Group #9614 — subgroups

Subgroups — #9614

Order 256.

191 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 95 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	#1947	C₂	1	2	yes	maximal
2	#1845	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#335	C₂	1	1	yes	maximal, characteristic
5	#1296	C₂	1	2	yes	maximal
6	C₂×#107	—	2	1	no	—
7	C₂×#107	—	2	1	no	—
8	C₂×#107	—	2	2	no	—
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#340	—	2	2	no	—
16	#336	—	2	1	no	—
17	#335	C₄	1	4	yes	complemented
18	C₂²×C₈	—	2	1	no	—
19	C₂²×C₈	—	2	2	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
27	C₂×C₄²	—	2	1	no	—
28	C₂×C₄²	C₂³	1	1	yes	characteristic
29	#107	—	4	2	no	—
30	#107	—	4	2	no	—
31	#107	—	4	4	no	—
32	C₄⋊C₈	—	2	2	no	—
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	C₂×C₄	1	4	yes	—
35	C₄⋊C₈	Q₈	1	2	yes	—
36	C₄×C₈	—	4	2	no	—
37	C₄×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈	Q₈	1	2	yes	—
39	#95	—	2	1	no	—
40	C₂²×C₄	—	2	1	no	—
41	C₂²×C₄	—	2	1	no	—
42	C₂²×C₄	—	4	2	no	—
43	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄	#41	1	1	yes	characteristic
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂×C₈	—	4	2	no	—
53	C₂×C₈	—	4	1	no	—
54	C₂×C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₈	D₈	1	2	yes	—
57	C₂×C₈	Q₁₆	1	2	yes	—
58	C₄⋊C₄	—	2	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	8	2	no	—
65	C₄⋊C₄	D₈	1	2	yes	—
66	C₄⋊C₄	Q₁₆	1	2	yes	—
67	C₄²	—	4	1	no	—
68	C₄²	—	4	1	no	—
69	C₄²	C₂²×C₄	1	1	yes	characteristic
70	C₄²	C₂×Q₈	1	1	yes	characteristic
71	C₄²	C₂×Q₈	1	1	yes	characteristic
72	C₄²	#41	1	1	yes	characteristic
73	C₂³	#121	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	8	2	no	—
84	C₂×C₄	—	8	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	C₂²⋊Q₈	1	1	yes	characteristic
89	C₂×C₄	#125	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄	#135	1	1	yes	characteristic
96	C₂×C₄	#135	1	1	yes	characteristic
97	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄	#137	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	4	4	no	—
101	C₈	—	4	2	no	—
102	C₈	#107	1	4	yes	complemented
103	C₂²	#389	1	1	yes	characteristic
104	C₂²	#459	1	1	yes	characteristic
105	C₂²	#462	1	1	yes	characteristic
106	C₂²	#466	1	1	yes	characteristic
107	C₂²	#467	1	1	yes	characteristic
108	C₂²	#472	1	1	yes	characteristic
109	C₂²	#473	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	4	2	no	—
113	C₄	—	4	2	no	—
114	C₄	—	16	2	no	—
115	C₄	C₂×#107	1	1	yes	characteristic
116	C₄	#428	1	1	yes	characteristic
117	C₄	C₄×D₈	1	1	yes	characteristic
118	C₄	C₄×Q₁₆	1	1	yes	characteristic
119	C₄	#474	1	1	yes	characteristic
120	C₄	#477	1	1	yes	characteristic
121	C₄	#499	1	1	yes	characteristic
122	C₄	#500	1	1	yes	characteristic
123	C₂	#1667	1	1	yes	characteristic
124	C₂	#1669	1	1	yes	characteristic
125	C₂	#1674	1	1	yes	characteristic
126	C₂	#1676	1	1	yes	characteristic
127	C₂	#1942	1	1	yes	characteristic
128	C₂	#1947	1	1	yes	characteristic
129	C₂	#1952	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1947

C₂

yes

maximal

#1845

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

#1296

C₂

yes

—

—

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

Q₈

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

Q₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₄⋊C₄

Q₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₂³

yes

C₂×C₄

—

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

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#125

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

100

C₈

—

101

C₈

—

102

C₈

#107

yes

complemented

103

C₂²

#389

yes

characteristic

104

C₂²

#459

yes

characteristic

105

C₂²

#462

yes

characteristic

106

C₂²

#466

yes

characteristic

107

C₂²

#467

yes

characteristic

108

C₂²

#472

yes

characteristic

109

C₂²

#473

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

C₂×#107

yes

characteristic

116

C₄

#428

yes

characteristic

117

C₄

C₄×D₈

yes

characteristic

118

C₄

C₄×Q₁₆

yes

characteristic

119

C₄

#474

yes

characteristic

120

C₄

#477

yes

characteristic

121

C₄

#499

yes

characteristic

122

C₄

#500

yes

characteristic

123

C₂

#1667

yes

characteristic

124

C₂

#1669

yes

characteristic

125

C₂

#1674

yes

characteristic

126

C₂

#1676

yes

characteristic

127

C₂

#1942

yes

characteristic

128

C₂

#1947

yes

characteristic

129

C₂

#1952

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
17. #335	C₄	semidirect
102. C₈	#107	semidirect

H ≅ G/N

type

17. #335

C₄

semidirect

102. C₈

#107

semidirect

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

subgroup	index
#1947	2
#1845	2
C₈×C₄⋊C₄	2
C₂×#335	2
#1296	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #9614

Conjugacy classes of subgroups

Complements

Maximal subgroups