Group #99310 — subgroups

Subgroups — #99310

Order 512.

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

normal subgroups only

This group has 346 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	#8806	C₂	1	1	yes	maximal, characteristic
2	#8805	C₂	1	1	yes	maximal, characteristic
3	#8805	C₂	1	2	yes	maximal
4	#8784	C₂	1	2	yes	maximal
5	#8131	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	2	no	—
12	C₂×#334	—	2	4	no	—
13	C₂×#334	—	2	2	no	—
14	#1277	—	2	4	no	—
15	#1273	C₄	1	1	yes	characteristic
16	#1273	C₄	1	2	yes	—
17	#1273	C₄	1	1	yes	characteristic
18	C₂×C₄⋊C₈	—	2	2	no	—
19	C₂×C₄⋊C₈	—	2	2	no	—
20	C₂×C₄⋊C₈	—	2	2	no	—
21	C₂×C₄⋊C₈	—	2	1	no	—
22	C₂×C₄⋊C₈	—	2	1	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	1	yes	characteristic
28	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₈	D₄	1	4	yes	—
30	C₂×C₄×C₈	D₄	1	2	yes	—
31	C₂×C₄×C₈	Q₈	1	4	yes	—
32	C₂×C₄×C₈	Q₈	1	2	yes	—
33	C₄³	C₂³	1	1	yes	characteristic
34	#335	—	4	4	no	—
35	#334	—	4	8	no	—
36	#334	—	4	4	no	—
37	C₂²×C₈	—	4	1	no	—
38	C₂²×C₈	—	4	1	no	—
39	C₂²×C₈	—	4	2	no	—
40	C₂²×C₈	C₄⋊C₄	1	4	yes	—
41	C₂²×C₈	C₄⋊C₄	1	2	yes	—
42	C₂²×C₈	C₄⋊C₄	1	2	yes	—
43	C₂²×C₈	C₄⋊C₄	1	4	yes	—
44	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄²	C₂×D₄	1	2	yes	—
47	C₂×C₄²	C₂×Q₈	1	2	yes	—
48	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₈	—	4	4	no	—
50	C₄⋊C₈	—	4	2	no	—
51	C₄⋊C₈	—	4	2	no	—
52	C₄⋊C₈	—	4	4	no	—
53	C₄⋊C₈	—	4	4	no	—
54	C₄⋊C₈	—	4	4	no	—
55	C₄⋊C₈	—	4	4	no	—
56	C₄×C₈	—	2	8	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	4	no	—
65	C₄×C₈	—	2	4	no	—
66	C₄×C₈	—	4	2	no	—
67	C₄×C₈	SD₁₆	1	4	yes	—
68	C₄×C₈	SD₁₆	1	4	yes	—
69	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
71	C₂²×C₄	#127	1	1	yes	characteristic
72	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
73	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	2	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	—	8	2	no	—
84	C₂×C₈	—	8	1	no	—
85	C₂×C₈	—	8	1	no	—
86	C₂×C₈	—	8	2	no	—
87	C₂×C₈	—	8	2	no	—
88	C₂×C₈	—	8	2	no	—
89	C₂×C₈	—	8	2	no	—
90	C₂×C₈	#106	1	4	yes	—
91	C₂×C₈	#106	1	4	yes	—
92	C₂×C₈	#108	1	8	yes	—
93	C₄²	—	2	4	no	—
94	C₄²	—	2	2	no	—
95	C₄²	#131	1	1	yes	characteristic
96	C₄²	#131	1	2	yes	—
97	C₄²	#131	1	1	yes	characteristic
98	C₄²	C₂×SD₁₆	1	2	yes	—
99	C₄²	C₂×SD₁₆	1	2	yes	—
100	C₄²	AGL(1,ℤ/8)	1	4	yes	—
101	C₄²	#137	1	4	yes	—
102	C₂³	#384	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	#424	1	1	yes	characteristic
107	C₂×C₄	#424	1	1	yes	characteristic
108	C₂×C₄	#424	1	2	yes	—
109	C₂×C₄	#424	1	2	yes	—
110	C₂×C₄	C₂×#106	1	2	yes	—
111	C₂×C₄	#428	1	4	yes	—
112	C₂×C₄	C₂×#108	1	4	yes	—
113	C₂×C₄	#430	1	2	yes	—
114	C₂×C₄	#492	1	1	yes	characteristic
115	C₂×C₄	#492	1	1	yes	characteristic
116	C₂×C₄	#496	1	2	yes	—
117	C₂×C₄	#497	1	2	yes	—
118	C₂×C₄	#498	1	2	yes	—
119	C₂×C₄	#501	1	4	yes	—
120	C₈	—	2	4	no	—
121	C₈	—	2	4	no	—
122	C₈	—	2	8	no	—
123	C₈	—	4	4	no	—
124	C₈	—	16	2	no	—
125	C₈	—	16	1	no	—
126	C₈	—	16	1	no	—
127	C₂²	#1843	1	1	yes	characteristic
128	C₂²	#1844	1	1	yes	characteristic
129	C₂²	#1847	1	2	yes	—
130	C₂²	#1848	1	2	yes	—
131	C₂²	#1849	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	4	no	—
134	C₄	#1570	1	2	yes	—
135	C₄	#1570	1	2	yes	—
136	C₄	#1573	1	4	yes	—
137	C₄	#1574	1	4	yes	—
138	C₄	#1714	1	2	yes	—
139	C₄	#1715	1	1	yes	characteristic
140	C₄	#1716	1	1	yes	characteristic
141	C₂	#8809	1	1	yes	characteristic
142	C₂	#8812	1	2	yes	—
143	C₂	#8819	1	1	yes	characteristic
144	C₂	#8822	1	1	yes	characteristic
145	C₂	#8828	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#8806

C₂

yes

maximal, characteristic

#8805

C₂

yes

maximal, characteristic

#8805

C₂

yes

maximal

#8784

C₂

yes

maximal

#8131

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

#1273

C₄

yes

—

#1273

C₄

yes

characteristic

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

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

—

—

—

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

—

C₂×C₄²

C₂×Q₈

yes

characteristic

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

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

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

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

#131

yes

characteristic

C₄²

#131

yes

—

C₄²

#131

yes

characteristic

C₄²

C₂×SD₁₆

yes

—

C₄²

C₂×SD₁₆

yes

—

100

C₄²

AGL(1,ℤ/8)

yes

—

101

C₄²

#137

yes

—

102

C₂³

#384

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#424

yes

characteristic

107

C₂×C₄

#424

yes

characteristic

108

C₂×C₄

#424

yes

—

109

C₂×C₄

#424

yes

—

110

C₂×C₄

C₂×#106

yes

—

111

C₂×C₄

#428

yes

—

112

C₂×C₄

C₂×#108

yes

—

113

C₂×C₄

#430

yes

—

114

C₂×C₄

#492

yes

characteristic

115

C₂×C₄

#492

yes

characteristic

116

C₂×C₄

#496

yes

—

117

C₂×C₄

#497

yes

—

118

C₂×C₄

#498

yes

—

119

C₂×C₄

#501

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₂²

#1843

yes

characteristic

128

C₂²

#1844

yes

characteristic

129

C₂²

#1847

yes

—

130

C₂²

#1848

yes

—

131

C₂²

#1849

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

#1570

yes

—

135

C₄

#1570

yes

—

136

C₄

#1573

yes

—

137

C₄

#1574

yes

—

138

C₄

#1714

yes

—

139

C₄

#1715

yes

characteristic

140

C₄

#1716

yes

characteristic

141

C₂

#8809

yes

characteristic

142

C₂

#8812

yes

—

143

C₂

#8819

yes

characteristic

144

C₂

#8822

yes

characteristic

145

C₂

#8828

yes

—

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#384
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#127

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

socle, prime radical

C₂³

#384

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

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
#8131	2

subgroup

index

Subgroups — #99310

Conjugacy classes of subgroups

Maximal subgroups