Group #8261 — subgroups

Subgroups — #8261

Order 256.

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

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#322	C₂	1	2	yes	maximal
5	#1273	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈	C₄	1	2	yes	—
7	C₂×C₄⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₈	C₄	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄³	C₂²	1	1	yes	characteristic
15	#336	C₄	1	2	yes	—
16	#336	C₄	1	2	yes	—
17	#322	—	2	4	no	—
18	C₂²×C₈	C₂×C₄	1	4	yes	—
19	C₂²×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²	C₂³	1	1	yes	characteristic
29	C₄⋊C₈	—	2	4	no	—
30	C₄⋊C₈	C₈	1	8	yes	complemented
31	C₄×C₈	—	2	4	no	—
32	C₄×C₈	—	2	4	no	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	—	2	1	no	—
37	C₄×C₈	—	2	2	no	—
38	C₄×C₈	—	2	2	no	—
39	C₄×C₈	D₄	1	2	yes	—
40	C₄×C₈	Q₈	1	2	yes	—
41	C₂²×C₄	C₄²	1	1	yes	characteristic
42	C₂²×C₄	C₄²	1	1	yes	characteristic
43	C₂²×C₄	C₄²	1	1	yes	characteristic
44	C₂²×C₄	C₄²	1	1	yes	characteristic
45	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	—	2	4	no	—
56	C₂×C₈	—	2	4	no	—
57	C₂×C₈	C₄⋊C₄	1	4	yes	—
58	C₂×C₈	C₂×C₈	1	8	yes	—
59	C₂×C₈	M₁₆	1	4	yes	—
60	C₄²	—	2	2	no	—
61	C₄²	—	2	2	no	—
62	C₄²	—	2	2	no	—
63	C₄²	C₂×C₈	1	4	yes	—
64	C₄²	M₁₆	1	4	yes	—
65	C₄²	M₁₆	1	4	yes	—
66	C₄²	C₂×D₄	1	1	yes	characteristic
67	C₄²	C₂×Q₈	1	1	yes	characteristic
68	C₄²	#41	1	1	yes	characteristic
69	C₄²	#41	1	1	yes	characteristic
70	C₂³	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	C₄×C₈	1	2	yes	—
78	C₂×C₄	C₄×C₈	1	2	yes	—
79	C₂×C₄	#97	1	2	yes	—
80	C₂×C₄	#97	1	2	yes	—
81	C₂×C₄	#97	1	2	yes	—
82	C₂×C₄	#97	1	2	yes	—
83	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄	#117	1	1	yes	characteristic
85	C₂×C₄	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄	C₂²×C₈	1	2	yes	—
90	C₂×C₄	C₂×M₁₆	1	2	yes	—
91	C₂×C₄	C₂×M₁₆	1	2	yes	—
92	C₂×C₄	C₂×M₁₆	1	2	yes	—
93	C₂×C₄	#131	1	2	yes	—
94	C₂×C₄	#131	1	2	yes	—
95	C₈	—	2	4	no	—
96	C₈	—	2	4	no	—
97	C₈	—	2	8	no	—
98	C₈	—	4	4	no	—
99	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
101	C₂²	C₂×#97	1	1	yes	characteristic
102	C₂²	C₂×#97	1	1	yes	characteristic
103	C₂²	C₄×M₁₆	1	1	yes	characteristic
104	C₂²	#405	1	1	yes	characteristic
105	C₂²	#405	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	2	no	—
109	C₄	#322	1	4	yes	complemented
110	C₄	#423	1	2	yes	—
111	C₄	#432	1	2	yes	—
112	C₄	C₈×D₄	1	2	yes	—
113	C₄	#436	1	2	yes	—
114	C₄	C₈×Q₈	1	2	yes	—
115	C₄	#446	1	2	yes	—
116	C₂	C₂×#322	1	1	yes	characteristic
117	C₂	C₈×M₁₆	1	1	yes	characteristic
118	C₂	#1451	1	1	yes	characteristic
119	C₂	#1452	1	1	yes	characteristic
120	C₂	#1767	1	1	yes	characteristic
121	C₂	C₈×C₄⋊C₄	1	1	yes	characteristic
122	C₂	#1770	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₄²×C₈

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal

#1273

C₂

yes

maximal

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#336

C₄

yes

—

#336

C₄

yes

—

#322

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₈

yes

complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

—

C₂²×C₄

C₄²

yes

characteristic

C₂²×C₄

C₄²

yes

characteristic

C₂²×C₄

C₄²

yes

characteristic

C₂²×C₄

C₄²

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₄²

M₁₆

yes

—

C₄²

M₁₆

yes

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₄²

yes

C₂³

C₂×C₄²

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

#97

yes

—

C₂×C₄

#97

yes

—

C₂×C₄

#97

yes

—

C₂×C₄

#97

yes

—

C₂×C₄

C₂×C₄⋊C₄

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

C₂²×C₈

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

C₄×C₄⋊C₄

yes

characteristic

100

C₂²

C₂×C₄×C₈

yes

characteristic

101

C₂²

C₂×#97

yes

characteristic

102

C₂²

C₂×#97

yes

characteristic

103

C₂²

C₄×M₁₆

yes

characteristic

104

C₂²

#405

yes

characteristic

105

C₂²

#405

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#322

yes

complemented

110

C₄

#423

yes

—

111

C₄

#432

yes

—

112

C₄

C₈×D₄

yes

—

113

C₄

#436

yes

—

114

C₄

C₈×Q₈

yes

—

115

C₄

#446

yes

—

116

C₂

C₂×#322

yes

characteristic

117

C₂

C₈×M₁₆

yes

characteristic

118

C₂

#1451

yes

characteristic

119

C₂

#1452

yes

characteristic

120

C₂

#1767

yes

characteristic

121

C₂

C₈×C₄⋊C₄

yes

characteristic

122

C₂

#1770

yes

characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂³
commutator	C₂²	C₂×C₄×C₈
socle, prime radical	C₂³	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂×C₄²

C₂³

commutator

C₂²

C₂×C₄×C₈

socle, prime radical

C₂³

C₂×C₄²

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
30. C₄⋊C₈	C₈	semidirect
109. C₄	#322	semidirect

H ≅ G/N

type

30. C₄⋊C₈

C₈

semidirect

109. C₄

#322

semidirect

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

subgroup	index
C₄×C₄⋊C₈	2
C₄²×C₈	2
C₂×#322	2
#1273	2

subgroup

index

C₄×C₄⋊C₈

C₄²×C₈

C₂×#322

#1273

Subgroups — #8261

Conjugacy classes of subgroups

Complements

Maximal subgroups