Group #124240 — subgroups

Subgroups — #124240

Order 512.

298 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 110 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	#21732	C₂	1	1	yes	maximal, characteristic
2	#8266	C₂	1	2	yes	maximal
3	#7134	C₂	1	4	yes	maximal
4	#3160	C₄	1	1	yes	characteristic
5	#2960	C₄	1	2	yes	—
6	#1994	C₂²	1	2	yes	—
7	#1945	—	2	2	no	—
8	#1848	C₄	1	1	yes	characteristic
9	#1772	—	2	2	no	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	#1564	—	2	2	no	—
12	#1450	—	2	2	no	—
13	#1276	C₂²	1	4	yes	—
14	#533	C₂×C₄	1	2	yes	—
15	#500	—	2	1	no	—
16	#499	C₈	1	4	yes	complemented
17	#498	—	2	1	no	—
18	#446	—	4	2	no	—
19	C₈×Q₈	—	4	2	no	—
20	#431	—	2	4	no	—
21	C₂×#108	—	2	2	no	—
22	#427	—	2	2	no	—
23	#424	—	2	2	no	—
24	#423	C₂×C₄	1	2	yes	—
25	#405	—	2	2	no	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	2	yes	—
29	C₂×C₄×C₈	D₄	1	2	yes	—
30	C₄³	C₂³	1	1	yes	characteristic
31	#339	—	2	4	no	—
32	#335	—	2	4	no	—
33	#334	—	2	4	no	—
34	#322	—	4	2	no	—
35	C₈²	—	4	2	no	—
36	C₂×M₁₆	—	2	2	no	—
37	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₄⋊Q₈	—	2	2	no	—
41	#125	C₂×C₈	1	4	yes	—
42	C₂²⋊Q₈	—	4	2	no	—
43	C₄×Q₈	—	4	2	no	—
44	#117	—	2	2	no	—
45	C₂×C₄²	—	4	1	no	—
46	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄²	C₂×D₄	1	2	yes	—
48	#108	—	4	2	no	—
49	#108	—	4	2	no	—
50	#107	—	2	4	no	—
51	#106	—	2	4	no	—
52	C₄⋊C₈	—	2	8	no	—
53	C₄⋊C₈	—	4	2	no	—
54	C₄⋊C₈	—	4	4	no	—
55	C₂²⋊C₈	—	4	4	no	—
56	#97	—	4	2	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	—	2	1	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	4	2	no	—
62	C₄×C₈	—	4	4	no	—
63	C₄×C₈	C₂×C₈	1	2	yes	—
64	C₄×C₈	M₁₆	1	1	yes	characteristic
65	C₄×C₈	M₁₆	1	1	yes	characteristic
66	C₄×C₈	SD₁₆	1	2	yes	—
67	C₄×C₈	SD₁₆	1	2	yes	—
68	C₄×C₈	Q₁₆	1	2	yes	—
69	C₄×C₈	Q₁₆	1	2	yes	—
70	C₂×Q₈	—	4	2	no	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	M₁₆	—	4	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	8	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	C₂²⋊C₈	1	2	yes	—
80	C₂×C₈	C₂²⋊C₈	1	2	yes	—
81	C₂×C₈	#100	1	1	yes	characteristic
82	C₂×C₈	#100	1	1	yes	characteristic
83	C₂×C₈	#103	1	2	yes	—
84	C₂×C₈	#103	1	2	yes	—
85	C₂×C₈	#103	1	2	yes	—
86	C₂×C₈	#103	1	2	yes	—
87	C₂×C₈	#104	1	2	yes	—
88	C₂×C₈	#104	1	2	yes	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₂²⋊C₄	—	4	2	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	2	no	—
95	C₄²	—	4	4	no	—
96	C₄²	—	4	2	no	—
97	C₄²	C₂²×C₈	1	1	yes	characteristic
98	C₄²	C₂×M₁₆	1	1	yes	characteristic
99	C₄²	C₂×SD₁₆	1	2	yes	—
100	C₄²	C₂×Q₁₆	1	2	yes	—
101	C₂³	#410	1	1	yes	characteristic
102	Q₈	—	8	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄	C₂×#100	1	1	yes	characteristic
112	C₂×C₄	C₂×#103	1	2	yes	—
113	C₂×C₄	C₂×#104	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	8	8	no	—
116	C₈	—	8	2	no	—
117	C₈	Q₈⋊C₈	1	2	yes	—
118	C₈	Q₈⋊C₈	1	2	yes	—
119	C₈	Q₈⋊C₈	1	2	yes	—
120	C₈	Q₈⋊C₈	1	2	yes	—
121	C₈	#333	1	2	yes	—
122	C₈	#333	1	1	yes	characteristic
123	C₈	#333	1	1	yes	characteristic
124	C₂²	—	4	1	no	—
125	C₂²	#1460	1	1	yes	characteristic
126	C₂²	#1503	1	2	yes	—
127	C₄	—	2	4	no	—
128	C₄	—	2	2	no	—
129	C₄	—	4	1	no	—
130	C₄	—	4	2	no	—
131	C₄	—	8	4	no	—
132	C₄	C₂×Q₈⋊C₈	1	2	yes	—
133	C₄	#1525	1	1	yes	characteristic
134	C₄	C₂×#333	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	#7571	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21732

C₂

yes

maximal, characteristic

#8266

C₂

yes

maximal

#7134

C₂

yes

maximal

#3160

C₄

yes

characteristic

#2960

C₄

yes

—

#1994

C₂²

yes

—

#1945

—

#1848

C₄

yes

characteristic

#1772

—

C₄²×C₈

C₂²

yes

—

—

C₂²

yes

—

#533

C₂×C₄

yes

—

#500

—

#499

C₈

yes

complemented

#498

—

#446

—

C₈×Q₈

—

—

—

—

—

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₄³

C₂³

yes

—

—

—

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊Q₈

—

#125

C₂×C₈

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

#104

yes

—

C₂×C₈

#104

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₄²

C₂×SD₁₆

yes

—

100

C₄²

C₂×Q₁₆

yes

—

101

C₂³

#410

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

111

C₂×C₄

C₂×#100

yes

characteristic

112

C₂×C₄

C₂×#103

yes

—

113

C₂×C₄

C₂×#104

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

Q₈⋊C₈

yes

—

118

C₈

Q₈⋊C₈

yes

—

119

C₈

Q₈⋊C₈

yes

—

120

C₈

Q₈⋊C₈

yes

—

121

C₈

#333

yes

—

122

C₈

#333

yes

characteristic

123

C₈

#333

yes

characteristic

124

C₂²

—

125

C₂²

#1460

yes

characteristic

126

C₂²

#1503

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂×Q₈⋊C₈

yes

—

133

C₄

#1525

yes

characteristic

134

C₄

C₂×#333

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#7571

yes

characteristic

	subgroup	G/H
center	C₈	#333
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂	#7571
prime radical	C₂³	#410
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#333

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₄³

C₂³

C₂

C₂³

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #499	C₈	semidirect

H ≅ G/N

type

16. #499

C₈

semidirect

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

subgroup	index
#21732	2
#8266	2
#7134	2

subgroup

index

#21732

#8266

#7134

Subgroups — #124240

Conjugacy classes of subgroups

Complements

Maximal subgroups