Group #8649 — subgroups

Subgroups — #8649

Order 256.

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

normal subgroups only

This group has 219 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	C₂×C₈×Q₈	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	C₈×Q₈	—	2	2	no	—
8	C₈×Q₈	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×C₄×C₈	—	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	C₄×C₈	—	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	1	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×Q₈	C₄⋊C₄	1	2	yes	—
49	C₂×Q₈	D₈	1	2	yes	—
50	C₂×Q₈	Q₁₆	1	2	yes	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	—	4	2	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄	#41	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	4	2	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈	C₂²⋊C₄	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	1	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	8	2	no	—
74	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	—	2	2	no	—
77	C₄²	C₂²×C₄	1	1	yes	characteristic
78	C₄²	C₂×D₄	1	1	yes	characteristic
79	C₄²	C₂×Q₈	1	1	yes	characteristic
80	C₄²	#41	1	1	yes	characteristic
81	C₂³	#121	1	1	yes	characteristic
82	Q₈	—	2	2	no	—
83	Q₈	—	2	2	no	—
84	Q₈	—	2	2	no	—
85	Q₈	#107	1	4	yes	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	8	2	no	—
93	C₂×C₄	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#117	1	1	yes	characteristic
98	C₂×C₄	#121	1	1	yes	characteristic
99	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄	#135	1	2	yes	—
104	C₂×C₄	#135	1	1	yes	characteristic
105	C₂×C₄	#135	1	1	yes	characteristic
106	C₂×C₄	#137	1	2	yes	—
107	C₈	—	2	2	no	—
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	8	2	no	—
111	C₂²	#380	1	1	yes	characteristic
112	C₂²	#464	1	2	yes	—
113	C₂²	#466	1	1	yes	characteristic
114	C₂²	#467	1	1	yes	characteristic
115	C₂²	#471	1	2	yes	—
116	C₄	—	2	4	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	4	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#416	1	1	yes	characteristic
122	C₄	#419	1	1	yes	characteristic
123	C₄	C₂×#107	1	1	yes	characteristic
124	C₄	#427	1	1	yes	characteristic
125	C₄	#466	1	1	yes	characteristic
126	C₄	#467	1	1	yes	characteristic
127	C₄	#479	1	2	yes	—
128	C₂	#1682	1	2	yes	—
129	C₂	#1684	1	2	yes	—
130	C₂	#1805	1	1	yes	characteristic
131	C₂	#1808	1	1	yes	characteristic
132	C₂	#1815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

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

C₈×Q₈

—

C₈×Q₈

C₄

yes

complemented

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

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

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂×Q₈

D₈

yes

—

C₂×Q₈

Q₁₆

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

—

Q₈

#107

yes

—

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

100

C₂×C₄

C₂²⋊Q₈

yes

characteristic

101

C₂×C₄

C₂×D₈

yes

characteristic

102

C₂×C₄

C₂×Q₁₆

yes

characteristic

103

C₂×C₄

#135

yes

—

104

C₂×C₄

#135

yes

characteristic

105

C₂×C₄

#135

yes

characteristic

106

C₂×C₄

#137

yes

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂²

#380

yes

characteristic

112

C₂²

#464

yes

—

113

C₂²

#466

yes

characteristic

114

C₂²

#467

yes

characteristic

115

C₂²

#471

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#416

yes

characteristic

122

C₄

#419

yes

characteristic

123

C₄

C₂×#107

yes

characteristic

124

C₄

#427

yes

characteristic

125

C₄

#466

yes

characteristic

126

C₄

#467

yes

characteristic

127

C₄

#479

yes

—

128

C₂

#1682

yes

—

129

C₂

#1684

yes

—

130

C₂

#1805

yes

characteristic

131

C₂

#1808

yes

characteristic

132

C₂

#1815

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. C₈×Q₈	C₄	semidirect

H ≅ G/N

type

8. C₈×Q₈

C₄

semidirect

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

subgroup	index
C₂×C₈×Q₈	2
#1834	2
#1803	2
C₂×#335	2
#1296	2

subgroup

index

C₂×C₈×Q₈

Subgroups — #8649

Conjugacy classes of subgroups

Complements

Maximal subgroups