Group #96887 — subgroups

Subgroups — C₈×#335 #96887

Order 512.

434 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 258 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×#335	C₂	1	2	yes	maximal
2	C₈×C₄⋊C₈	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×#335	C₄	1	4	yes	—
10	C₂×C₈²	—	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	2	yes	—
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	#335	C₈	1	16	yes	direct product factor
30	C₈²	—	2	4	no	—
31	C₈²	—	4	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	2	yes	—
36	C₂²×C₈	C₄²	1	1	yes	characteristic
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	C₂×C₈	1	16	yes	—
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	8	yes	—
58	C₄×C₈	C₂×C₈	1	4	yes	—
59	C₄×C₈	M₁₆	1	4	yes	—
60	C₄×C₈	D₈	1	2	yes	—
61	C₄×C₈	Q₁₆	1	2	yes	—
62	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
63	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄	#117	1	1	yes	characteristic
65	C₂²×C₄	C₄×D₄	1	2	yes	—
66	C₂²×C₄	C₄×Q₈	1	2	yes	—
67	C₂×C₈	—	2	8	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	2	no	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	16	no	—
75	C₂×C₈	—	4	8	no	—
76	C₂×C₈	—	4	8	no	—
77	C₂×C₈	C₄×C₈	1	4	yes	—
78	C₂×C₈	C₄×C₈	1	2	yes	—
79	C₂×C₈	C₄×C₈	1	4	yes	—
80	C₂×C₈	C₄×C₈	1	2	yes	—
81	C₂×C₈	C₄⋊C₈	1	4	yes	—
82	C₂×C₈	C₄⋊C₈	1	4	yes	—
83	C₂×C₈	#107	1	4	yes	—
84	C₂×C₈	#108	1	4	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	4	no	—
87	C₄²	C₂²×C₈	1	2	yes	—
88	C₄²	C₂²×C₈	1	4	yes	—
89	C₄²	C₂×M₁₆	1	2	yes	—
90	C₄²	#131	1	4	yes	—
91	C₄²	C₂×D₈	1	1	yes	characteristic
92	C₄²	C₂×Q₁₆	1	1	yes	characteristic
93	C₄²	#135	1	2	yes	—
94	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
99	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
101	C₂×C₄	#405	1	2	yes	—
102	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
103	C₂×C₄	C₂×#107	1	1	yes	characteristic
104	C₂×C₄	#427	1	1	yes	characteristic
105	C₂×C₄	C₂×#108	1	2	yes	—
106	C₂×C₄	#431	1	2	yes	—
107	C₂×C₄	C₈×D₄	1	4	yes	—
108	C₂×C₄	C₄×D₈	1	2	yes	—
109	C₂×C₄	C₄×Q₁₆	1	2	yes	—
110	C₂×C₄	#443	1	4	yes	—
111	C₂×C₄	C₈×Q₈	1	4	yes	—
112	C₈	—	2	8	no	—
113	C₈	—	2	4	no	—
114	C₈	—	2	4	no	—
115	C₈	—	2	2	no	—
116	C₈	—	4	16	no	—
117	C₈	C₈²	1	4	yes	complemented
118	C₈	#335	1	8	yes	direct product factor
119	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
120	C₂²	C₈×C₄⋊C₄	1	2	yes	—
121	C₂²	#1772	1	2	yes	—
122	C₂²	C₄×#107	1	1	yes	characteristic
123	C₂²	C₄×#108	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	4	no	—
126	C₄	C₂×C₈²	1	1	yes	characteristic
127	C₄	C₈×M₁₆	1	2	yes	—
128	C₄	#1450	1	1	yes	characteristic
129	C₄	C₂×#335	1	2	yes	—
130	C₄	#1564	1	2	yes	—
131	C₄	C₈×D₈	1	4	yes	—
132	C₄	C₈×Q₁₆	1	4	yes	—
133	C₂	C₈×C₄⋊C₈	1	1	yes	characteristic
134	C₂	#8266	1	1	yes	characteristic
135	C₂	C₄×#335	1	1	yes	characteristic
136	C₂	C₈×#107	1	2	yes	—
137	C₂	C₈×#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#335

C₂

yes

maximal

C₈×C₄⋊C₈

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×#335

C₄

yes

—

C₂×C₈²

—

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

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

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

characteristic

#336

—

#335

C₈

yes

direct product factor

C₈²

—

C₈²

—

C₈²

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄²

yes

—

C₂²×C₈

C₄²

yes

characteristic

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

C₂×C₈

yes

—

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

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

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

C₄×C₈

yes

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₈

yes

—

C₄²

C₂²×C₈

yes

—

C₄²

C₂×M₁₆

yes

—

C₄²

#131

yes

—

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₄²

#135

yes

—

C₂³

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×C₈

yes

—

C₂×C₄

C₂×C₄×C₈

yes

characteristic

100

C₂×C₄

C₄×M₁₆

yes

characteristic

101

C₂×C₄

#405

yes

—

102

C₂×C₄

C₂×C₄⋊C₈

yes

—

103

C₂×C₄

C₂×#107

yes

characteristic

104

C₂×C₄

#427

yes

characteristic

105

C₂×C₄

C₂×#108

yes

—

106

C₂×C₄

#431

yes

—

107

C₂×C₄

C₈×D₄

yes

—

108

C₂×C₄

C₄×D₈

yes

—

109

C₂×C₄

C₄×Q₁₆

yes

—

110

C₂×C₄

#443

yes

—

111

C₂×C₄

C₈×Q₈

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

C₈²

yes

complemented

118

C₈

#335

yes

direct product factor

119

C₂²

C₄×C₄⋊C₈

yes

characteristic

120

C₂²

C₈×C₄⋊C₄

yes

—

121

C₂²

#1772

yes

—

122

C₂²

C₄×#107

yes

characteristic

123

C₂²

C₄×#108

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

C₂×C₈²

yes

characteristic

127

C₄

C₈×M₁₆

yes

—

128

C₄

#1450

yes

characteristic

129

C₄

C₂×#335

yes

—

130

C₄

#1564

yes

—

131

C₄

C₈×D₈

yes

—

132

C₄

C₈×Q₁₆

yes

—

133

C₂

C₈×C₄⋊C₈

yes

characteristic

134

C₂

#8266

yes

characteristic

135

C₂

C₄×#335

yes

characteristic

136

C₂

C₈×#107

yes

—

137

C₂

C₈×#108

yes

—

	subgroup	G/H
center	C₂×C₄×C₈	D₄
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂×C₄×C₈

D₄

commutator

C₄

C₂×C₈²

Frattini, cosocle

C₄³

C₂³

socle, prime radical

C₂³

C₄×C₄⋊C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
29. #335	C₈	direct (N × H)
32. C₈²	C₈	semidirect
117. C₈	C₈²	semidirect
118. C₈	#335	direct (N × H)

H ≅ G/N

type

29. #335

C₈

direct (N × H)

32. C₈²

C₈

semidirect

117. C₈

C₈²

semidirect

118. C₈

#335

direct (N × H)

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

subgroup	index
C₄×#335	2
C₈×C₄⋊C₈	2
C₄×C₈²	2
#7015	2

subgroup

index

C₄×#335

C₈×C₄⋊C₈

C₄×C₈²

#7015

Subgroups — C₈×#335 #96887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×#335 #96887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#335 #96887