Group #12677 — subgroups

Subgroups — #12677

Order 256.

159 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 87 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	#2159	C₂	1	1	yes	maximal, characteristic
2	C₂×#365	C₂	1	1	yes	maximal, characteristic
3	C₂×#347	C₂	1	1	yes	maximal, characteristic
4	#1776	C₂	1	1	yes	maximal, characteristic
5	#1773	C₂	1	1	yes	maximal, characteristic
6	#1387	C₂	1	1	yes	maximal, characteristic
7	#1386	C₂	1	1	yes	maximal, characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#108	C₂²	1	1	yes	characteristic
11	#427	C₂²	1	1	yes	characteristic
12	C₂×#107	C₂²	1	1	yes	characteristic
13	C₂×#106	C₂²	1	1	yes	characteristic
14	#405	—	2	1	no	—
15	C₂×#97	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄	—	2	1	no	—
17	#367	C₄	1	2	yes	complemented
18	#366	C₄	1	2	yes	complemented
19	#365	C₄	1	4	yes	complemented
20	#347	C₄	1	4	yes	complemented
21	#340	—	2	1	no	—
22	#339	—	2	1	no	—
23	C₂×M₁₆	—	2	1	no	—
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₈	C₂³	1	1	yes	characteristic
26	#117	—	2	1	no	—
27	C₂×C₄⋊C₄	—	2	1	no	—
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄²	—	4	1	no	—
30	C₂×C₄²	D₄	1	1	yes	characteristic
31	M₃₂	—	2	1	no	—
32	M₃₂	C₂×C₄	1	2	yes	—
33	M₃₂	C₂×C₄	1	2	yes	—
34	M₃₂	C₂×C₄	1	2	yes	—
35	C₂×C₁₆	—	2	1	no	—
36	C₂×C₁₆	C₂×C₄	1	2	yes	—
37	#108	—	2	1	no	—
38	#108	C₂×C₄	1	2	yes	—
39	#107	—	2	1	no	—
40	#107	C₂×C₄	1	1	yes	characteristic
41	#107	C₂×C₄	1	2	yes	—
42	#107	C₂×C₄	1	1	yes	characteristic
43	#106	—	2	1	no	—
44	#106	—	2	1	no	—
45	#106	C₂×C₄	1	2	yes	—
46	#97	—	4	1	no	—
47	#97	C₂×C₄	1	2	yes	—
48	#97	Q₈	1	2	yes	—
49	C₄×C₈	—	4	1	no	—
50	#95	—	4	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	4	1	no	—
53	C₂²×C₄	—	4	1	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	M₁₆	—	2	2	no	—
56	M₁₆	—	4	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈	#41	1	1	yes	characteristic
66	C₂×C₈	#41	1	1	yes	characteristic
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	—	2	1	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	4	1	no	—
72	C₄⋊C₄	—	4	1	no	—
73	C₂²⋊C₄	—	4	1	no	—
74	C₄²	—	2	1	no	—
75	C₄²	—	4	1	no	—
76	C₄²	—	8	2	no	—
77	C₄²	C₄⋊C₄	1	2	yes	—
78	C₁₆	—	2	2	no	—
79	C₁₆	C₄²	1	4	yes	complemented
80	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	8	1	no	—
89	C₂×C₄	—	8	1	no	—
90	C₂×C₄	—	8	1	no	—
91	C₂×C₄	—	8	1	no	—
92	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄	#137	1	1	yes	characteristic
97	C₂×C₄	#137	1	1	yes	characteristic
98	C₈	—	2	2	no	—
99	C₈	—	4	2	no	—
100	C₈	C₂×C₄²	1	1	yes	characteristic
101	C₈	#117	1	1	yes	characteristic
102	C₈	C₄×Q₈	1	1	yes	characteristic
103	C₈	C₄×Q₈	1	1	yes	characteristic
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	#428	1	1	yes	characteristic
107	C₂²	#440	1	1	yes	characteristic
108	C₂²	#442	1	1	yes	characteristic
109	C₄	—	4	2	no	—
110	C₄	—	8	2	no	—
111	C₄	—	8	1	no	—
112	C₄	—	8	1	no	—
113	C₄	—	8	2	no	—
114	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₄	#428	1	1	yes	characteristic
116	C₄	#440	1	1	yes	characteristic
117	C₄	#441	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#1776	1	1	yes	characteristic
121	C₂	#2180	1	1	yes	characteristic
122	C₂	AGL(1,ℤ/16)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2159

C₂

yes

maximal, characteristic

C₂×#365

C₂

yes

maximal, characteristic

C₂×#347

C₂

yes

maximal, characteristic

#1776

C₂

yes

maximal, characteristic

#1773

C₂

yes

maximal, characteristic

#1387

C₂

yes

maximal, characteristic

#1386

C₂

yes

maximal, characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

C₂×#107

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

#405

—

C₂×#97

C₂²

yes

characteristic

C₄×C₄⋊C₄

—

#367

C₄

yes

complemented

#366

C₄

yes

complemented

#365

C₄

yes

complemented

#347

C₄

yes

complemented

#340

—

#339

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

C₂×C₄

yes

—

M₃₂

C₂×C₄

yes

—

M₃₂

C₂×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

#108

—

#108

C₂×C₄

yes

—

#107

—

#107

C₂×C₄

yes

characteristic

#107

C₂×C₄

yes

—

#107

C₂×C₄

yes

—

—

C₂×C₄

yes

—

#97

—

#97

C₂×C₄

yes

—

#97

Q₈

yes

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₄

yes

—

C₁₆

—

C₁₆

C₄²

yes

complemented

C₂³

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

100

C₈

C₂×C₄²

yes

characteristic

101

C₈

#117

yes

characteristic

102

C₈

C₄×Q₈

yes

characteristic

103

C₈

C₄×Q₈

yes

characteristic

104

C₂²

—

105

C₂²

—

106

C₂²

#428

yes

characteristic

107

C₂²

#440

yes

characteristic

108

C₂²

#442

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

C₄×C₄⋊C₄

yes

characteristic

115

C₄

#428

yes

characteristic

116

C₄

#440

yes

characteristic

117

C₄

#441

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#1776

yes

characteristic

121

C₂

#2180

yes

characteristic

122

C₂

AGL(1,ℤ/16)

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#442
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#442

commutator

C₈

C₂×C₄²

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

AGL(1,ℤ/8)

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
17. #367	C₄	semidirect
18. #366	C₄	semidirect
19. #365	C₄	semidirect
20. #347	C₄	semidirect
79. C₁₆	C₄²	semidirect

H ≅ G/N

type

17. #367

C₄

semidirect

18. #366

C₄

semidirect

19. #365

C₄

semidirect

20. #347

C₄

semidirect

79. C₁₆

C₄²

semidirect

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

subgroup	index
#2159	2
C₂×#365	2
C₂×#347	2
#1776	2
#1773	2
#1387	2
#1386	2

subgroup

index

Subgroups — #12677

Conjugacy classes of subgroups

Complements

Maximal subgroups