Group #10608 — subgroups

Subgroups — #10608

Order 256.

161 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 57 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	#1966	C₂	1	1	yes	maximal, characteristic
2	#1950	C₂	1	1	yes	maximal, characteristic
3	#1570	C₂	1	1	yes	maximal, characteristic
4	#1565	C₂	1	1	yes	maximal, characteristic
5	#1546	C₂	1	1	yes	maximal, characteristic
6	C₂×#332	C₂	1	1	yes	maximal, characteristic
7	#1292	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	C₂×#108	—	2	1	no	—
11	#423	C₂²	1	1	yes	characteristic
12	#423	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×#101	—	2	1	no	—
16	C₂×#101	—	2	1	no	—
17	C₄×M₁₆	C₂²	1	1	yes	characteristic
18	#341	—	2	1	no	—
19	#335	—	2	1	no	—
20	#334	—	2	1	no	—
21	#334	—	2	2	no	—
22	#332	—	2	2	no	—
23	#332	—	2	1	no	—
24	#332	—	2	1	no	—
25	C₂²×Q₈	—	2	1	no	—
26	C₂×M₁₆	—	2	1	no	—
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	D₄	1	1	yes	characteristic
29	C₂×M₁₆	D₄	1	1	yes	characteristic
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	—	2	1	no	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄⋊C₄	Q₈	1	2	yes	—
38	C₂×C₄²	C₂³	1	1	yes	characteristic
39	#108	—	4	2	no	—
40	C₄⋊C₈	—	2	1	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	—	2	1	no	—
47	C₄⋊C₈	D₄	1	2	yes	—
48	C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₄⋊C₈	D₄	1	1	yes	characteristic
50	#101	—	4	2	no	—
51	#101	—	4	2	no	—
52	C₂²⋊C₈	—	2	1	no	—
53	#97	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄	#41	1	1	yes	characteristic
66	M₁₆	—	2	1	no	—
67	M₁₆	—	2	1	no	—
68	M₁₆	—	2	1	no	—
69	M₁₆	—	4	1	no	—
70	M₁₆	—	4	1	no	—
71	M₁₆	—	4	1	no	—
72	M₁₆	—	4	1	no	—
73	M₁₆	SD₁₆	1	2	yes	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	8	1	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄²	C₂×D₄	1	1	yes	characteristic
90	C₄²	C₂×D₄	1	1	yes	characteristic
91	C₄²	#41	1	1	yes	characteristic
92	C₄²	#41	1	1	yes	characteristic
93	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
94	Q₈	—	8	1	no	—
95	Q₈	—	8	1	no	—
96	Q₈	—	8	1	no	—
97	Q₈	—	8	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	#121	1	1	yes	characteristic
105	C₂×C₄	#121	1	1	yes	characteristic
106	C₂×C₄	#121	1	1	yes	characteristic
107	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄	#126	1	1	yes	characteristic
110	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄	#135	1	1	yes	characteristic
112	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	2	no	—
116	C₈	—	4	1	no	—
117	C₈	—	4	1	no	—
118	C₈	—	4	1	no	—
119	C₈	—	8	1	no	—
120	C₈	—	8	1	no	—
121	C₈	—	8	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#396	1	1	yes	characteristic
125	C₂²	#476	1	1	yes	characteristic
126	C₂²	#478	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	8	2	no	—
131	C₄	#460	1	1	yes	characteristic
132	C₄	#463	1	1	yes	characteristic
133	C₄	#465	1	1	yes	characteristic
134	C₄	#469	1	1	yes	characteristic
135	C₄	#471	1	1	yes	characteristic
136	C₄	#473	1	1	yes	characteristic
137	C₂	—	2	1	no	—
138	C₂	—	2	1	no	—
139	C₂	#2064	1	1	yes	characteristic
140	C₂	#2065	1	1	yes	characteristic
141	C₂	#2070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1966

C₂

yes

maximal, characteristic

#1950

C₂

yes

maximal, characteristic

#1570

C₂

yes

maximal, characteristic

#1565

C₂

yes

maximal, characteristic

#1546

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal, characteristic

#1292

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

C₂×#108

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#101

—

C₂×#101

—

C₄×M₁₆

C₂²

yes

—

—

—

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

#101

—

#101

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

SD₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

C₂²⋊Q₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#121

yes

characteristic

105

C₂×C₄

#121

yes

characteristic

106

C₂×C₄

#121

yes

characteristic

107

C₂×C₄

C₂²⋊Q₈

yes

characteristic

108

C₂×C₄

C₂²⋊Q₈

yes

characteristic

109

C₂×C₄

#126

yes

characteristic

110

C₂×C₄

C₂×SD₁₆

yes

characteristic

111

C₂×C₄

#135

yes

characteristic

112

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

113

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#396

yes

characteristic

125

C₂²

#476

yes

characteristic

126

C₂²

#478

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#460

yes

characteristic

132

C₄

#463

yes

characteristic

133

C₄

#465

yes

characteristic

134

C₄

#469

yes

characteristic

135

C₄

#471

yes

characteristic

136

C₄

#473

yes

characteristic

137

C₂

—

138

C₂

—

139

C₂

#2064

yes

characteristic

140

C₂

#2065

yes

characteristic

141

C₂

#2070

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#396

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ 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
#1966	2
#1950	2
#1570	2
#1565	2
#1546	2
C₂×#332	2
#1292	2

subgroup

index

Subgroups — #10608

Conjugacy classes of subgroups

Maximal subgroups