Group #9632 — subgroups

Subgroups — #9632

Order 256.

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

normal subgroups only

This group has 147 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	#1837	C₂	1	1	yes	maximal, characteristic
2	#1770	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	1	yes	maximal, characteristic
4	#1570	C₂	1	1	yes	maximal, characteristic
5	#1564	C₂	1	1	yes	maximal, characteristic
6	#1303	C₂	1	1	yes	maximal, characteristic
7	#1303	C₂	1	1	yes	maximal, characteristic
8	#431	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	#423	C₂²	1	1	yes	characteristic
11	#423	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×#97	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#341	—	2	1	no	—
17	#341	—	2	1	no	—
18	#336	—	2	1	no	—
19	#335	—	2	2	no	—
20	#335	C₄	1	2	yes	complemented
21	#334	—	2	2	no	—
22	#334	C₄	1	2	yes	complemented
23	C₂×M₁₆	—	2	1	no	—
24	C₂×M₁₆	—	2	1	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄²	—	2	1	no	—
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	C₄⋊C₈	—	2	1	no	—
32	C₄⋊C₈	—	2	1	no	—
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₄⋊C₈	D₄	1	2	yes	—
39	C₂²⋊C₈	—	2	1	no	—
40	C₂²⋊C₈	—	2	1	no	—
41	#97	—	4	2	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈	Q₈	1	1	yes	characteristic
47	C₄×C₈	Q₈	1	1	yes	characteristic
48	#95	Q₈	1	2	yes	—
49	C₂²×C₄	—	2	2	no	—
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄	#41	1	1	yes	characteristic
53	M₁₆	—	4	1	no	—
54	M₁₆	—	4	1	no	—
55	M₁₆	—	4	1	no	—
56	M₁₆	—	4	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₄⋊C₄	—	4	2	no	—
73	C₄²	—	4	1	no	—
74	C₄²	—	4	1	no	—
75	C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₄²	C₂×D₄	1	1	yes	characteristic
77	C₄²	C₂×Q₈	1	1	yes	characteristic
78	C₄²	#41	1	1	yes	characteristic
79	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	C₂×C₄⋊C₄	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	#121	1	1	yes	characteristic
89	C₂×C₄	#125	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄	#135	1	1	yes	characteristic
92	C₂×C₄	#135	1	1	yes	characteristic
93	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₄	#137	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	2	1	no	—
99	C₈	—	4	2	no	—
100	C₈	—	4	2	no	—
101	C₈	—	4	2	no	—
102	C₈	—	8	1	no	—
103	C₈	—	8	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	#389	1	1	yes	characteristic
107	C₂²	#478	1	1	yes	characteristic
108	C₂²	#479	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	8	2	no	—
113	C₄	#427	1	1	yes	characteristic
114	C₄	#428	1	1	yes	characteristic
115	C₄	#444	1	2	yes	—
116	C₄	#463	1	1	yes	characteristic
117	C₄	#464	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#1946	1	1	yes	characteristic
121	C₂	#1949	1	1	yes	characteristic
122	C₂	#1953	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1837

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal, characteristic

#1571

C₂

yes

maximal, characteristic

#1570

C₂

yes

maximal, characteristic

#1564

C₂

yes

maximal, characteristic

#1303

C₂

yes

maximal, characteristic

#1303

C₂

yes

maximal, characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

C₄

yes

complemented

#334

—

#334

C₄

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

D₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

#95

Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₂²⋊Q₈

yes

characteristic

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×Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#389

yes

characteristic

107

C₂²

#478

yes

characteristic

108

C₂²

#479

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#427

yes

characteristic

114

C₄

#428

yes

characteristic

115

C₄

#444

yes

—

116

C₄

#463

yes

characteristic

117

C₄

#464

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#1946

yes

characteristic

121

C₂

#1949

yes

characteristic

122

C₂

#1953

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#389
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#389

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
20. #335	C₄	semidirect
22. #334	C₄	semidirect

H ≅ G/N

type

20. #335

C₄

semidirect

22. #334

C₄

semidirect

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

subgroup	index
#1837	2
#1770	2
#1571	2
#1570	2
#1564	2
#1303	2

subgroup

index

Subgroups — #9632

Conjugacy classes of subgroups

Complements

Maximal subgroups