Group #9792 — subgroups

Subgroups — #9792

Order 256.

179 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 65 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×#499	C₂	1	1	yes	maximal, characteristic
2	#1568	C₂	1	1	yes	maximal, characteristic
3	#1528	C₂	1	2	yes	maximal
4	#1470	C₂	1	2	yes	maximal
5	#1451	C₂	1	1	yes	maximal, characteristic
6	C₂×#125	C₂²	1	2	yes	—
7	#499	—	2	1	no	—
8	#499	—	2	1	no	—
9	#499	—	2	1	no	—
10	#499	—	2	1	no	—
11	#432	C₂²	1	2	yes	—
12	#431	C₂²	1	1	yes	characteristic
13	C₂×#107	C₄	1	1	yes	characteristic
14	C₂×#107	C₄	1	1	yes	characteristic
15	C₂×#106	C₄	1	2	yes	—
16	C₄×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#335	—	2	1	no	—
19	#334	—	2	1	no	—
20	#330	—	2	4	no	—
21	#324	—	2	4	no	—
22	#322	—	2	1	no	—
23	#322	—	2	1	no	—
24	C₂×M₁₆	—	2	1	no	—
25	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	#125	—	2	2	no	—
28	#125	—	2	2	no	—
29	#125	—	2	2	no	—
30	#125	—	2	2	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	#107	—	2	1	no	—
37	#107	—	2	1	no	—
38	#107	—	2	1	no	—
39	#107	—	2	1	no	—
40	#106	—	2	2	no	—
41	#106	—	2	2	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	—	2	1	no	—
44	C₂²⋊C₈	—	2	2	no	—
45	C₂²⋊C₈	D₄	1	2	yes	—
46	#97	—	2	2	no	—
47	#97	D₄	1	2	yes	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	D₄	1	1	yes	characteristic
51	C₄×C₈	D₄	1	1	yes	characteristic
52	C₄×C₈	D₄	1	1	yes	characteristic
53	C₄×C₈	D₄	1	1	yes	characteristic
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄	#41	1	1	yes	characteristic
59	M₁₆	—	4	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄²	C₂×D₄	1	1	yes	characteristic
78	C₄²	C₂×D₄	1	1	yes	characteristic
79	C₄²	C₂×D₄	1	1	yes	characteristic
80	C₄²	#41	1	1	yes	characteristic
81	C₂³	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	4	no	—
89	C₂×C₄	—	8	2	no	—
90	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄	#121	1	1	yes	characteristic
93	C₂×C₄	#121	1	1	yes	characteristic
94	C₂×C₄	#124	1	1	yes	characteristic
95	C₂×C₄	#127	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	4	2	no	—
99	C₈	—	4	2	no	—
100	C₈	—	8	2	no	—
101	C₈	#101	1	2	yes	—
102	C₈	#101	1	2	yes	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#390	1	1	yes	characteristic
106	C₂²	#444	1	2	yes	—
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	8	2	no	—
113	C₄	—	8	2	no	—
114	C₄	C₂×#101	1	1	yes	characteristic
115	C₄	C₂×#101	1	1	yes	characteristic
116	C₂	—	2	2	no	—
117	C₂	#1966	1	1	yes	characteristic
118	C₂	#1975	1	1	yes	characteristic
119	C₂	#1976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#499

C₂

yes

maximal, characteristic

#1568

C₂

yes

maximal, characteristic

#1528

C₂

yes

maximal

#1470

C₂

yes

maximal

#1451

C₂

yes

maximal, characteristic

C₂×#125

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#431

C₂²

yes

characteristic

C₂×#107

C₄

yes

characteristic

C₂×#107

C₄

yes

characteristic

C₂×#106

C₄

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

D₄

yes

—

#97

—

#97

D₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

#101

yes

—

102

C₈

#101

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

#390

yes

characteristic

106

C₂²

#444

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

C₂×#101

yes

characteristic

115

C₄

C₂×#101

yes

characteristic

116

C₂

—

117

C₂

#1966

yes

characteristic

118

C₂

#1975

yes

characteristic

119

C₂

#1976

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#390
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂²

C₂²×C₄

C₂×C₄²

C₂³

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
C₂×#499	2
#1568	2
#1528	2
#1470	2
#1451	2

subgroup

index

Subgroups — #9792

Conjugacy classes of subgroups

Maximal subgroups