Group #93084 — subgroups

Subgroups — #93084

Order 512.

166 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 74 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	#12012	C₂	1	1	yes	maximal, characteristic
2	#11996	C₂	1	1	yes	maximal, characteristic
3	#8817	C₂	1	1	yes	maximal, characteristic
4	#2162	—	2	1	no	—
5	#2150	—	2	1	no	—
6	#1834	—	2	1	no	—
7	#1834	—	2	1	no	—
8	#1834	C₄	1	2	yes	—
9	C₂×C₈²	C₂²	1	1	yes	characteristic
10	#1370	—	2	1	no	—
11	#1367	—	2	1	no	—
12	#1367	C₄	1	2	yes	complemented
13	#1367	C₄	1	2	yes	complemented
14	C₂×M₃₂	—	4	1	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	#384	—	2	2	no	—
19	C₄⋊C₁₆	—	2	2	no	—
20	C₄⋊C₁₆	—	2	2	no	—
21	C₄⋊C₁₆	—	4	1	no	—
22	C₂²⋊C₁₆	—	4	1	no	—
23	C₄×C₁₆	—	4	1	no	—
24	#340	—	4	1	no	—
25	#340	—	4	1	no	—
26	#340	—	4	1	no	—
27	#340	—	4	1	no	—
28	#340	—	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	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	D₈	1	1	yes	characteristic
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	C₂×C₄⋊C₄	—	4	2	no	—
39	C₂×C₄⋊C₄	—	4	2	no	—
40	C₂×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
42	M₃₂	—	8	2	no	—
43	C₂×C₁₆	—	4	2	no	—
44	C₂×C₁₆	—	4	2	no	—
45	C₄×C₈	—	2	1	no	—
46	C₄×C₈	C₄²	1	1	yes	characteristic
47	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₈	C₂²⋊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	C₄⋊C₄	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	C₄×C₈	Q₁₆	1	1	yes	characteristic
56	C₂²×C₄	—	8	2	no	—
57	C₂²×C₄	#100	1	1	yes	characteristic
58	C₂²×C₄	#102	1	1	yes	characteristic
59	C₂²×C₄	#102	1	1	yes	characteristic
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	#102	1	1	yes	characteristic
68	C₂×C₈	#102	1	1	yes	characteristic
69	C₂×C₈	#103	1	1	yes	characteristic
70	C₂×C₈	#103	1	1	yes	characteristic
71	C₂×C₈	#106	1	1	yes	characteristic
72	C₂×C₈	#106	1	1	yes	characteristic
73	C₂×C₈	#107	1	1	yes	characteristic
74	C₂×C₈	#107	1	1	yes	characteristic
75	C₄⋊C₄	—	8	4	no	—
76	C₄⋊C₄	—	8	2	no	—
77	C₄⋊C₄	—	8	2	no	—
78	C₄⋊C₄	—	8	2	no	—
79	C₄⋊C₄	—	8	2	no	—
80	C₄²	#95	1	1	yes	characteristic
81	C₄²	#101	1	1	yes	characteristic
82	C₄²	#103	1	1	yes	characteristic
83	C₄²	#103	1	1	yes	characteristic
84	C₁₆	—	8	2	no	—
85	C₁₆	—	8	2	no	—
86	C₂³	#331	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	16	2	no	—
91	C₂×C₄	—	16	2	no	—
92	C₂×C₄	—	16	2	no	—
93	C₂×C₄	#332	1	1	yes	characteristic
94	C₂×C₄	#332	1	1	yes	characteristic
95	C₂×C₄	#333	1	1	yes	characteristic
96	C₂×C₄	#340	1	1	yes	characteristic
97	C₂×C₄	#340	1	1	yes	characteristic
98	C₂×C₄	#343	1	1	yes	characteristic
99	C₂×C₄	#359	1	2	yes	—
100	C₂×C₄	#360	1	2	yes	—
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	2	1	no	—
105	C₈	—	2	1	no	—
106	C₈	—	2	1	no	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	4	1	no	—
110	C₈	—	4	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#1296	1	1	yes	characteristic
114	C₂²	#1360	1	2	yes	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	32	2	no	—
118	C₄	#1363	1	2	yes	—
119	C₄	#1366	1	2	yes	—
120	C₄	#1382	1	1	yes	characteristic
121	C₄	#1384	1	1	yes	characteristic
122	C₄	#1386	1	1	yes	characteristic
123	C₄	#1389	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#7271	1	1	yes	characteristic
126	C₂	#7275	1	1	yes	characteristic
127	C₂	#7279	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12012

C₂

yes

maximal, characteristic

#11996

C₂

yes

maximal, characteristic

#8817

C₂

yes

maximal, characteristic

—

—

—

—

C₄

yes

—

C₂×C₈²

C₂²

yes

—

—

C₄

yes

complemented

#1367

C₄

yes

complemented

C₂×M₃₂

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

#384

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

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

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

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

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₂³

#331

yes

characteristic

C₂×C₄

—

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

C₂×C₄

yes

—

100

C₂×C₄

#360

yes

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

#1296

yes

characteristic

114

C₂²

#1360

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#1363

yes

—

119

C₄

#1366

yes

—

120

C₄

#1382

yes

characteristic

121

C₄

#1384

yes

characteristic

122

C₄

#1386

yes

characteristic

123

C₄

#1389

yes

characteristic

124

C₂

—

125

C₂

#7271

yes

characteristic

126

C₂

#7275

yes

characteristic

127

C₂

#7279

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1296
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
prime radical	C₂³	#331
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

#331

LCS

G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
12. #1367	C₄	semidirect
13. #1367	C₄	semidirect

H ≅ G/N

type

12. #1367

C₄

semidirect

13. #1367

C₄

semidirect

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

subgroup	index
#12012	2
#11996	2
#8817	2

subgroup

index

#12012

#11996

#8817

Subgroups — #93084

Conjugacy classes of subgroups

Complements

Maximal subgroups