Group #140421 — subgroups

Subgroups — #140421

Order 512.

250 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 106 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	#13183	C₂	1	2	yes	maximal
2	#11996	C₂	1	1	yes	maximal, characteristic
3	#8813	C₂	1	1	yes	maximal, characteristic
4	C₈×#108	C₂	1	1	yes	maximal, characteristic
5	#7278	C₂	1	2	yes	maximal
6	C₂×#365	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	#1844	—	2	1	no	—
9	#1844	—	2	1	no	—
10	#1844	C₂²	1	2	yes	—
11	C₄×#108	C₂²	1	1	yes	characteristic
12	#1564	C₂²	1	1	yes	characteristic
13	C₈×M₁₆	—	2	1	no	—
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1383	—	2	2	no	—
16	#1367	C₄	1	4	yes	complemented
17	#1278	—	2	1	no	—
18	C₂×M₃₂	—	2	2	no	—
19	#431	—	2	1	no	—
20	C₂×#108	D₄	1	2	yes	—
21	C₂×#106	—	2	2	no	—
22	C₂×#106	—	2	2	no	—
23	C₂×#106	—	4	2	no	—
24	C₂×#106	—	4	2	no	—
25	C₄×M₁₆	—	2	1	no	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	#384	—	2	2	no	—
30	#365	—	4	4	no	—
31	C₄⋊C₁₆	—	2	2	no	—
32	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
33	#341	—	2	1	no	—
34	#335	Q₈	1	2	yes	—
35	#334	—	2	1	no	—
36	#322	—	4	1	no	—
37	C₈²	—	4	1	no	—
38	C₈²	C₂×C₄	1	2	yes	—
39	C₈²	Q₈	1	2	yes	—
40	C₂×M₁₆	—	2	2	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	SD₁₆	1	2	yes	—
43	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₈	#41	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	4	2	no	—
46	C₂×C₄⋊C₄	—	4	2	no	—
47	C₂×C₄⋊C₄	—	4	2	no	—
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	M₃₂	—	4	2	no	—
50	M₃₂	—	4	2	no	—
51	C₂×C₁₆	—	2	4	no	—
52	#108	—	2	2	no	—
53	#108	SD₁₆	1	4	yes	—
54	#106	—	4	2	no	—
55	#106	—	4	2	no	—
56	#106	—	8	4	no	—
57	#106	—	8	4	no	—
58	#106	—	8	2	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₂²⋊C₈	—	4	1	no	—
61	#97	—	4	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	4	1	no	—
64	C₄×C₈	—	4	1	no	—
65	C₄×C₈	C₄⋊C₄	1	2	yes	—
66	C₄×C₈	C₄⋊C₄	1	2	yes	—
67	C₄×C₈	SD₁₆	1	1	yes	characteristic
68	C₄×C₈	SD₁₆	1	1	yes	characteristic
69	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₄×C₈	#41	1	1	yes	characteristic
73	C₂²×C₄	—	8	2	no	—
74	C₂²×C₄	#121	1	1	yes	characteristic
75	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
76	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	M₁₆	—	4	2	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	#106	1	2	yes	—
86	C₂×C₈	#106	1	2	yes	—
87	C₂×C₈	C₄×D₄	1	1	yes	characteristic
88	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₈	#125	1	1	yes	characteristic
90	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₈	C₂×SD₁₆	1	2	yes	—
92	C₂×C₈	#135	1	2	yes	—
93	C₄⋊C₄	—	8	4	no	—
94	C₄⋊C₄	—	8	4	no	—
95	C₄⋊C₄	—	8	2	no	—
96	C₄⋊C₄	—	8	2	no	—
97	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
99	C₄²	C₂×SD₁₆	1	1	yes	characteristic
100	C₄²	#137	1	1	yes	characteristic
101	C₁₆	—	4	4	no	—
102	C₂³	#460	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	16	2	no	—
106	C₂×C₄	—	16	2	no	—
107	C₂×C₄	—	16	2	no	—
108	C₂×C₄	#389	1	1	yes	characteristic
109	C₂×C₄	C₂×#106	1	1	yes	characteristic
110	C₂×C₄	#428	1	1	yes	characteristic
111	C₂×C₄	#461	1	1	yes	characteristic
112	C₂×C₄	#465	1	2	yes	—
113	C₂×C₄	#471	1	2	yes	—
114	C₂×C₄	#475	1	1	yes	characteristic
115	C₂×C₄	#476	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₈	—	2	2	no	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	—	4	4	no	—
121	C₈	—	8	2	no	—
122	C₈	#365	1	4	yes	complemented
123	C₈	C₄×SD₁₆	1	2	yes	—
124	C₈	#498	1	1	yes	characteristic
125	C₈	#499	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	#1666	1	2	yes	—
129	C₂²	#1948	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	32	2	no	—
133	C₄	#1671	1	2	yes	—
134	C₄	#1941	1	1	yes	characteristic
135	C₄	#1951	1	1	yes	characteristic
136	C₄	C₂×#365	1	2	yes	—
137	C₄	#2230	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	#9612	1	1	yes	characteristic
140	C₂	#13183	1	1	yes	characteristic
141	C₂	#13186	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13183

C₂

yes

maximal

#11996

C₂

yes

maximal, characteristic

#8813

C₂

yes

maximal, characteristic

C₈×#108

C₂

yes

maximal, characteristic

#7278

C₂

yes

maximal

C₂×#365

—

#2150

C₂²

yes

—

#1844

—

#1844

—

#1844

C₂²

yes

—

C₄×#108

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

C₈×M₁₆

—

C₂×C₈²

C₂²

yes

characteristic

#1383

—

#1367

C₄

yes

complemented

#1278

—

C₂×M₃₂

—

#431

—

C₂×#108

D₄

yes

—

—

—

—

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#365

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

#341

—

#335

Q₈

yes

—

#334

—

#322

—

C₈²

—

C₈²

C₂×C₄

yes

—

C₈²

Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

SD₁₆

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

#135

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

100

C₄²

#137

yes

characteristic

101

C₁₆

—

102

C₂³

#460

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#389

yes

characteristic

109

C₂×C₄

C₂×#106

yes

characteristic

110

C₂×C₄

#428

yes

characteristic

111

C₂×C₄

#461

yes

characteristic

112

C₂×C₄

#465

yes

—

113

C₂×C₄

#471

yes

—

114

C₂×C₄

#475

yes

characteristic

115

C₂×C₄

#476

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

#365

yes

complemented

123

C₈

C₄×SD₁₆

yes

—

124

C₈

#498

yes

characteristic

125

C₈

#499

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

#1666

yes

—

129

C₂²

#1948

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#1671

yes

—

134

C₄

#1941

yes

characteristic

135

C₄

#1951

yes

characteristic

136

C₄

C₂×#365

yes

—

137

C₄

#2230

yes

—

138

C₂

—

139

C₂

#9612

yes

characteristic

140

C₂

#13183

yes

characteristic

141

C₂

#13186

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1948
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#460
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1948

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#460

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #1367	C₄	semidirect
122. C₈	#365	semidirect

H ≅ G/N

type

16. #1367

C₄

semidirect

122. C₈

#365

semidirect

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

subgroup	index
#13183	2
#11996	2
#8813	2
C₈×#108	2
#7278	2

subgroup

index

Subgroups — #140421

Conjugacy classes of subgroups

Complements

Maximal subgroups