Group #9585 — subgroups

Subgroups — #9585

Order 256.

145 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 145 conjugacy classes of subgroups — too many to draw. Only its 55 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	#1574	C₂	1	1	yes	maximal, characteristic
2	#1544	C₂	1	1	yes	maximal, characteristic
3	#1493	C₂	1	1	yes	maximal, characteristic
4	#1485	C₂	1	1	yes	maximal, characteristic
5	#1450	C₂	1	1	yes	maximal, characteristic
6	#1299	C₂	1	1	yes	maximal, characteristic
7	#1278	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	#431	C₂²	1	1	yes	characteristic
11	#431	C₂²	1	1	yes	characteristic
12	#423	C₂²	1	1	yes	characteristic
13	C₄×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#335	—	2	1	no	—
16	#334	—	2	1	no	—
17	#332	C₄	1	1	yes	characteristic
18	#332	C₄	1	1	yes	characteristic
19	#332	C₄	1	1	yes	characteristic
20	#332	C₄	1	1	yes	characteristic
21	Q₈⋊C₈	—	2	1	no	—
22	Q₈⋊C₈	—	2	1	no	—
23	Q₈⋊C₈	—	2	1	no	—
24	Q₈⋊C₈	—	2	1	no	—
25	#322	—	2	1	no	—
26	C₈²	—	2	1	no	—
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂²×C₈	—	2	1	no	—
30	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	#125	—	2	1	no	—
33	C₂²⋊Q₈	—	2	1	no	—
34	C₂²⋊Q₈	—	2	1	no	—
35	C₄×Q₈	—	2	1	no	—
36	C₄×Q₈	—	2	1	no	—
37	#117	—	2	1	no	—
38	C₂×C₄²	C₂³	1	1	yes	characteristic
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₄⋊C₈	D₄	1	1	yes	characteristic
47	C₂²⋊C₈	—	2	1	no	—
48	C₂²⋊C₈	—	2	1	no	—
49	C₂²⋊C₈	—	2	1	no	—
50	#97	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	D₄	1	1	yes	characteristic
57	C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×Q₈	—	2	1	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂²×C₄	#41	1	1	yes	characteristic
61	C₂²×C₄	#41	1	1	yes	characteristic
62	C₂²×C₄	#41	1	1	yes	characteristic
63	M₁₆	—	2	2	no	—
64	M₁₆	—	4	1	no	—
65	M₁₆	—	8	1	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	—	4	1	no	—
78	C₄⋊C₄	—	2	1	no	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₂²⋊C₄	—	4	1	no	—
85	C₄²	—	4	1	no	—
86	C₄²	C₂²×C₄	1	1	yes	characteristic
87	C₄²	C₂×D₄	1	1	yes	characteristic
88	C₄²	C₂×D₄	1	1	yes	characteristic
89	C₄²	#41	1	1	yes	characteristic
90	C₂³	#126	1	1	yes	characteristic
91	Q₈	—	4	1	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	4	1	no	—
95	C₂×C₄	—	2	1	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	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	#117	1	1	yes	characteristic
104	C₂×C₄	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄	#121	1	1	yes	characteristic
107	C₂×C₄	#123	1	1	yes	characteristic
108	C₂×C₄	#124	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	4	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	4	1	no	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	4	1	no	—
117	C₈	—	4	1	no	—
118	C₈	—	8	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#388	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	8	1	no	—
128	C₄	#443	1	1	yes	characteristic
129	C₄	#443	1	1	yes	characteristic
130	C₄	#443	1	1	yes	characteristic
131	C₄	#444	1	1	yes	characteristic
132	C₄	#471	1	1	yes	characteristic
133	C₄	#473	1	1	yes	characteristic
134	C₂	—	4	1	no	—
135	C₂	#1929	1	1	yes	characteristic
136	C₂	#1930	1	1	yes	characteristic
137	C₂	#1937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1574

C₂

yes

maximal, characteristic

#1544

C₂

yes

maximal, characteristic

#1493

C₂

yes

maximal, characteristic

#1485

C₂

yes

maximal, characteristic

#1450

C₂

yes

maximal, characteristic

#1299

C₂

yes

maximal, characteristic

#1278

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

C₄

yes

characteristic

#332

C₄

yes

characteristic

#332

C₄

yes

characteristic

#332

C₄

yes

characteristic

Q₈⋊C₈

—

Q₈⋊C₈

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×C₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#117

yes

characteristic

104

C₂×C₄

C₄×D₄

yes

characteristic

105

C₂×C₄

C₄×D₄

yes

characteristic

106

C₂×C₄

#121

yes

characteristic

107

C₂×C₄

#123

yes

characteristic

108

C₂×C₄

#124

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#388

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#443

yes

characteristic

129

C₄

#443

yes

characteristic

130

C₄

#443

yes

characteristic

131

C₄

#444

yes

characteristic

132

C₄

#471

yes

characteristic

133

C₄

#473

yes

characteristic

134

C₂

—

135

C₂

#1929

yes

characteristic

136

C₂

#1930

yes

characteristic

137

C₂

#1937

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#388
prime radical	C₂³	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

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
#1574	2
#1544	2
#1493	2
#1485	2
#1450	2
#1299	2
#1278	2

subgroup

index

Subgroups — #9585

Conjugacy classes of subgroups

Maximal subgroups