Group #12827 — subgroups

Subgroups — #12827

Order 256.

188 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 69 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	#2950	C₂	1	1	yes	maximal, characteristic
2	#2153	C₂	1	1	yes	maximal, characteristic
3	C₂×#362	C₂	1	1	yes	maximal, characteristic
4	#2141	C₂	1	1	yes	maximal, characteristic
5	#1833	C₂	1	1	yes	maximal, characteristic
6	#1389	C₂	1	1	yes	maximal, characteristic
7	#1384	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
9	#519	—	2	1	no	—
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	#441	C₄	1	4	yes	complemented
13	C₄×Q₁₆	—	2	2	no	—
14	C₂×#108	C₂²	1	1	yes	characteristic
15	#427	C₂²	1	1	yes	characteristic
16	#427	C₂²	1	1	yes	characteristic
17	#419	—	2	1	no	—
18	#405	C₂²	1	1	yes	characteristic
19	#364	—	2	2	no	—
20	#362	—	2	2	no	—
21	#358	—	2	2	no	—
22	#344	—	2	1	no	—
23	H(3,ℤ/4)	—	2	1	no	—
24	C₂²×Q₈	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	1	no	—
28	C₂×Q₁₆	C₂×C₄	1	2	yes	—
29	C₂×Q₁₆	D₄	1	2	yes	—
30	C₂×Q₁₆	Q₈	1	2	yes	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	D₄	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	1	yes	characteristic
34	C₄×Q₈	—	2	4	no	—
35	C₄×Q₈	—	4	2	no	—
36	#117	—	2	1	no	—
37	#117	—	2	1	no	—
38	#117	—	4	2	no	—
39	#117	D₄	1	1	yes	characteristic
40	M₃₂	—	2	1	no	—
41	M₃₂	—	2	1	no	—
42	M₃₂	—	2	1	no	—
43	M₃₂	—	2	1	no	—
44	C₂×C₁₆	—	2	1	no	—
45	C₂×C₁₆	—	2	1	no	—
46	#108	—	2	1	no	—
47	#108	—	2	1	no	—
48	#107	—	2	1	no	—
49	#107	—	2	1	no	—
50	#106	—	2	1	no	—
51	#106	C₂×C₄	1	2	yes	—
52	#103	—	2	4	no	—
53	#97	C₂×C₄	1	2	yes	—
54	C₄×C₈	D₄	1	2	yes	—
55	C₂×Q₈	—	2	2	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	2	2	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	4	1	no	—
61	C₂²×C₄	—	4	1	no	—
62	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
63	Q₁₆	—	2	2	no	—
64	Q₁₆	—	2	2	no	—
65	Q₁₆	—	2	2	no	—
66	Q₁₆	C₄⋊C₄	1	4	yes	—
67	M₁₆	—	2	1	no	—
68	M₁₆	—	2	1	no	—
69	M₁₆	—	4	1	no	—
70	M₁₆	—	4	1	no	—
71	C₂×C₈	—	4	1	no	—
72	C₂×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈	#41	1	1	yes	characteristic
78	C₂×C₈	#41	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	4	4	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₂²⋊C₄	—	2	1	no	—
84	C₂²⋊C₄	—	4	1	no	—
85	C₂²⋊C₄	—	4	1	no	—
86	C₄²	—	4	4	no	—
87	C₄²	—	4	1	no	—
88	C₄²	—	4	1	no	—
89	C₄²	C₂²⋊C₄	1	2	yes	—
90	C₁₆	—	4	1	no	—
91	C₁₆	—	4	1	no	—
92	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	4	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	8	1	no	—
103	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄	#121	1	1	yes	characteristic
105	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄	#135	1	1	yes	characteristic
107	C₂×C₄	#135	1	1	yes	characteristic
108	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	8	1	no	—
111	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈	#117	1	1	yes	characteristic
113	C₈	#121	1	1	yes	characteristic
114	C₈	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²	—	4	1	no	—
116	C₂²	#418	1	1	yes	characteristic
117	C₂²	#463	1	1	yes	characteristic
118	C₂²	#478	1	1	yes	characteristic
119	C₄	—	4	2	no	—
120	C₄	—	4	4	no	—
121	C₄	—	8	2	no	—
122	C₄	—	16	1	no	—
123	C₄	#380	1	1	yes	characteristic
124	C₄	#416	1	1	yes	characteristic
125	C₄	#463	1	1	yes	characteristic
126	C₄	#478	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#1804	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2950

C₂

yes

maximal, characteristic

#2153

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal, characteristic

#2141

C₂

yes

maximal, characteristic

#1833

C₂

yes

maximal, characteristic

#1389

C₂

yes

maximal, characteristic

#1384

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

#519

—

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

#441

C₄

yes

complemented

C₄×Q₁₆

—

C₂×#108

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

#427

C₂²

yes

characteristic

#419

—

#405

C₂²

yes

—

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×Q₁₆

D₄

yes

—

C₂×Q₁₆

Q₈

yes

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

—

—

—

D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

C₂×C₄

yes

—

#103

—

#97

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₄⋊C₄

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

—

C₁₆

—

C₁₆

—

C₂³

AGL(1,ℤ/8)

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₄

#121

yes

characteristic

105

C₂×C₄

C₂²⋊Q₈

yes

characteristic

106

C₂×C₄

#135

yes

characteristic

107

C₂×C₄

#135

yes

characteristic

108

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

C₂×C₄⋊C₄

yes

characteristic

112

C₈

#117

yes

characteristic

113

C₈

#121

yes

characteristic

114

C₈

C₂²⋊Q₈

yes

characteristic

115

C₂²

—

116

C₂²

#418

yes

characteristic

117

C₂²

#463

yes

characteristic

118

C₂²

#478

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#380

yes

characteristic

124

C₄

#416

yes

characteristic

125

C₄

#463

yes

characteristic

126

C₄

#478

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#1804

yes

characteristic

	subgroup	G/H
center, socle	C₂	#1804
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#1804

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

AGL(1,ℤ/8)

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
12. #441	C₄	semidirect

H ≅ G/N

type

12. #441

C₄

semidirect

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

subgroup	index
#2950	2
#2153	2
C₂×#362	2
#2141	2
#1833	2
#1389	2
#1384	2

subgroup

index

Subgroups — #12827

Conjugacy classes of subgroups

Complements

Maximal subgroups