Group #93097 — subgroups

Subgroups — #93097

Order 512.

144 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 66 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	#12013	C₂	1	1	yes	maximal, characteristic
2	#12012	C₂	1	1	yes	maximal, characteristic
3	#8824	C₂	1	1	yes	maximal, characteristic
4	#2163	—	2	1	no	—
5	#2162	—	2	1	no	—
6	#1837	—	2	1	no	—
7	#1567	—	2	1	no	—
8	#1566	C₄	1	2	yes	—
9	C₂×C₈²	C₂²	1	1	yes	characteristic
10	#1370	—	2	1	no	—
11	#1370	—	2	1	no	—
12	#1367	C₄	1	2	yes	—
13	#1367	C₄	1	2	yes	—
14	#423	—	2	2	no	—
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×C₈	D₄	1	1	yes	characteristic
18	C₄⋊C₁₆	—	2	2	no	—
19	C₄⋊C₁₆	—	2	2	no	—
20	C₂²⋊C₁₆	—	4	1	no	—
21	C₂²⋊C₁₆	—	4	1	no	—
22	#346	—	4	1	no	—
23	C₄×C₁₆	—	4	1	no	—
24	#341	—	4	1	no	—
25	#341	—	4	1	no	—
26	#335	—	4	1	no	—
27	#335	—	4	1	no	—
28	#334	—	4	2	no	—
29	C₈²	C₂×C₄	1	1	yes	characteristic
30	C₈²	C₂×C₄	1	1	yes	characteristic
31	C₈²	D₄	1	1	yes	characteristic
32	C₈²	Q₈	1	1	yes	characteristic
33	C₂×M₁₆	—	4	2	no	—
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	D₈	1	1	yes	characteristic
36	C₂²×C₈	SD₁₆	1	1	yes	characteristic
37	C₂²×C₈	SD₁₆	1	1	yes	characteristic
38	C₂²×C₈	Q₁₆	1	1	yes	characteristic
39	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₁₆	—	4	2	no	—
41	C₂×C₁₆	—	4	2	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄⋊C₈	—	4	2	no	—
44	C₄×C₈	—	2	1	no	—
45	C₄×C₈	C₄²	1	1	yes	characteristic
46	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₈	D₈	1	1	yes	characteristic
52	C₄×C₈	SD₁₆	1	1	yes	characteristic
53	C₄×C₈	SD₁₆	1	1	yes	characteristic
54	C₄×C₈	Q₁₆	1	1	yes	characteristic
55	C₂²×C₄	#101	1	1	yes	characteristic
56	C₂²×C₄	#102	1	1	yes	characteristic
57	C₂²×C₄	#103	1	1	yes	characteristic
58	M₁₆	—	8	2	no	—
59	M₁₆	—	8	2	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	—	8	2	no	—
68	C₂×C₈	#102	1	1	yes	characteristic
69	C₂×C₈	#102	1	1	yes	characteristic
70	C₂×C₈	#103	1	1	yes	characteristic
71	C₂×C₈	#103	1	1	yes	characteristic
72	C₂×C₈	#106	1	1	yes	characteristic
73	C₂×C₈	#106	1	1	yes	characteristic
74	C₂×C₈	#107	1	1	yes	characteristic
75	C₂×C₈	#107	1	1	yes	characteristic
76	C₄²	#95	1	1	yes	characteristic
77	C₄²	#100	1	1	yes	characteristic
78	C₄²	#102	1	1	yes	characteristic
79	C₄²	#103	1	1	yes	characteristic
80	C₁₆	—	8	2	no	—
81	C₁₆	—	8	2	no	—
82	C₂³	#332	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	#331	1	1	yes	characteristic
89	C₂×C₄	#332	1	1	yes	characteristic
90	C₂×C₄	#333	1	1	yes	characteristic
91	C₂×C₄	#340	1	1	yes	characteristic
92	C₂×C₄	#340	1	1	yes	characteristic
93	C₂×C₄	#343	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	—	2	1	no	—
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	2	1	no	—
101	C₈	—	2	1	no	—
102	C₈	—	4	1	no	—
103	C₈	—	16	2	no	—
104	C₈	#368	1	2	yes	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#1296	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	#1381	1	1	yes	characteristic
114	C₄	#1384	1	1	yes	characteristic
115	C₄	#1389	1	1	yes	characteristic
116	C₄	#1389	1	1	yes	characteristic
117	C₂	—	4	1	no	—
118	C₂	#7273	1	1	yes	characteristic
119	C₂	#7276	1	1	yes	characteristic
120	C₂	#7279	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12013

C₂

yes

maximal, characteristic

#12012

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal, characteristic

—

—

—

—

C₄

yes

—

C₂×C₈²

C₂²

yes

—

—

C₄

yes

—

#1367

C₄

yes

—

#423

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

Q₁₆

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

C₂²×C₄

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

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂³

#332

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

#368

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#1296

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#1381

yes

characteristic

114

C₄

#1384

yes

characteristic

115

C₄

#1389

yes

characteristic

116

C₄

#1389

yes

characteristic

117

C₂

—

118

C₂

#7273

yes

characteristic

119

C₂

#7276

yes

characteristic

120

C₂

#7279

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1296
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
prime radical	C₂³	#332
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#1296

commutator

C₄×C₈

C₄²

Frattini, cosocle

C₂×C₈²

C₂²

prime radical

C₂³

#332

LCS

G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#12013	2
#12012	2
#8824	2

subgroup

index

#12013

#12012

#8824

Subgroups — #93097

Conjugacy classes of subgroups

Maximal subgroups