Group #8233 — subgroups

Subgroups — #8233

Order 256.

258 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#1767	C₂	1	1	yes	maximal, characteristic
3	#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×#332	C₂	1	2	yes	maximal
5	#1296	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈	—	2	1	no	—
8	C₂×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×#97	—	2	1	no	—
11	C₂×#97	—	2	1	no	—
12	#391	—	2	1	no	—
13	#389	—	2	1	no	—
14	#389	C₄	1	2	yes	—
15	#384	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄	—	2	1	no	—
17	C₄×C₄⋊C₄	C₄	1	2	yes	—
18	C₄³	C₂²	1	1	yes	characteristic
19	#336	—	2	1	no	—
20	#336	—	2	1	no	—
21	#332	C₄	1	8	yes	complemented
22	C₂²×Q₈	—	2	1	no	—
23	C₂²×C₈	—	2	2	no	—
24	C₂²×C₈	—	2	2	no	—
25	C₄⋊Q₈	—	2	2	no	—
26	C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₄×Q₈	—	4	4	no	—
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄²	—	2	2	no	—
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²	D₄	1	1	yes	characteristic
38	C₂×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₄²	D₄	1	1	yes	characteristic
40	C₂×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₄⋊C₈	C₂×C₄	1	4	yes	—
46	#97	—	4	2	no	—
47	#97	—	4	2	no	—
48	#95	—	2	2	no	—
49	#95	—	2	2	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	—	4	1	no	—
53	C₂×Q₈	—	4	1	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	4	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	4	4	no	—
73	C₄⋊C₄	—	4	1	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	1	no	—
76	C₄⋊C₄	C₄²	1	4	yes	—
77	C₄²	—	2	2	no	—
78	C₄²	—	2	1	no	—
79	C₄²	—	2	1	no	—
80	C₄²	—	2	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	—	2	1	no	—
83	C₄²	—	2	1	no	—
84	C₄²	—	4	1	no	—
85	C₄²	—	4	4	no	—
86	C₄²	C₂²×C₄	1	2	yes	—
87	C₄²	#41	1	2	yes	—
88	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈	—	4	4	no	—
90	Q₈	—	4	4	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	#101	1	2	yes	—
111	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄	#117	1	1	yes	characteristic
113	C₂×C₄	C₄×D₄	1	2	yes	—
114	C₂×C₄	C₄×D₄	1	2	yes	—
115	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄	#137	1	1	yes	characteristic
119	C₂×C₄	#137	1	2	yes	—
120	C₂×C₄	#137	1	1	yes	characteristic
121	C₈	—	4	4	no	—
122	C₈	—	4	4	no	—
123	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²	C₂×#101	1	1	yes	characteristic
125	C₂²	#413	1	1	yes	characteristic
126	C₂²	#417	1	1	yes	characteristic
127	C₂²	#417	1	1	yes	characteristic
128	C₂²	#418	1	1	yes	characteristic
129	C₂²	#419	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	4	4	no	—
137	C₄	—	8	2	no	—
138	C₄	#440	1	2	yes	—
139	C₄	#440	1	2	yes	—
140	C₄	#441	1	2	yes	—
141	C₄	#442	1	2	yes	—
142	C₂	#1556	1	1	yes	characteristic
143	C₂	#1557	1	1	yes	characteristic
144	C₂	#1559	1	1	yes	characteristic
145	C₂	#1561	1	1	yes	characteristic
146	C₂	C₄×#101	1	1	yes	characteristic
147	C₂	#1762	1	1	yes	characteristic
148	C₂	#1763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊Q₈

C₂

yes

maximal, characteristic

#1767

C₂

yes

maximal, characteristic

#1767

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal

#1296

C₂

yes

maximal

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#97

—

C₂×#97

—

#391

—

#389

—

#389

C₄

yes

—

#384

C₂²

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₄

yes

—

C₄³

C₂²

yes

—

—

C₄

yes

complemented

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₄²

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₂²⋊C₄

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#101

yes

—

111

C₂×C₄

C₂×C₄²

yes

characteristic

112

C₂×C₄

#117

yes

characteristic

113

C₂×C₄

C₄×D₄

yes

—

114

C₂×C₄

C₄×D₄

yes

—

115

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

116

C₂×C₄

AGL(1,ℤ/8)

yes

—

117

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

118

C₂×C₄

#137

yes

characteristic

119

C₂×C₄

#137

yes

—

120

C₂×C₄

#137

yes

characteristic

121

C₈

—

122

C₈

—

123

C₂²

C₄×C₂²⋊C₄

yes

characteristic

124

C₂²

C₂×#101

yes

characteristic

125

C₂²

#413

yes

characteristic

126

C₂²

#417

yes

characteristic

127

C₂²

#417

yes

characteristic

128

C₂²

#418

yes

characteristic

129

C₂²

#419

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#440

yes

—

139

C₄

#440

yes

—

140

C₄

#441

yes

—

141

C₄

#442

yes

—

142

C₂

#1556

yes

characteristic

143

C₂

#1557

yes

characteristic

144

C₂

#1559

yes

characteristic

145

C₂

#1561

yes

characteristic

146

C₂

C₄×#101

yes

characteristic

147

C₂

#1762

yes

characteristic

148

C₂

#1763

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
21. #332	C₄	semidirect

H ≅ G/N

type

21. #332

C₄

semidirect

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

subgroup	index
C₄×C₄⋊Q₈	2
#1767	2
C₂×#332	2
#1296	2

subgroup

index

C₄×C₄⋊Q₈

#1767

C₂×#332

#1296

Subgroups — #8233

Conjugacy classes of subgroups

Complements

Maximal subgroups