Group #21726 — subgroups

Subgroups — #21726

Order 256.

427 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₂×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈	C₂	1	2	yes	maximal
3	#1990	C₂	1	2	yes	maximal
4	#1943	C₂	1	4	yes	maximal
5	#1844	C₂	1	1	yes	maximal, characteristic
6	#1776	C₂	1	4	yes	maximal
7	C₄×#97	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×Q₈	—	2	2	no	—
10	#500	C₄	1	8	yes	complemented
11	C₂×#107	C₂²	1	4	yes	—
12	C₂×#106	C₂²	1	4	yes	—
13	C₂×#97	C₂²	1	2	yes	—
14	C₂×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#391	—	2	2	no	—
18	#389	—	2	2	no	—
19	#389	C₂²	1	4	yes	—
20	#384	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄	—	2	2	no	—
22	C₄×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄³	C₂²	1	1	yes	characteristic
24	#340	C₂²	1	8	yes	—
25	C₂²×Q₈	—	2	2	no	—
26	C₂²×C₈	C₂³	1	2	yes	—
27	C₂²×C₈	C₂³	1	2	yes	—
28	C₄⋊Q₈	—	2	4	no	—
29	C₄⋊Q₈	C₂×C₄	1	8	yes	—
30	C₄×Q₈	—	4	8	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	—	2	4	no	—
34	C₂×C₄⋊C₄	—	2	2	no	—
35	C₂×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₄²	—	2	4	no	—
38	C₂×C₄²	D₄	1	2	yes	—
39	C₂×C₄²	D₄	1	2	yes	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	#107	C₂×C₄	1	16	yes	—
44	#106	—	2	8	no	—
45	#97	—	2	4	no	—
46	#97	Q₈	1	8	yes	complemented
47	C₄×C₈	—	2	2	no	—
48	C₄×C₈	C₂×C₄	1	4	yes	—
49	#95	—	2	4	no	—
50	#95	—	2	4	no	—
51	C₂×Q₈	—	2	8	no	—
52	C₂×Q₈	—	4	4	no	—
53	C₂×Q₈	—	4	4	no	—
54	C₂²×C₄	—	2	4	no	—
55	C₂²×C₄	—	2	4	no	—
56	C₂²×C₄	C₂×D₄	1	2	yes	—
57	C₂²×C₄	C₂×D₄	1	2	yes	—
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	C₂⁴	1	1	yes	characteristic
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₈	C₂×Q₈	1	4	yes	—
65	C₂×C₈	C₂×Q₈	1	4	yes	—
66	C₂×C₈	#41	1	4	yes	—
67	C₄⋊C₄	—	2	8	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	2	8	no	—
70	C₄⋊C₄	—	4	8	no	—
71	C₄⋊C₄	—	4	4	no	—
72	C₄⋊C₄	—	4	4	no	—
73	C₄⋊C₄	C₂²×C₄	1	8	yes	—
74	C₄²	—	2	4	no	—
75	C₄²	—	2	4	no	—
76	C₄²	—	2	2	no	—
77	C₄²	—	4	8	no	—
78	C₄²	C₂²×C₄	1	2	yes	—
79	C₄²	C₂×Q₈	1	4	yes	—
80	C₄²	#41	1	2	yes	—
81	C₂³	C₂²×D₄	1	1	yes	characteristic
82	Q₈	—	4	16	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	8	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	4	4	no	—
92	C₂×C₄	—	4	4	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	C₄×D₄	1	2	yes	—
95	C₂×C₄	C₄×D₄	1	2	yes	—
96	C₂×C₄	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×C₄	#137	1	4	yes	—
99	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×Q₈	1	2	yes	—
101	C₂×C₄	C₂×#41	1	2	yes	—
102	C₂×C₄	C₂×#41	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	C₄×Q₈	1	8	yes	complemented
105	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
106	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²	#532	1	1	yes	characteristic
108	C₂²	C₂×AGL(1,ℤ/8)	1	2	yes	—
109	C₂²	C₂×#137	1	2	yes	—
110	C₄	—	2	4	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	8	no	—
113	C₄	—	8	4	no	—
114	C₄	#441	1	4	yes	—
115	C₄	#442	1	4	yes	—
116	C₄	C₂×C₄×Q₈	1	2	yes	—
117	C₄	#533	1	2	yes	—
118	C₂	C₄×C₄⋊Q₈	1	1	yes	characteristic
119	C₂	C₂×#441	1	1	yes	characteristic
120	C₂	C₂×#442	1	1	yes	characteristic
121	C₂	#3165	1	2	yes	—
122	C₂	#3182	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

C₄×C₄⋊Q₈

C₂

yes

maximal

#1990

C₂

yes

maximal

#1943

C₂

yes

maximal

#1844

C₂

yes

maximal, characteristic

#1776

C₂

yes

maximal

C₄×#97

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈

—

#500

C₄

yes

complemented

C₂×#107

C₂²

yes

—

C₂×#106

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

C₂²

yes

—

#384

C₂²

yes

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄³

C₂²

yes

characteristic

#340

C₂²

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

#107

C₂×C₄

yes

—

#106

—

#97

—

#97

Q₈

yes

complemented

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₂³

C₂²×D₄

yes

characteristic

Q₈

—

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×D₄

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄⋊Q₈

yes

—

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₄

#137

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

100

C₂×C₄

C₂²×Q₈

yes

—

101

C₂×C₄

C₂×#41

yes

—

102

C₂×C₄

C₂×#41

yes

characteristic

103

C₈

—

104

C₈

C₄×Q₈

yes

complemented

105

C₂²

C₂×C₄×D₄

yes

characteristic

106

C₂²

C₂×C₄⋊Q₈

yes

characteristic

107

C₂²

#532

yes

characteristic

108

C₂²

C₂×AGL(1,ℤ/8)

yes

—

109

C₂²

C₂×#137

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#441

yes

—

115

C₄

#442

yes

—

116

C₄

C₂×C₄×Q₈

yes

—

117

C₄

#533

yes

—

118

C₂

C₄×C₄⋊Q₈

yes

characteristic

119

C₂

C₂×#441

yes

characteristic

120

C₂

C₂×#442

yes

characteristic

121

C₂

#3165

yes

—

122

C₂

#3182

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
10. #500	C₄	semidirect
46. #97	Q₈	semidirect
104. C₈	C₄×Q₈	semidirect

H ≅ G/N

type

10. #500

C₄

semidirect

46. #97

Q₈

semidirect

104. C₈

C₄×Q₈

semidirect

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

subgroup	index
C₂×#500	2
C₄×C₄⋊Q₈	2
#1990	2
#1943	2
#1844	2
#1776	2
C₄×#97	2

subgroup

index

C₄×C₄⋊Q₈

C₄×#97

Subgroups — #21726

Conjugacy classes of subgroups

Complements

Maximal subgroups