Group #8821 — subgroups

Subgroups — #8821

Order 256.

203 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 103 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	#1845	C₂	1	1	yes	maximal, characteristic
2	#1844	C₂	1	1	yes	maximal, characteristic
3	#1834	C₂	1	1	yes	maximal, characteristic
4	#1834	C₂	1	1	yes	maximal, characteristic
5	#1834	C₂	1	1	yes	maximal, characteristic
6	#1834	C₂	1	1	yes	maximal, characteristic
7	C₂×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×#107	—	2	2	no	—
9	C₂×#106	—	2	2	no	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	C₂²	1	1	yes	characteristic
14	#384	C₂²	1	1	yes	characteristic
15	#384	C₂²	1	1	yes	characteristic
16	#384	C₂²	1	1	yes	characteristic
17	#340	—	2	2	no	—
18	#340	—	2	2	no	—
19	#340	—	2	2	no	—
20	#340	—	2	2	no	—
21	#322	C₄	1	4	yes	complemented
22	C₂²×C₈	D₄	1	2	yes	—
23	C₂²×C₈	D₄	1	2	yes	—
24	C₂²×C₈	D₄	1	2	yes	—
25	C₂×C₄⋊C₄	—	2	1	no	—
26	C₂×C₄⋊C₄	—	2	1	no	—
27	C₂×C₄⋊C₄	—	2	1	no	—
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄⋊C₄	—	2	1	no	—
30	C₂×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄⋊C₄	—	2	1	no	—
35	C₂×C₄⋊C₄	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	#107	—	4	4	no	—
39	#106	—	4	4	no	—
40	C₄×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₈	Q₈	1	2	yes	—
44	C₄×C₈	Q₈	1	2	yes	—
45	C₄×C₈	Q₈	1	2	yes	—
46	C₂²×C₄	—	4	1	no	—
47	C₂²×C₄	—	4	1	no	—
48	C₂²×C₄	—	4	1	no	—
49	C₂²×C₄	—	4	1	no	—
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	C₄⋊C₄	1	4	yes	—
60	C₂×C₈	C₄⋊C₄	1	4	yes	—
61	C₂×C₈	C₄⋊C₄	1	4	yes	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	—	4	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	—	4	2	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₄²	C₂×Q₈	1	1	yes	characteristic
76	C₄²	C₂×Q₈	1	1	yes	characteristic
77	C₄²	C₂×Q₈	1	1	yes	characteristic
78	C₂³	#127	1	1	yes	characteristic
79	C₂×C₄	—	8	1	no	—
80	C₂×C₄	—	8	1	no	—
81	C₂×C₄	—	8	1	no	—
82	C₂×C₄	—	8	1	no	—
83	C₂×C₄	—	8	1	no	—
84	C₂×C₄	—	8	1	no	—
85	C₂×C₄	—	8	1	no	—
86	C₂×C₄	—	8	1	no	—
87	C₂×C₄	—	8	1	no	—
88	C₂×C₄	—	8	1	no	—
89	C₂×C₄	—	8	1	no	—
90	C₂×C₄	—	8	1	no	—
91	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄	#135	1	2	yes	—
98	C₂×C₄	#135	1	2	yes	—
99	C₂×C₄	AGL(1,ℤ/8)	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	4	no	—
104	C₈	—	2	4	no	—
105	C₈	—	2	4	no	—
106	C₂²	#384	1	1	yes	characteristic
107	C₂²	#495	1	1	yes	characteristic
108	C₂²	#495	1	1	yes	characteristic
109	C₂²	#496	1	1	yes	characteristic
110	C₂²	#496	1	1	yes	characteristic
111	C₂²	#497	1	1	yes	characteristic
112	C₂²	#497	1	1	yes	characteristic
113	C₄	—	16	1	no	—
114	C₄	—	16	1	no	—
115	C₄	—	16	1	no	—
116	C₄	—	16	1	no	—
117	C₄	#427	1	2	yes	—
118	C₄	#428	1	2	yes	—
119	C₄	#428	1	2	yes	—
120	C₄	#499	1	2	yes	—
121	C₄	#501	1	2	yes	—
122	C₄	#501	1	2	yes	—
123	C₂	#1717	1	1	yes	characteristic
124	C₂	#1718	1	1	yes	characteristic
125	C₂	#1719	1	1	yes	characteristic
126	C₂	#1720	1	1	yes	characteristic
127	C₂	#1846	1	1	yes	characteristic
128	C₂	#1847	1	1	yes	characteristic
129	C₂	#1847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1845

C₂

yes

maximal, characteristic

#1844

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#384

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

Q₈

yes

—

C₄×C₈

Q₈

yes

—

C₄×C₈

Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₂³

#127

yes

characteristic

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

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

#135

yes

—

C₂×C₄

#135

yes

—

C₂×C₄

AGL(1,ℤ/8)

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

C₂²

#384

yes

characteristic

107

C₂²

#495

yes

characteristic

108

C₂²

#495

yes

characteristic

109

C₂²

#496

yes

characteristic

110

C₂²

#496

yes

characteristic

111

C₂²

#497

yes

characteristic

112

C₂²

#497

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#427

yes

—

118

C₄

#428

yes

—

119

C₄

#428

yes

—

120

C₄

#499

yes

—

121

C₄

#501

yes

—

122

C₄

#501

yes

—

123

C₂

#1717

yes

characteristic

124

C₂

#1718

yes

characteristic

125

C₂

#1719

yes

characteristic

126

C₂

#1720

yes

characteristic

127

C₂

#1846

yes

characteristic

128

C₂

#1847

yes

characteristic

129

C₂

#1847

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#127
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

#127

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
21. #322	C₄	semidirect

H ≅ G/N

type

21. #322

C₄

semidirect

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

subgroup	index
#1845	2
#1844	2
#1834	2
C₂×#322	2

subgroup

index

Subgroups — #8821

Conjugacy classes of subgroups

Complements

Maximal subgroups