Group #8279 — subgroups

Subgroups — #8279

Order 256.

197 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 107 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	#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×#334	C₂	1	1	yes	maximal, characteristic
6	#1277	C₂	1	1	yes	maximal, characteristic
7	#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#97	—	2	1	no	—
13	C₂×C₄×C₈	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄³	C₂²	1	1	yes	characteristic
17	#336	—	2	1	no	—
18	#336	—	2	1	no	—
19	#336	C₄	1	4	yes	—
20	#335	C₄	1	4	yes	complemented
21	#334	C₄	1	4	yes	complemented
22	C₂²×C₈	—	2	1	no	—
23	C₂²×C₈	—	2	1	no	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄²	C₂×C₄	1	2	yes	—
29	C₂×C₄²	D₄	1	1	yes	characteristic
30	C₂×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₄²	Q₈	1	2	yes	—
32	C₂×C₄²	C₂³	1	1	yes	characteristic
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₄⋊C₈	C₂×C₄	1	2	yes	—
41	#97	—	4	2	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	—	4	2	no	—
45	C₄×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₈	C₂×C₄	1	2	yes	—
47	C₂²×C₄	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	1	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	C₄²	1	4	yes	—
68	C₄²	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₄²	—	2	1	no	—
73	C₄²	—	2	1	no	—
74	C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₄²	#41	1	1	yes	characteristic
77	C₄²	#41	1	1	yes	characteristic
78	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	#108	1	4	yes	—
86	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄	#117	1	1	yes	characteristic
88	C₂×C₄	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄	#131	1	4	yes	—
93	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄	#137	1	1	yes	characteristic
96	C₂×C₄	#137	1	1	yes	characteristic
97	C₈	—	2	4	no	—
98	C₈	—	4	2	no	—
99	C₈	—	4	2	no	—
100	C₈	—	4	2	no	—
101	C₈	—	4	2	no	—
102	C₈	—	4	2	no	—
103	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²	#424	1	2	yes	—
105	C₂²	#428	1	1	yes	characteristic
106	C₂²	#428	1	1	yes	characteristic
107	C₂²	C₂×#108	1	2	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	#405	1	2	yes	—
113	C₄	#433	1	2	yes	—
114	C₄	#440	1	1	yes	characteristic
115	C₄	#440	1	1	yes	characteristic
116	C₄	#441	1	1	yes	characteristic
117	C₄	#442	1	1	yes	characteristic
118	C₂	#1573	1	2	yes	—
119	C₂	#1574	1	2	yes	—
120	C₂	#1768	1	1	yes	characteristic
121	C₂	#1776	1	1	yes	characteristic
122	C₂	C₄×#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1767

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

#1744

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

C₂×#334

C₂

yes

maximal, characteristic

#1277

C₂

yes

maximal, characteristic

#1277

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

—

—

C₄

yes

—

#335

C₄

yes

complemented

#334

C₄

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

C₄²

yes

C₄²

yes

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#108

yes

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

#131

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₂²

C₄×C₄⋊C₄

yes

characteristic

104

C₂²

#424

yes

—

105

C₂²

#428

yes

characteristic

106

C₂²

#428

yes

characteristic

107

C₂²

C₂×#108

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#405

yes

—

113

C₄

#433

yes

—

114

C₄

#440

yes

characteristic

115

C₄

#440

yes

characteristic

116

C₄

#441

yes

characteristic

117

C₄

#442

yes

characteristic

118

C₂

#1573

yes

—

119

C₂

#1574

yes

—

120

C₂

#1768

yes

characteristic

121

C₂

#1776

yes

characteristic

122

C₂

C₄×#108

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₂×C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
20. #335	C₄	semidirect
21. #334	C₄	semidirect

H ≅ G/N

type

20. #335

C₄

semidirect

21. #334

C₄

semidirect

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

subgroup	index
#1767	2
C₄×C₄⋊C₈	2
#1744	2
C₂×#335	2
C₂×#334	2
#1277	2

subgroup

index

C₄×C₄⋊C₈

Subgroups — #8279

Conjugacy classes of subgroups

Complements

Maximal subgroups