Group #8655 — subgroups

Subgroups — #8655

Order 256.

209 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 87 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×#446	C₂	1	1	yes	maximal, characteristic
2	#1834	C₂	1	1	yes	maximal, characteristic
3	#1803	C₂	1	2	yes	maximal
4	C₂×#335	C₂	1	1	yes	maximal, characteristic
5	#1296	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	#446	—	2	2	no	—
8	#446	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#103	—	2	2	no	—
13	C₂×#97	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#384	C₂²	1	2	yes	—
16	#340	—	2	1	no	—
17	#340	—	2	1	no	—
18	#340	—	2	2	no	—
19	#335	—	2	2	no	—
20	C₂²×Q₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	C₄×Q₈	—	2	1	no	—
26	C₄×Q₈	—	2	1	no	—
27	C₄×Q₈	C₂×C₄	1	2	yes	—
28	C₄×Q₈	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄	—	2	1	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₂×C₄²	—	2	1	no	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	C₄⋊C₈	—	2	4	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	2	no	—
40	C₄⋊C₈	C₂×C₄	1	2	yes	—
41	#103	—	4	4	no	—
42	#97	—	2	4	no	—
43	C₄×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	2	2	no	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	—	4	2	no	—
52	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄	#41	1	1	yes	characteristic
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	4	2	no	—
59	C₂×C₈	—	4	2	no	—
60	C₂×C₈	—	4	2	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈	C₂²⋊C₄	1	2	yes	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	8	2	no	—
72	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
73	C₄²	—	2	2	no	—
74	C₄²	—	2	2	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	#121	1	1	yes	characteristic
80	Q₈	—	2	4	no	—
81	Q₈	—	4	1	no	—
82	Q₈	—	4	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	8	2	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	—	8	2	no	—
93	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄	#117	1	1	yes	characteristic
96	C₂×C₄	#121	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄	#135	1	2	yes	—
100	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄	#137	1	2	yes	—
102	C₂×C₄	#137	1	2	yes	—
103	C₈	—	2	2	no	—
104	C₈	—	2	2	no	—
105	C₈	—	4	2	no	—
106	C₈	—	8	2	no	—
107	C₂²	#380	1	1	yes	characteristic
108	C₂²	#464	1	2	yes	—
109	C₂²	#468	1	2	yes	—
110	C₂²	#472	1	1	yes	characteristic
111	C₂²	#473	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	16	2	no	—
117	C₄	#416	1	1	yes	characteristic
118	C₄	#419	1	1	yes	characteristic
119	C₄	#428	1	2	yes	—
120	C₄	#469	1	1	yes	characteristic
121	C₄	#470	1	1	yes	characteristic
122	C₄	#479	1	2	yes	—
123	C₂	#1698	1	2	yes	—
124	C₂	#1700	1	2	yes	—
125	C₂	#1805	1	1	yes	characteristic
126	C₂	#1809	1	1	yes	characteristic
127	C₂	#1816	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#446

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1803

C₂

yes

maximal

C₂×#335

C₂

yes

maximal, characteristic

#1296

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

characteristic

#446

—

#446

C₄

yes

complemented

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

—

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

—

—

—

—

C₂²×Q₈

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

Q₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

#103

—

#97

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

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

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

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

C₂×C₄⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

#135

yes

—

100

C₂×C₄

AGL(1,ℤ/8)

yes

—

101

C₂×C₄

#137

yes

—

102

C₂×C₄

#137

yes

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₂²

#380

yes

characteristic

108

C₂²

#464

yes

—

109

C₂²

#468

yes

—

110

C₂²

#472

yes

characteristic

111

C₂²

#473

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#416

yes

characteristic

118

C₄

#419

yes

characteristic

119

C₄

#428

yes

—

120

C₄

#469

yes

characteristic

121

C₄

#470

yes

characteristic

122

C₄

#479

yes

—

123

C₂

#1698

yes

—

124

C₂

#1700

yes

—

125

C₂

#1805

yes

characteristic

126

C₂

#1809

yes

characteristic

127

C₂

#1816

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
8. #446	C₄	semidirect

H ≅ G/N

type

8. #446

C₄

semidirect

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

subgroup	index
C₂×#446	2
#1834	2
#1803	2
C₂×#335	2
#1296	2

subgroup

index

Subgroups — #8655

Conjugacy classes of subgroups

Complements

Maximal subgroups