Group #102272 — subgroups

Subgroups — #102272

Order 512.

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

normal subgroups only

This group has 464 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	#33312	C₂	1	1	yes	maximal, characteristic
2	#9692	C₂	1	4	yes	maximal
3	#8805	C₂	1	1	yes	maximal, characteristic
4	#8260	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	2	no	—
6	#2868	C₂²	1	2	yes	—
7	#1845	—	2	1	no	—
8	#1845	—	2	1	no	—
9	#1845	C₄	1	4	yes	—
10	#1842	C₂²	1	2	yes	—
11	C₄×C₄⋊C₈	C₂²	1	2	yes	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	C₂×#334	—	2	2	no	—
14	C₂×#332	—	2	8	no	—
15	C₂×#322	—	2	1	no	—
16	#1273	—	2	1	no	—
17	C₂×C₄⋊Q₈	—	2	4	no	—
18	C₂×C₄⋊Q₈	—	2	2	no	—
19	#500	—	4	4	no	—
20	#500	—	4	4	no	—
21	C₂×#107	—	2	4	no	—
22	C₂×#107	—	2	4	no	—
23	C₂×C₄⋊C₈	—	2	4	no	—
24	C₂×C₄⋊C₈	—	2	2	no	—
25	C₂×C₄⋊C₈	D₄	1	4	yes	—
26	C₂×C₄×C₈	—	2	2	no	—
27	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	2	yes	—
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	#384	—	2	2	no	—
32	#384	—	2	4	no	—
33	#384	C₂×C₄	1	4	yes	—
34	C₄³	C₂³	1	1	yes	characteristic
35	#336	—	2	2	no	—
36	#334	—	4	4	no	—
37	#332	—	4	16	no	—
38	#322	—	4	2	no	—
39	C₂²×Q₈	—	4	2	no	—
40	C₂²×C₈	—	2	4	no	—
41	C₂²×C₈	—	4	2	no	—
42	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
44	C₄⋊Q₈	—	4	4	no	—
45	C₄⋊Q₈	—	4	4	no	—
46	C₄⋊Q₈	—	4	8	no	—
47	C₄⋊Q₈	—	4	8	no	—
48	C₂×C₄⋊C₄	—	2	4	no	—
49	C₂×C₄⋊C₄	—	2	8	no	—
50	C₂×C₄⋊C₄	—	4	4	no	—
51	C₂×C₄⋊C₄	—	4	2	no	—
52	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄²	C₂×D₄	1	2	yes	—
56	C₂×C₄²	#41	1	2	yes	—
57	#107	—	4	8	no	—
58	#107	—	4	8	no	—
59	C₄⋊C₈	—	2	8	no	—
60	C₄⋊C₈	—	4	4	no	—
61	C₄⋊C₈	—	4	8	no	—
62	C₄×C₈	—	2	4	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	4	2	no	—
65	C₄×C₈	—	4	2	no	—
66	C₄×C₈	D₈	1	4	yes	—
67	C₄×C₈	SD₁₆	1	4	yes	—
68	C₄×C₈	SD₁₆	1	2	yes	—
69	C₄×C₈	Q₁₆	1	2	yes	—
70	C₂×Q₈	—	8	2	no	—
71	C₂×Q₈	—	8	2	no	—
72	C₂×Q₈	—	8	4	no	—
73	C₂×Q₈	—	8	4	no	—
74	C₂²×C₄	—	4	4	no	—
75	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₄	1	2	yes	—
77	C₂²×C₄	#121	1	2	yes	—
78	C₂²×C₄	#124	1	1	yes	characteristic
79	C₂²×C₄	#127	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	4	4	no	—
83	C₂×C₈	—	4	4	no	—
84	C₂×C₈	—	4	4	no	—
85	C₂×C₈	—	8	2	no	—
86	C₂×C₈	—	8	4	no	—
87	C₂×C₈	#101	1	4	yes	—
88	C₂×C₈	#102	1	4	yes	—
89	C₂×C₈	#102	1	4	yes	—
90	C₂×C₈	#103	1	4	yes	—
91	C₄⋊C₄	—	4	8	no	—
92	C₄⋊C₄	—	4	16	no	—
93	C₄⋊C₄	—	8	8	no	—
94	C₄⋊C₄	—	8	4	no	—
95	C₄²	—	2	2	no	—
96	C₄²	—	2	4	no	—
97	C₄²	C₂×D₈	1	2	yes	—
98	C₄²	C₂×SD₁₆	1	2	yes	—
99	C₄²	C₂×SD₁₆	1	1	yes	characteristic
100	C₄²	C₂×Q₁₆	1	1	yes	characteristic
101	C₄²	AGL(1,ℤ/8)	1	8	yes	—
102	C₄²	#137	1	2	yes	—
103	C₂³	#390	1	1	yes	characteristic
104	Q₈	—	16	2	no	—
105	Q₈	—	16	2	no	—
106	Q₈	—	16	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	4	no	—
111	C₂×C₄	—	8	8	no	—
112	C₂×C₄	C₂×#101	1	2	yes	—
113	C₂×C₄	C₂×#102	1	2	yes	—
114	C₂×C₄	C₂×#103	1	1	yes	characteristic
115	C₂×C₄	#419	1	1	yes	characteristic
116	C₂×C₄	#442	1	4	yes	—
117	C₂×C₄	#461	1	2	yes	—
118	C₂×C₄	#462	1	2	yes	—
119	C₂×C₄	#469	1	4	yes	—
120	C₂×C₄	#473	1	4	yes	—
121	C₂×C₄	#486	1	2	yes	—
122	C₂×C₄	#492	1	1	yes	characteristic
123	C₂×C₄	#493	1	1	yes	characteristic
124	C₂×C₄	#496	1	4	yes	—
125	C₈	—	2	4	no	—
126	C₈	—	8	4	no	—
127	C₈	—	16	2	no	—
128	C₈	#332	1	8	yes	complemented
129	C₂²	#1957	1	1	yes	characteristic
130	C₂²	#1966	1	1	yes	characteristic
131	C₂²	#1972	1	2	yes	—
132	C₂²	#1979	1	2	yes	—
133	C₂²	#1981	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	4	no	—
136	C₄	—	16	4	no	—
137	C₄	C₂×#332	1	2	yes	—
138	C₄	#1546	1	2	yes	—
139	C₄	#1693	1	4	yes	—
140	C₄	#1694	1	4	yes	—
141	C₄	#1715	1	2	yes	—
142	C₄	#1721	1	2	yes	—
143	C₂	#9692	1	1	yes	characteristic
144	C₂	#9740	1	2	yes	—
145	C₂	#9772	1	1	yes	characteristic
146	C₂	#9808	1	2	yes	—
147	C₂	#9837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33312

C₂

yes

maximal, characteristic

#9692

C₂

yes

maximal

#8805

C₂

yes

maximal, characteristic

#8260

C₂

yes

maximal, characteristic

C₂×#500

—

#2868

C₂²

yes

—

#1845

—

#1845

—

#1845

C₄

yes

—

#1842

C₂²

yes

—

C₄×C₄⋊C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

—

—

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

—

—

—

—

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

—

—

C₂×C₄

yes

—

C₄³

C₂³

yes

—

—

—

—

C₂²×Q₈

—

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×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

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂×Q₈

—

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

#101

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

#103

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₈

yes

—

C₄²

C₂×SD₁₆

yes

—

C₄²

C₂×SD₁₆

yes

characteristic

100

C₄²

C₂×Q₁₆

yes

characteristic

101

C₄²

AGL(1,ℤ/8)

yes

—

102

C₄²

#137

yes

—

103

C₂³

#390

yes

characteristic

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×#101

yes

—

113

C₂×C₄

C₂×#102

yes

—

114

C₂×C₄

C₂×#103

yes

characteristic

115

C₂×C₄

#419

yes

characteristic

116

C₂×C₄

#442

yes

—

117

C₂×C₄

#461

yes

—

118

C₂×C₄

#462

yes

—

119

C₂×C₄

#469

yes

—

120

C₂×C₄

#473

yes

—

121

C₂×C₄

#486

yes

—

122

C₂×C₄

#492

yes

characteristic

123

C₂×C₄

#493

yes

characteristic

124

C₂×C₄

#496

yes

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

#332

yes

complemented

129

C₂²

#1957

yes

characteristic

130

C₂²

#1966

yes

characteristic

131

C₂²

#1972

yes

—

132

C₂²

#1979

yes

—

133

C₂²

#1981

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

C₂×#332

yes

—

138

C₄

#1546

yes

—

139

C₄

#1693

yes

—

140

C₄

#1694

yes

—

141

C₄

#1715

yes

—

142

C₄

#1721

yes

—

143

C₂

#9692

yes

characteristic

144

C₂

#9740

yes

—

145

C₂

#9772

yes

characteristic

146

C₂

#9808

yes

—

147

C₂

#9837

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
128. C₈	#332	semidirect

H ≅ G/N

type

128. C₈

#332

semidirect

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

subgroup	index
#33312	2
#9692	2
#8805	2
#8260	2

subgroup

index

Subgroups — #102272

Conjugacy classes of subgroups

Complements

Maximal subgroups