Group #8619 — subgroups

Subgroups — #8619

Order 256.

189 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 75 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	#1801	C₂	1	2	yes	maximal
3	#1567	C₂	1	1	yes	maximal, characteristic
4	#1566	C₂	1	1	yes	maximal, characteristic
5	#1292	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	#446	—	2	1	no	—
8	#446	—	2	1	no	—
9	#446	—	2	1	no	—
10	#446	—	2	1	no	—
11	#423	C₂²	1	2	yes	—
12	#423	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×#101	—	2	2	no	—
16	C₂×#97	C₄	1	2	yes	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#343	—	2	2	no	—
19	#335	—	2	1	no	—
20	#335	—	2	1	no	—
21	#334	—	2	1	no	—
22	#334	—	2	1	no	—
23	C₂²×Q₈	D₄	1	1	yes	characteristic
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	—	2	2	no	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈	—	2	1	no	—
30	C₄×Q₈	—	2	1	no	—
31	C₄×Q₈	—	2	1	no	—
32	C₄×Q₈	—	2	1	no	—
33	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄	Q₈	1	2	yes	—
35	C₂×C₄²	C₂×C₄	1	2	yes	—
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	—	2	2	no	—
45	#101	—	4	4	no	—
46	#97	—	2	2	no	—
47	#97	—	2	2	no	—
48	C₄×C₈	D₄	1	1	yes	characteristic
49	C₄×C₈	D₄	1	1	yes	characteristic
50	C₄×C₈	D₄	1	1	yes	characteristic
51	C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×Q₈	—	2	1	no	—
53	C₂×Q₈	—	2	1	no	—
54	C₂×Q₈	—	2	2	no	—
55	C₂×Q₈	—	2	2	no	—
56	C₂²×C₄	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
61	M₁₆	—	4	2	no	—
62	M₁₆	—	4	2	no	—
63	M₁₆	—	4	2	no	—
64	M₁₆	—	4	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
75	C₄⋊C₄	—	2	1	no	—
76	C₄⋊C₄	—	2	1	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄²	—	2	2	no	—
80	C₄²	—	2	2	no	—
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₄²	C₂×D₄	1	1	yes	characteristic
83	C₄²	#41	1	1	yes	characteristic
84	C₄²	#41	1	1	yes	characteristic
85	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
86	Q₈	—	4	2	no	—
87	Q₈	—	4	1	no	—
88	Q₈	—	4	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#117	1	1	yes	characteristic
98	C₂×C₄	#121	1	1	yes	characteristic
99	C₂×C₄	#121	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄	#137	1	2	yes	—
104	C₈	—	2	1	no	—
105	C₈	—	2	1	no	—
106	C₈	—	2	1	no	—
107	C₈	—	4	2	no	—
108	C₈	—	8	2	no	—
109	C₈	—	8	2	no	—
110	C₈	#101	1	2	yes	complemented
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#380	1	1	yes	characteristic
114	C₂²	#428	1	2	yes	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	4	2	no	—
119	C₄	—	4	2	no	—
120	C₄	C₂×#101	1	1	yes	characteristic
121	C₄	#413	1	1	yes	characteristic
122	C₄	#468	1	2	yes	—
123	C₄	#469	1	1	yes	characteristic
124	C₄	#470	1	1	yes	characteristic
125	C₂	—	2	2	no	—
126	C₂	#1801	1	1	yes	characteristic
127	C₂	#1809	1	1	yes	characteristic
128	C₂	#1813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#446

C₂

yes

maximal, characteristic

#1801

C₂

yes

maximal

#1567

C₂

yes

maximal, characteristic

#1566

C₂

yes

maximal, characteristic

#1292

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

—

—

—

—

C₂²

yes

—

#423

C₂²

yes

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#101

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

C₂²×Q₈

D₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄⋊C₄

Q₈

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

#97

—

#97

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

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

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

C₂×C₄⋊C₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²⋊Q₈

yes

characteristic

101

C₂×C₄

C₂²⋊Q₈

yes

characteristic

102

C₂×C₄

AGL(1,ℤ/8)

yes

—

103

C₂×C₄

#137

yes

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

#101

yes

complemented

111

C₂²

—

112

C₂²

—

113

C₂²

#380

yes

characteristic

114

C₂²

#428

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#101

yes

characteristic

121

C₄

#413

yes

characteristic

122

C₄

#468

yes

—

123

C₄

#469

yes

characteristic

124

C₄

#470

yes

characteristic

125

C₂

—

126

C₂

#1801

yes

characteristic

127

C₂

#1809

yes

characteristic

128

C₂

#1813

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#380
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₂×C₄⋊C₄
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

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
110. C₈	#101	semidirect

H ≅ G/N

type

110. C₈

#101

semidirect

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

subgroup	index
C₂×#446	2
#1801	2
#1567	2
#1566	2
#1292	2

subgroup

index

Subgroups — #8619

Conjugacy classes of subgroups

Complements

Maximal subgroups