Group #9756 — subgroups

Subgroups — #9756

Order 256.

251 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 83 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	#3147	C₂	1	1	yes	maximal, characteristic
2	#1846	C₂	1	1	yes	maximal, characteristic, complemented
3	#1517	C₂	1	4	yes	maximal, complemented
4	#1450	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135	—	2	1	no	—
6	C₂×#135	—	2	1	no	—
7	#532	C₂²	1	2	yes	—
8	#495	—	2	1	no	—
9	#495	—	2	1	no	—
10	#494	C₄	1	1	yes	characteristic, complemented
11	#493	C₄	1	1	yes	characteristic, complemented
12	#492	C₄	1	2	yes	complemented
13	#431	C₂²	1	2	yes	—
14	C₂×#107	—	2	1	no	—
15	C₂×#106	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#382	C₂²	1	2	yes	—
18	#327	—	2	8	no	—
19	#322	—	2	1	no	—
20	C₈²	—	2	1	no	—
21	C₂×#41	—	2	2	no	—
22	#135	—	4	2	no	—
23	#135	—	4	2	no	—
24	#135	—	4	2	no	—
25	#135	—	4	2	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	1	no	—
28	C₂×SD₁₆	—	2	2	no	—
29	C₂×SD₁₆	—	2	2	no	—
30	C₂×D₈	—	2	1	no	—
31	C₂×D₈	—	2	1	no	—
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₈	D₄	1	1	yes	characteristic
34	C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	#127	C₂×C₄	1	2	yes	—
36	#124	—	2	2	no	—
37	#121	—	2	4	no	—
38	C₄×D₄	—	2	4	no	—
39	C₂×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	#107	—	4	2	no	—
42	#106	—	4	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₂²⋊C₈	D₄	1	4	yes	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈	D₄	1	1	yes	characteristic
49	C₄×C₈	D₄	1	1	yes	characteristic
50	#95	—	2	4	no	—
51	#41	—	4	2	no	—
52	#41	—	4	2	no	—
53	#41	—	4	2	no	—
54	#41	—	4	2	no	—
55	C₂×Q₈	—	2	2	no	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	4	no	—
58	C₂×D₄	—	4	2	no	—
59	C₂²×C₄	—	4	2	no	—
60	C₂²×C₄	—	4	2	no	—
61	C₂²×C₄	C₂×D₄	1	2	yes	—
62	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
63	Q₁₆	—	4	2	no	—
64	Q₁₆	—	4	2	no	—
65	SD₁₆	—	4	4	no	—
66	SD₁₆	—	4	4	no	—
67	D₈	—	4	2	no	—
68	D₈	—	4	2	no	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈	D₈	1	2	yes	—
75	C₂×C₈	D₈	1	2	yes	—
76	C₂×C₈	SD₁₆	1	2	yes	—
77	C₂×C₈	SD₁₆	1	2	yes	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₂²⋊C₄	—	4	4	no	—
82	C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₄²	C₂×D₄	1	1	yes	characteristic
84	C₄²	#41	1	2	yes	—
85	C₂³	—	4	2	no	—
86	C₂³	#127	1	1	yes	characteristic
87	Q₈	—	4	2	no	—
88	Q₈	—	4	2	no	—
89	D₄	—	4	2	no	—
90	D₄	—	4	2	no	—
91	D₄	—	8	4	no	—
92	D₄	—	8	2	no	—
93	D₄	—	8	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	8	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	8	4	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄	C₄×D₄	1	2	yes	—
102	C₂×C₄	#121	1	2	yes	—
103	C₂×C₄	#124	1	1	yes	characteristic
104	C₂×C₄	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
108	C₈	—	4	4	no	—
109	C₈	#102	1	2	yes	—
110	C₈	#102	1	2	yes	—
111	C₈	#102	1	2	yes	—
112	C₈	#102	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	—	8	2	no	—
115	C₂²	—	8	4	no	—
116	C₂²	#390	1	1	yes	characteristic
117	C₂²	#492	1	1	yes	characteristic
118	C₂²	#493	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	2	2	no	—
121	C₄	—	8	2	no	—
122	C₄	—	16	2	no	—
123	C₄	C₂×#102	1	1	yes	characteristic
124	C₄	C₂×#102	1	1	yes	characteristic
125	C₄	#486	1	1	yes	characteristic
126	C₄	#486	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	16	2	no	—
130	C₂	#1957	1	1	yes	characteristic
131	C₂	#1976	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3147

C₂

yes

maximal, characteristic

#1846

C₂

yes

maximal, characteristic, complemented

#1517

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

C₂×#135

—

#532

C₂²

yes

—

#495

—

#495

—

#494

C₄

yes

characteristic, complemented

#493

C₄

yes

characteristic, complemented

#492

C₄

yes

complemented

#431

C₂²

yes

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

C₂²

yes

characteristic

#382

C₂²

yes

—

#327

—

#322

—

C₈²

—

C₂×#41

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

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

C₂×C₈

D₈

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

—

C₂³

—

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

101

C₂×C₄

C₄×D₄

yes

—

102

C₂×C₄

#121

yes

—

103

C₂×C₄

#124

yes

characteristic

104

C₂×C₄

C₂×D₈

yes

characteristic

105

C₂×C₄

C₂×D₈

yes

characteristic

106

C₂×C₄

C₂×SD₁₆

yes

characteristic

107

C₂×C₄

C₂×SD₁₆

yes

characteristic

108

C₈

—

109

C₈

#102

yes

—

110

C₈

#102

yes

—

111

C₈

#102

yes

—

112

C₈

#102

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#390

yes

characteristic

117

C₂²

#492

yes

characteristic

118

C₂²

#493

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂×#102

yes

characteristic

124

C₄

C₂×#102

yes

characteristic

125

C₄

#486

yes

characteristic

126

C₄

#486

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#1957

yes

characteristic

131

C₂

#1976

yes

—

	subgroup	G/H
center, socle	C₂²	#390
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#390

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1846	C₂	semidirect
3. #1517	C₂	semidirect
4. #1450	C₂	semidirect
10. #494	C₄	semidirect
11. #493	C₄	semidirect
12. #492	C₄	semidirect

H ≅ G/N

type

2. #1846

C₂

semidirect

3. #1517

C₂

semidirect

4. #1450

C₂

semidirect

10. #494

C₄

semidirect

11. #493

C₄

semidirect

12. #492

C₄

semidirect

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

subgroup	index
#3147	2
#1846	2
#1517	2
#1450	2

subgroup

index

Subgroups — #9756

Conjugacy classes of subgroups

Complements

Maximal subgroups