Group #93018 — subgroups

Subgroups — #93018

Order 512.

194 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 62 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	#13263	C₂	1	1	yes	maximal, characteristic
2	#8617	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
4	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
6	#1801	—	2	2	no	—
7	#1567	—	2	2	no	—
8	#1374	—	2	2	no	—
9	#1292	C₄	1	4	yes	—
10	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
11	C₂²×C₁₆	—	2	1	no	—
12	C₂²×C₁₆	—	2	1	no	—
13	C₈×Q₈	—	4	1	no	—
14	C₈×Q₈	—	4	1	no	—
15	#423	—	2	4	no	—
16	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄⋊C₈	Q₈	1	1	yes	characteristic
18	C₂×#101	—	4	2	no	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	D₄	1	2	yes	—
21	C₄⋊C₁₆	—	4	1	no	—
22	C₄×C₁₆	—	4	1	no	—
23	#343	—	4	2	no	—
24	#335	—	4	2	no	—
25	#335	—	4	2	no	—
26	C₂²×Q₈	D₈	1	1	yes	characteristic
27	C₂×M₁₆	—	4	4	no	—
28	C₂²×C₈	C₄²	1	1	yes	characteristic
29	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₈	C₄⋊C₄	1	2	yes	—
31	C₄×Q₈	—	4	1	no	—
32	C₄×Q₈	—	4	1	no	—
33	C₂×C₄⋊C₄	SD₁₆	1	2	yes	—
34	C₂×C₄⋊C₄	Q₁₆	1	1	yes	characteristic
35	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₁₆	—	2	2	no	—
38	C₂×C₁₆	—	4	1	no	—
39	C₂×C₁₆	—	4	1	no	—
40	C₂×C₁₆	—	4	2	no	—
41	C₂×C₁₆	—	4	1	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄⋊C₈	—	4	1	no	—
44	C₄⋊C₈	—	4	4	no	—
45	C₄⋊C₈	—	4	4	no	—
46	#101	—	8	2	no	—
47	#101	—	8	2	no	—
48	C₄×C₈	—	2	2	no	—
49	C₄×C₈	—	2	2	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₂×Q₈	—	4	1	no	—
53	C₂×Q₈	—	4	1	no	—
54	C₂×Q₈	D₁₆	1	1	yes	characteristic
55	C₂×Q₈	SD₃₂	1	2	yes	—
56	C₂×Q₈	Q₃₂	1	1	yes	characteristic
57	C₂²×C₄	#95	1	1	yes	characteristic
58	C₂²×C₄	#102	1	2	yes	—
59	C₂²×C₄	#103	1	2	yes	—
60	C₂²×C₄	#106	1	1	yes	characteristic
61	C₂²×C₄	#107	1	1	yes	characteristic
62	M₁₆	—	8	4	no	—
63	M₁₆	—	8	4	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	4	1	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	—	8	4	no	—
71	C₂×C₈	#99	1	1	yes	characteristic
72	C₂×C₈	#99	1	1	yes	characteristic
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄²	—	2	2	no	—
76	C₄²	—	2	1	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	2	1	no	—
79	C₁₆	—	4	2	no	—
80	C₁₆	—	8	1	no	—
81	C₂³	#340	1	1	yes	characteristic
82	Q₈	—	4	1	no	—
83	Q₈	—	4	2	no	—
84	Q₈	—	4	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
96	C₂×C₄	#342	1	1	yes	characteristic
97	C₂×C₄	#357	1	2	yes	—
98	C₂×C₄	#358	1	2	yes	—
99	C₂×C₄	#366	1	1	yes	characteristic
100	C₂×C₄	#367	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	4	1	no	—
103	C₈	—	4	2	no	—
104	C₈	—	16	4	no	—
105	C₈	#355	1	1	yes	characteristic
106	C₈	#356	1	1	yes	characteristic
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#1289	1	1	yes	characteristic
110	C₂²	#1380	1	1	yes	characteristic
111	C₂²	#1384	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	4	2	no	—
118	C₄	—	4	1	no	—
119	C₄	#1392	1	1	yes	characteristic
120	C₄	#1394	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	#7229	1	1	yes	characteristic
124	C₂	#7230	1	1	yes	characteristic
125	C₂	#7238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13263

C₂

yes

maximal, characteristic

#8617

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

—

—

—

C₄

yes

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×Q₈

—

C₈×Q₈

—

#423

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄⋊C₈

Q₈

yes

characteristic

C₂×#101

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#343

—

#335

—

#335

—

C₂²×Q₈

D₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

SD₁₆

yes

—

C₂×C₄⋊C₄

Q₁₆

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

#101

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₆

yes

characteristic

C₂×Q₈

SD₃₂

yes

—

C₂×Q₈

Q₃₂

yes

C₂²×C₄

yes

C₂²×C₄

yes

—

C₂²×C₄

#103

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

#340

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#358

yes

—

C₂×C₄

#366

yes

characteristic

100

C₂×C₄

#367

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

#355

yes

characteristic

106

C₈

#356

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

#1289

yes

characteristic

110

C₂²

#1380

yes

characteristic

111

C₂²

#1384

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#1392

yes

characteristic

120

C₄

#1394

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

#7229

yes

characteristic

124

C₂

#7230

yes

characteristic

125

C₂

#7238

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1289
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×C₈×Q₈	C₂²
prime radical	C₂³	#340
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#1289

commutator

C₂²×C₈

C₄²

Frattini, cosocle

C₂×C₈×Q₈

C₂²

prime radical

C₂³

#340

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

subgroup	index
#13263	2
#8617	2

subgroup

index

#13263

#8617

Subgroups — #93018

Conjugacy classes of subgroups

Maximal subgroups