Group #8879 — subgroups

Subgroups — #8879

Order 256.

182 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 69 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	#2961	C₂	1	1	yes	maximal, characteristic
2	#1940	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	1	yes	maximal, characteristic
4	#1503	C₂	1	1	yes	maximal, characteristic
5	#1481	C₂	1	1	yes	maximal, characteristic
6	#1303	C₂	1	1	yes	maximal, characteristic
7	#1281	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	#446	—	2	2	no	—
10	#446	—	2	2	no	—
11	#431	C₂²	1	1	yes	characteristic
12	#431	C₂²	1	1	yes	characteristic
13	#424	—	2	1	no	—
14	#423	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈	—	2	1	no	—
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	#405	—	2	1	no	—
18	C₂×#97	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	#336	—	2	1	no	—
21	#335	—	2	2	no	—
22	#328	C₄	1	2	yes	—
23	#328	C₄	1	2	yes	—
24	Q₈⋊C₈	—	2	2	no	—
25	C₂×M₁₆	—	2	1	no	—
26	C₂²×C₈	—	2	1	no	—
27	C₂²×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₄⋊Q₈	Q₈	1	1	yes	characteristic
30	C₄⋊Q₈	Q₈	1	1	yes	characteristic
31	#125	—	2	1	no	—
32	C₂²⋊Q₈	—	2	1	no	—
33	C₂²⋊Q₈	C₂×C₄	1	2	yes	—
34	C₄×Q₈	—	2	1	no	—
35	C₄×Q₈	D₄	1	2	yes	—
36	#117	—	2	1	no	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄²	—	2	1	no	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	C₄⋊C₈	—	2	4	no	—
41	C₄⋊C₈	—	2	1	no	—
42	C₄⋊C₈	—	2	1	no	—
43	C₄⋊C₈	—	2	1	no	—
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	4	2	no	—
47	C₄⋊C₈	D₄	1	2	yes	—
48	C₂²⋊C₈	—	2	1	no	—
49	C₂²⋊C₈	C₂×C₄	1	2	yes	—
50	#97	—	2	2	no	—
51	#97	—	2	1	no	—
52	#97	—	2	1	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	#95	C₂×C₄	1	1	yes	characteristic
57	#95	C₂×C₄	1	1	yes	characteristic
58	C₂×Q₈	—	2	1	no	—
59	C₂×Q₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄	—	2	1	no	—
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄	#41	1	1	yes	characteristic
65	M₁₆	—	4	1	no	—
66	M₁₆	—	4	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	4	1	no	—
78	C₂×C₈	Q₁₆	1	2	yes	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	4	2	no	—
85	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
86	C₂²⋊C₄	—	2	2	no	—
87	C₄²	—	2	2	no	—
88	C₄²	—	4	1	no	—
89	C₄²	—	4	1	no	—
90	C₄²	C₂×D₄	1	1	yes	characteristic
91	C₄²	C₂×D₄	1	1	yes	characteristic
92	C₄²	C₂×Q₈	1	1	yes	characteristic
93	C₄²	#41	1	1	yes	characteristic
94	C₂³	C₄×D₄	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	Q₈	—	8	1	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄	#120	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄	#123	1	1	yes	characteristic
113	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₄	#135	1	1	yes	characteristic
115	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄	#137	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	4	1	no	—
120	C₈	—	4	2	no	—
121	C₈	—	8	1	no	—
122	C₈	—	8	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	#385	1	1	yes	characteristic
126	C₂²	C₄×Q₁₆	1	1	yes	characteristic
127	C₂²	#440	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	4	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	8	1	no	—
136	C₄	#449	1	1	yes	characteristic
137	C₄	#451	1	1	yes	characteristic
138	C₄	#477	1	1	yes	characteristic
139	C₄	#479	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#1861	1	1	yes	characteristic
143	C₂	#1863	1	1	yes	characteristic
144	C₂	#1868	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2961

C₂

yes

maximal, characteristic

#1940

C₂

yes

maximal, characteristic

#1571

C₂

yes

maximal, characteristic

#1503

C₂

yes

maximal, characteristic

#1481

C₂

yes

maximal, characteristic

#1303

C₂

yes

maximal, characteristic

#1281

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

—

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#424

—

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

#405

—

C₂×#97

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

C₄

yes

—

#328

C₄

yes

—

Q₈⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

Q₈

yes

characteristic

C₄⋊Q₈

Q₈

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×C₄

yes

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

C₂×C₄

yes

characteristic

#95

C₂×C₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

Q₈

—

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

C₂×C₄⋊C₄

yes

characteristic

108

C₂×C₄

C₄×D₄

yes

characteristic

109

C₂×C₄

#120

yes

characteristic

110

C₂×C₄

C₂²⋊Q₈

yes

characteristic

111

C₂×C₄

C₂²⋊Q₈

yes

characteristic

112

C₂×C₄

#123

yes

characteristic

113

C₂×C₄

C₂×Q₁₆

yes

characteristic

114

C₂×C₄

#135

yes

characteristic

115

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

116

C₂×C₄

#137

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

#385

yes

characteristic

126

C₂²

C₄×Q₁₆

yes

characteristic

127

C₂²

#440

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#449

yes

characteristic

137

C₄

#451

yes

characteristic

138

C₄

#477

yes

characteristic

139

C₄

#479

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#1861

yes

characteristic

143

C₂

#1863

yes

characteristic

144

C₂

#1868

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#385
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂×C₄

C₂²×C₄

C₂×C₄²

C₂³

socle

C₂²

#385

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#2961	2
#1940	2
#1571	2
#1503	2
#1481	2
#1303	2
#1281	2

subgroup

index

Subgroups — #8879

Conjugacy classes of subgroups

Maximal subgroups