

characteristic

147

C₂

#8826

yes

characteristic

148

C₂

#8828

yes

—

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#384
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#127

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₄³

C₂³

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
39. #322	C₈	semidirect
129. C₈	#335	semidirect

H ≅ G/N

type

39. #322

C₈

semidirect

129. C₈

#335

semidirect

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

subgroup	index
#8806	2
#8805	2
#8784	2
C₄×#322	2

subgroup

index

Subgroups — #99308

Conjugacy classes of subgroups

Complements

Maximal subgroups