Group #22256 — subgroups

Subgroups — #22256

Order 256.

307 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 155 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	#3160	C₂	1	1	yes	maximal, characteristic
2	#2960	C₂	1	2	yes	maximal
3	#1994	C₂	1	2	yes	maximal
4	#1922	C₂	1	8	yes	maximal
5	#1837	C₂	1	1	yes	maximal, characteristic
6	#1744	C₂	1	1	yes	maximal, characteristic
7	#533	C₂²	1	2	yes	—
8	#500	C₄	1	1	yes	characteristic, complemented
9	#500	C₄	1	1	yes	characteristic, complemented
10	#499	C₄	1	4	yes	complemented
11	#498	C₄	1	2	yes	complemented
12	#446	—	2	2	no	—
13	#446	—	2	2	no	—
14	C₈×Q₈	—	2	2	no	—
15	C₈×Q₈	—	2	2	no	—
16	#427	C₂²	1	4	yes	—
17	#424	C₂²	1	4	yes	—
18	#423	C₂²	1	2	yes	—
19	#405	C₂²	1	4	yes	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄³	C₂²	1	1	yes	characteristic
23	#341	C₂²	1	4	yes	—
24	#339	C₂²	1	8	yes	—
25	#336	C₂²	1	4	yes	—
26	C₂×M₁₆	C₂³	1	4	yes	—
27	C₂²×C₈	C₂³	1	2	yes	—
28	C₂²×C₈	C₂³	1	2	yes	—
29	C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	#125	C₂×C₄	1	4	yes	—
32	C₂²⋊Q₈	—	2	2	no	—
33	C₂²⋊Q₈	—	2	2	no	—
34	C₄×Q₈	—	2	2	no	—
35	C₄×Q₈	—	2	2	no	—
36	#117	C₂³	1	4	yes	—
37	C₂×C₄²	—	2	1	no	—
38	C₂×C₄²	—	2	1	no	—
39	C₂×C₄²	C₂³	1	2	yes	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	#107	C₂×C₄	1	4	yes	—
42	#107	C₂×C₄	1	4	yes	—
43	#106	C₂×C₄	1	8	yes	—
44	C₄⋊C₈	—	2	4	no	—
45	C₄⋊C₈	—	2	4	no	—
46	C₄⋊C₈	—	2	2	no	—
47	C₄⋊C₈	—	2	2	no	—
48	#97	—	2	4	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈	C₂×C₄	1	2	yes	—
52	C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈	D₄	1	1	yes	characteristic
54	C₄×C₈	D₄	1	2	yes	—
55	C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	2	2	no	—
58	C₂²×C₄	—	2	2	no	—
59	C₂²×C₄	#41	1	1	yes	characteristic
60	C₂²×C₄	#41	1	1	yes	characteristic
61	C₂²×C₄	C₂⁴	1	1	yes	characteristic
62	M₁₆	—	2	8	no	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₈	C₂×D₄	1	2	yes	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	—	2	4	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄	C₂²×C₄	1	4	yes	—
76	C₂²⋊C₄	—	2	4	no	—
77	C₄²	—	2	4	no	—
78	C₄²	—	2	4	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	4	2	no	—
81	C₄²	C₂²×C₄	1	1	yes	characteristic
82	C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₄²	C₂×D₄	1	1	yes	characteristic
84	C₄²	C₂×D₄	1	1	yes	characteristic
85	C₂³	C₂×#41	1	1	yes	characteristic
86	Q₈	—	4	4	no	—
87	Q₈	—	4	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄	2₋¹⁺⁴	1	4	yes	—
102	C₈	—	2	2	no	—
103	C₈	—	2	2	no	—
104	C₈	—	2	2	no	—
105	C₈	—	2	2	no	—
106	C₈	—	4	4	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂²	#541	1	2	yes	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	4	4	no	—
114	C₄	—	4	4	no	—
115	C₄	—	4	4	no	—
116	C₄	#443	1	2	yes	—
117	C₄	#444	1	2	yes	—
118	C₄	#519	1	2	yes	—
119	C₄	#536	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	#2327	1	1	yes	characteristic
122	C₂	C₂×#443	1	1	yes	characteristic
123	C₂	C₂×#444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3160

C₂

yes

maximal, characteristic

#2960

C₂

yes

maximal

#1994

C₂

yes

maximal

#1922

C₂

yes

maximal

#1837

C₂

yes

maximal, characteristic

#1744

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#500

C₄

yes

characteristic, complemented

#500

C₄

yes

characteristic, complemented

#499

C₄

yes

complemented

#498

C₄

yes

complemented

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

#427

C₂²

yes

—

#424

C₂²

yes

—

#423

C₂²

yes

—

#405

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#341

C₂²

yes

—

#339

C₂²

yes

—

#336

C₂²

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#125

C₂×C₄

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#107

C₂×C₄

yes

—

#107

C₂×C₄

yes

—

#106

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₄⋊C₄

C₂²×C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₂³

C₂×#41

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₂³×C₄

yes

characteristic

100

C₂×C₄

C₂²×D₄

yes

characteristic

101

C₂×C₄

2₋¹⁺⁴

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₄×D₄

yes

characteristic

110

C₂²

#541

yes

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#443

yes

—

117

C₄

#444

yes

—

118

C₄

#519

yes

—

119

C₄

#536

yes

—

120

C₂

—

121

C₂

#2327

yes

characteristic

122

C₂

C₂×#443

yes

characteristic

123

C₂

C₂×#444

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle

C₂²

C₂×C₄×D₄

prime radical

C₂³

C₂×#41

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
8. #500	C₄	semidirect
9. #500	C₄	semidirect
10. #499	C₄	semidirect
11. #498	C₄	semidirect

H ≅ G/N

type

8. #500

C₄

semidirect

9. #500

C₄

semidirect

10. #499

C₄

semidirect

11. #498

C₄

semidirect

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

subgroup	index
#3160	2
#2960	2
#1994	2
#1922	2
#1837	2
#1744	2

subgroup

index

Subgroups — #22256

Conjugacy classes of subgroups

Complements

Maximal subgroups