Group #146272 — subgroups

Subgroups — #146272

Order 512.

309 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 96 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	#13210	C₂	1	2	yes	maximal
2	#12903	C₂	1	2	yes	maximal, complemented
3	#9856	C₂	1	1	yes	maximal, characteristic
4	#8815	C₂	1	1	yes	maximal, characteristic, complemented
5	#7138	C₂	1	1	yes	maximal, characteristic, complemented
6	#3147	C₂²	1	1	yes	characteristic
7	C₂×#365	—	2	4	no	—
8	C₂×#360	—	2	4	no	—
9	C₄×M₃₂	C₂²	1	2	yes	—
10	#1848	C₂²	1	1	yes	characteristic
11	#1844	C₂²	1	2	yes	—
12	#1834	—	2	2	no	—
13	#1492	—	2	2	no	—
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	C₂×#135	—	2	2	no	—
16	#532	—	2	1	no	—
17	C₂×M₃₂	D₄	1	4	yes	—
18	#495	—	2	2	no	—
19	#494	Q₈	1	1	yes	characteristic
20	#493	Q₈	1	1	yes	characteristic
21	#492	—	2	1	no	—
22	C₂×#108	—	2	2	no	—
23	C₂×#106	—	2	2	no	—
24	C₂×#106	—	2	2	no	—
25	#423	—	2	1	no	—
26	C₂×C₄×C₈	D₄	1	2	yes	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#384	—	2	2	no	—
29	#365	—	4	8	no	—
30	#360	—	4	8	no	—
31	#346	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	#340	—	4	4	no	—
34	D₄⋊C₈	—	4	4	no	—
35	C₈²	—	2	2	no	—
36	C₂×#41	—	4	1	no	—
37	#135	—	4	4	no	—
38	#135	—	4	4	no	—
39	C₂×Q₁₆	—	2	2	no	—
40	C₂×SD₁₆	—	4	2	no	—
41	C₂×D₈	—	2	2	no	—
42	C₂×M₁₆	—	4	1	no	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	C₂×D₄	1	2	yes	—
45	C₄⋊Q₈	—	2	1	no	—
46	#127	—	2	1	no	—
47	#124	—	4	1	no	—
48	#121	—	4	2	no	—
49	C₄×D₄	—	4	2	no	—
50	C₂×C₄⋊C₄	—	4	2	no	—
51	C₂×C₄⋊C₄	—	4	4	no	—
52	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
53	M₃₂	—	2	4	no	—
54	M₃₂	SD₁₆	1	8	yes	—
55	C₂×C₁₆	—	2	4	no	—
56	#108	—	4	4	no	—
57	#106	—	4	4	no	—
58	#106	—	4	4	no	—
59	C₄⋊C₈	—	4	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	SD₁₆	1	4	yes	—
62	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₄×C₈	#41	1	1	yes	characteristic
64	C₄×C₈	#41	1	2	yes	—
65	#41	—	8	2	no	—
66	#41	—	8	2	no	—
67	C₂×Q₈	—	4	1	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	2	no	—
70	C₂×D₄	—	8	1	no	—
71	C₂²×C₄	—	8	1	no	—
72	C₂²×C₄	—	8	2	no	—
73	C₂²×C₄	#127	1	1	yes	characteristic
74	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
75	Q₁₆	—	4	4	no	—
76	SD₁₆	—	8	4	no	—
77	D₈	—	4	4	no	—
78	M₁₆	—	8	2	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	4	4	no	—
81	C₂×C₈	—	8	1	no	—
82	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₈	#123	1	2	yes	—
84	C₂×C₈	C₂×SD₁₆	1	4	yes	—
85	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	8	8	no	—
88	C₄⋊C₄	—	8	2	no	—
89	C₄⋊C₄	—	8	2	no	—
90	C₂²⋊C₄	—	8	2	no	—
91	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
92	C₄²	#123	1	1	yes	characteristic
93	C₄²	C₂×SD₁₆	1	2	yes	—
94	C₁₆	—	2	8	no	—
95	C₂³	—	8	1	no	—
96	C₂³	#496	1	1	yes	characteristic
97	Q₈	—	8	2	no	—
98	D₄	—	8	2	no	—
99	D₄	—	8	4	no	—
100	D₄	—	16	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	8	1	no	—
103	C₂×C₄	—	16	1	no	—
104	C₂×C₄	—	16	1	no	—
105	C₂×C₄	—	16	4	no	—
106	C₂×C₄	—	16	2	no	—
107	C₂×C₄	#399	1	1	yes	characteristic
108	C₂×C₄	#476	1	2	yes	—
109	C₂×C₄	#481	1	2	yes	—
110	C₂×C₄	#492	1	2	yes	—
111	C₂×C₄	#492	1	1	yes	characteristic
112	C₂×C₄	#496	1	2	yes	—
113	C₈	—	4	4	no	—
114	C₈	—	16	1	no	—
115	C₈	#476	1	4	yes	—
116	C₈	#481	1	4	yes	—
117	C₂²	—	2	2	no	—
118	C₂²	—	16	1	no	—
119	C₂²	—	16	2	no	—
120	C₂²	#1715	1	2	yes	—
121	C₂²	#2094	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	16	1	no	—
124	C₄	—	32	2	no	—
125	C₄	#1705	1	1	yes	characteristic
126	C₄	#1715	1	1	yes	characteristic
127	C₄	#2094	1	2	yes	—
128	C₄	#2231	1	4	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	32	1	no	—
131	C₂	#10683	1	1	yes	characteristic
132	C₂	#13534	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13210

C₂

yes

maximal

#12903

C₂

yes

maximal, complemented

#9856

C₂

yes

maximal, characteristic

#8815

C₂

yes

maximal, characteristic, complemented

#7138

C₂

yes

maximal, characteristic, complemented

#3147

C₂²

yes

characteristic

C₂×#365

—

C₂×#360

—

C₄×M₃₂

C₂²

yes

—

#1848

C₂²

yes

characteristic

#1844

C₂²

yes

—

#1834

—

#1492

—

C₂×C₈²

C₂²

yes

characteristic

C₂×#135

—

#532

—

C₂×M₃₂

D₄

yes

—

#495

—

#494

Q₈

yes

characteristic

#493

Q₈

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄×C₁₆

—

#340

—

D₄⋊C₈

—

C₈²

—

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

SD₁₆

yes

—

C₂×C₁₆

—

#108

—

#106

—

#106

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

#123

yes

characteristic

C₄²

C₂×SD₁₆

yes

—

C₁₆

—

C₂³

—

C₂³

#496

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#399

yes

characteristic

108

C₂×C₄

#476

yes

—

109

C₂×C₄

#481

yes

—

110

C₂×C₄

#492

yes

—

111

C₂×C₄

#492

yes

characteristic

112

C₂×C₄

#496

yes

—

113

C₈

—

114

C₈

—

115

C₈

#476

yes

—

116

C₈

#481

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

#1715

yes

—

121

C₂²

#2094

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#1705

yes

characteristic

126

C₄

#1715

yes

characteristic

127

C₄

#2094

yes

—

128

C₄

#2231

yes

—

129

C₂

—

130

C₂

—

131

C₂

#10683

yes

characteristic

132

C₂

#13534

yes

—

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2094

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

#3147

C₂²

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #12903	C₂	semidirect
4. #8815	C₂	semidirect
5. #7138	C₂	semidirect

H ≅ G/N

type

2. #12903

C₂

semidirect

4. #8815

C₂

semidirect

5. #7138

C₂

semidirect

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

subgroup	index
#13210	2
#12903	2
#9856	2
#8815	2
#7138	2

subgroup

index

Subgroups — #146272

Conjugacy classes of subgroups

Complements

Maximal subgroups