Group #96889 — subgroups

Subgroups — #96889

Order 512.

402 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 226 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	C₄×#334	C₂	1	2	yes	maximal
2	#8260	C₂	1	2	yes	maximal
3	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
4	#7015	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈	C₂²	1	4	yes	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#334	C₄	1	4	yes	—
10	C₂×#322	—	2	2	no	—
11	C₂×C₈²	C₄	1	2	yes	—
12	C₂×C₈²	C₄	1	2	yes	—
13	#1277	C₄	1	4	yes	—
14	#1273	—	2	2	no	—
15	C₂×C₄⋊C₈	C₂×C₄	1	8	yes	—
16	C₂×C₄×C₈	—	2	2	no	—
17	C₂×C₄×C₈	—	2	4	no	—
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	Q₈	1	2	yes	—
27	C₄³	C₂³	1	1	yes	characteristic
28	#336	—	2	4	no	—
29	#334	—	2	8	no	—
30	#322	—	4	4	no	—
31	C₈²	—	2	4	no	—
32	C₈²	C₈	1	8	yes	complemented
33	C₂²×C₈	—	2	2	no	—
34	C₂²×C₈	—	2	8	no	—
35	C₂²×C₈	C₄²	1	1	yes	characteristic
36	C₂²×C₈	C₄²	1	2	yes	—
37	C₂²×C₈	C₄²	1	1	yes	characteristic
38	C₂²×C₈	C₄⋊C₄	1	2	yes	—
39	C₂²×C₈	C₄⋊C₄	1	2	yes	—
40	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂²×C₄	1	2	yes	—
42	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄²	#41	1	2	yes	—
45	C₄⋊C₈	—	2	8	no	—
46	C₄⋊C₈	—	2	8	no	—
47	C₄×C₈	—	2	8	no	—
48	C₄×C₈	—	2	4	no	—
49	C₄×C₈	—	2	2	no	—
50	C₄×C₈	—	2	2	no	—
51	C₄×C₈	—	2	4	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	4	4	no	—
54	C₄×C₈	—	4	4	no	—
55	C₄×C₈	C₂×C₈	1	4	yes	—
56	C₄×C₈	C₂×C₈	1	4	yes	—
57	C₄×C₈	C₂×C₈	1	4	yes	—
58	C₄×C₈	M₁₆	1	8	yes	—
59	C₄×C₈	M₁₆	1	4	yes	—
60	C₄×C₈	SD₁₆	1	4	yes	—
61	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
62	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄	#117	1	1	yes	characteristic
64	C₂²×C₄	C₄×D₄	1	2	yes	—
65	C₂²×C₄	C₄×Q₈	1	2	yes	—
66	C₂×C₈	—	2	8	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	4	8	no	—
74	C₂×C₈	—	4	8	no	—
75	C₂×C₈	—	4	8	no	—
76	C₂×C₈	C₄×C₈	1	2	yes	—
77	C₂×C₈	C₄×C₈	1	2	yes	—
78	C₂×C₈	#97	1	4	yes	—
79	C₂×C₈	#97	1	4	yes	—
80	C₂×C₈	C₄⋊C₈	1	4	yes	—
81	C₂×C₈	C₄⋊C₈	1	4	yes	—
82	C₂×C₈	#106	1	4	yes	—
83	C₂×C₈	#108	1	4	yes	—
84	C₄²	—	2	2	no	—
85	C₄²	—	2	4	no	—
86	C₄²	C₂²×C₈	1	2	yes	—
87	C₄²	C₂×M₁₆	1	4	yes	—
88	C₄²	C₂×M₁₆	1	2	yes	—
89	C₄²	#131	1	4	yes	—
90	C₄²	C₂×SD₁₆	1	2	yes	—
91	C₄²	#135	1	2	yes	—
92	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂×C₄	C₂×#97	1	2	yes	—
98	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
99	C₂×C₄	#405	1	2	yes	—
100	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
101	C₂×C₄	C₂×#106	1	1	yes	characteristic
102	C₂×C₄	#427	1	1	yes	characteristic
103	C₂×C₄	C₂×#108	1	2	yes	—
104	C₂×C₄	#431	1	2	yes	—
105	C₂×C₄	#436	1	4	yes	—
106	C₂×C₄	C₄×SD₁₆	1	4	yes	—
107	C₂×C₄	#443	1	4	yes	—
108	C₂×C₄	#446	1	4	yes	—
109	C₈	—	2	8	no	—
110	C₈	—	2	4	no	—
111	C₈	—	2	2	no	—
112	C₈	—	2	4	no	—
113	C₈	—	8	8	no	—
114	C₈	#322	1	4	yes	complemented
115	C₈	#334	1	8	yes	complemented
116	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
117	C₂²	#1770	1	2	yes	—
118	C₂²	#1772	1	2	yes	—
119	C₂²	C₄×#106	1	1	yes	characteristic
120	C₂²	C₄×#108	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	4	no	—
123	C₄	C₂×#322	1	1	yes	characteristic
124	C₄	#1450	1	1	yes	characteristic
125	C₄	#1451	1	2	yes	—
126	C₄	C₂×#334	1	2	yes	—
127	C₄	#1564	1	2	yes	—
128	C₄	#1590	1	8	yes	—
129	C₂	#8260	1	1	yes	characteristic
130	C₂	#8266	1	1	yes	characteristic
131	C₂	C₄×#334	1	1	yes	characteristic
132	C₂	#8283	1	2	yes	—
133	C₂	#8293	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#334

C₂

yes

maximal

#8260

C₂

yes

maximal

C₄×C₈²

C₂

yes

maximal, characteristic

#7015

C₂

yes

maximal

C₄×C₄⋊C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#334

C₄

yes

—

C₂×#322

—

C₂×C₈²

C₄

yes

—

C₂×C₈²

C₄

yes

—

#1277

C₄

yes

—

#1273

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

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

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

—

—

—

C₈²

—

C₈²

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₄²

yes

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

#41

yes

—

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

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₂²×C₄

C₂×C₄²

yes

characteristic

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#117

yes

characteristic

C₂²×C₄

C₄×D₄

yes

—

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

C₄×C₈

yes

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

#97

yes

—

C₂×C₈

#97

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₈

yes

—

C₄²

C₂×M₁₆

yes

—

C₄²

C₂×M₁₆

yes

—

C₄²

#131

yes

—

C₄²

C₂×SD₁₆

yes

—

C₄²

#135

yes

—

C₂³

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×C₈

yes

characteristic

C₂×C₄

C₂×#97

yes

—

C₂×C₄

C₄×M₁₆

yes

characteristic

C₂×C₄

#405

yes

—

100

C₂×C₄

C₂×C₄⋊C₈

yes

—

101

C₂×C₄

C₂×#106

yes

characteristic

102

C₂×C₄

#427

yes

characteristic

103

C₂×C₄

C₂×#108

yes

—

104

C₂×C₄

#431

yes

—

105

C₂×C₄

#436

yes

—

106

C₂×C₄

C₄×SD₁₆

yes

—

107

C₂×C₄

#443

yes

—

108

C₂×C₄

#446

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

#322

yes

complemented

115

C₈

#334

yes

complemented

116

C₂²

C₄×C₄⋊C₈

yes

characteristic

117

C₂²

#1770

yes

—

118

C₂²

#1772

yes

—

119

C₂²

C₄×#106

yes

characteristic

120

C₂²

C₄×#108

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

C₂×#322

yes

characteristic

124

C₄

#1450

yes

characteristic

125

C₄

#1451

yes

—

126

C₄

C₂×#334

yes

—

127

C₄

#1564

yes

—

128

C₄

#1590

yes

—

129

C₂

#8260

yes

characteristic

130

C₂

#8266

yes

characteristic

131

C₂

C₄×#334

yes

characteristic

132

C₂

#8283

yes

—

133

C₂

#8293

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂×C₄²

C₂×D₄

commutator

C₂×C₄

C₂×C₄×C₈

Frattini, cosocle

C₄³

C₂³

socle, prime radical

C₂³

C₄×C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
32. C₈²	C₈	semidirect
114. C₈	#322	semidirect
115. C₈	#334	semidirect

H ≅ G/N

type

32. C₈²

C₈

semidirect

114. C₈

#322

semidirect

115. C₈

#334

semidirect

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

subgroup	index
C₄×#334	2
#8260	2
C₄×C₈²	2
#7015	2

subgroup

index

C₄×#334

#8260

C₄×C₈²

#7015

Subgroups — #96889

Conjugacy classes of subgroups

Complements

Maximal subgroups