Group #144200 — subgroups

Subgroups — #144200

Order 512.

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

normal subgroups only

This group has 354 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	#33728	C₂	1	2	yes	maximal
2	#19618	C₂	1	1	yes	maximal, characteristic
3	#13286	C₂	1	2	yes	maximal
4	#9861	C₂	1	1	yes	maximal, characteristic
5	#7137	C₂	1	1	yes	maximal, characteristic
6	#3170	—	2	2	no	—
7	#3160	C₂²	1	1	yes	characteristic
8	#3160	C₂²	1	2	yes	—
9	#2255	—	2	4	no	—
10	#2255	—	2	4	no	—
11	#2158	—	2	4	no	—
12	C₂×#346	C₂²	1	2	yes	—
13	#1848	C₂²	1	1	yes	characteristic
14	#1723	—	2	4	no	—
15	#1493	—	2	2	no	—
16	C₂×#322	C₂²	1	1	yes	characteristic
17	#1382	—	2	4	no	—
18	#533	—	2	2	no	—
19	#533	—	2	1	no	—
20	C₂²×C₁₆	D₄	1	4	yes	—
21	#500	—	2	2	no	—
22	#500	D₄	1	4	yes	—
23	#500	D₄	1	2	yes	—
24	#499	—	2	2	no	—
25	#499	—	2	2	no	—
26	#499	—	2	2	no	—
27	#498	—	2	2	no	—
28	#498	—	2	1	no	—
29	#494	—	2	2	no	—
30	#487	—	2	4	no	—
31	#467	—	4	4	no	—
32	C₄×Q₁₆	—	4	4	no	—
33	C₂×#108	—	2	2	no	—
34	#427	—	2	2	no	—
35	#427	—	2	2	no	—
36	#427	—	2	2	no	—
37	#423	—	2	1	no	—
38	C₂×C₄×C₈	D₄	1	2	yes	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#367	—	2	8	no	—
41	#366	—	2	8	no	—
42	#346	—	2	2	no	—
43	#346	—	2	2	no	—
44	Q₈⋊C₈	—	4	4	no	—
45	#322	—	2	2	no	—
46	C₂×Q₁₆	—	4	4	no	—
47	C₂×M₁₆	—	4	1	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	C₂×D₄	1	2	yes	—
50	C₄⋊Q₈	—	2	4	no	—
51	C₄⋊Q₈	—	2	2	no	—
52	#125	—	4	1	no	—
53	#125	—	4	2	no	—
54	C₂²⋊Q₈	—	4	2	no	—
55	C₂²⋊Q₈	—	4	4	no	—
56	C₄×Q₈	—	4	2	no	—
57	C₄×Q₈	—	4	4	no	—
58	#117	—	4	2	no	—
59	#117	—	4	1	no	—
60	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
61	C₂×C₁₆	—	2	4	no	—
62	C₂×C₁₆	—	2	4	no	—
63	C₂×C₁₆	Q₁₆	1	8	yes	complemented
64	#108	—	4	4	no	—
65	#107	—	2	4	no	—
66	#107	—	2	4	no	—
67	#107	—	2	4	no	—
68	#107	—	4	4	no	—
69	#106	—	4	2	no	—
70	#106	—	4	2	no	—
71	#106	—	4	2	no	—
72	C₄⋊C₈	—	4	2	no	—
73	#103	—	4	8	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	D₈	1	4	yes	—
76	C₄×C₈	C₂×D₄	1	2	yes	—
77	C₄×C₈	C₂×D₄	1	1	yes	characteristic
78	C₄×C₈	#41	1	1	yes	characteristic
79	C₂×Q₈	—	4	4	no	—
80	C₂×Q₈	—	4	2	no	—
81	C₂²×C₄	#127	1	1	yes	characteristic
82	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
83	Q₁₆	—	8	8	no	—
84	M₁₆	—	8	2	no	—
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	—	8	2	no	—
87	C₂×C₈	—	8	1	no	—
88	C₂×C₈	#120	1	2	yes	—
89	C₂×C₈	#121	1	2	yes	—
90	C₂×C₈	C₂×Q₁₆	1	4	yes	—
91	C₂×C₈	#137	1	4	yes	—
92	C₄⋊C₄	—	4	4	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₄⋊C₄	—	8	2	no	—
97	C₄⋊C₄	—	8	4	no	—
98	C₂²⋊C₄	—	8	1	no	—
99	C₂²⋊C₄	—	8	2	no	—
100	C₄²	—	8	1	no	—
101	C₄²	—	8	2	no	—
102	C₄²	#120	1	1	yes	characteristic
103	C₄²	#121	1	1	yes	characteristic
104	C₄²	C₂×D₈	1	2	yes	—
105	C₁₆	—	2	8	no	—
106	C₂³	#496	1	1	yes	characteristic
107	Q₈	—	8	8	no	—
108	Q₈	—	8	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	4	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	#392	1	1	yes	characteristic
113	C₂×C₄	#447	1	2	yes	—
114	C₂×C₄	#459	1	2	yes	—
115	C₂×C₄	#493	1	1	yes	characteristic
116	C₂×C₄	#494	1	2	yes	—
117	C₂×C₄	#497	1	2	yes	—
118	C₈	—	4	4	no	—
119	C₈	—	16	1	no	—
120	C₈	#451	1	4	yes	—
121	C₈	#462	1	4	yes	—
122	C₂²	—	2	2	no	—
123	C₂²	#1723	1	2	yes	—
124	C₂²	#2011	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	—	16	4	no	—
128	C₄	—	16	2	no	—
129	C₄	#1713	1	1	yes	characteristic
130	C₄	#1722	1	1	yes	characteristic
131	C₄	#2016	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	#10309	1	1	yes	characteristic
134	C₂	#13340	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33728

C₂

yes

maximal

#19618

C₂

yes

maximal, characteristic

#13286

C₂

yes

maximal

#9861

C₂

yes

maximal, characteristic

#7137

C₂

yes

maximal, characteristic

#3170

—

#3160

C₂²

yes

characteristic

#3160

C₂²

yes

—

—

—

—

C₂²

yes

—

#1848

C₂²

yes

—

—

C₂²

yes

—

—

—

C₂²×C₁₆

D₄

yes

—

#500

—

#500

D₄

yes

—

#500

D₄

yes

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

Q₈⋊C₈

—

#322

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

—

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#120

yes

—

C₂×C₈

#121

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

#120

yes

characteristic

103

C₄²

#121

yes

characteristic

104

C₄²

C₂×D₈

yes

—

105

C₁₆

—

106

C₂³

#496

yes

characteristic

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#392

yes

characteristic

113

C₂×C₄

#447

yes

—

114

C₂×C₄

#459

yes

—

115

C₂×C₄

#493

yes

characteristic

116

C₂×C₄

#494

yes

—

117

C₂×C₄

#497

yes

—

118

C₈

—

119

C₈

—

120

C₈

#451

yes

—

121

C₈

#462

yes

—

122

C₂²

—

123

C₂²

#1723

yes

—

124

C₂²

#2011

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#1713

yes

characteristic

130

C₄

#1722

yes

characteristic

131

C₄

#2016

yes

—

132

C₂

—

133

C₂

#10309

yes

characteristic

134

C₂

#13340

yes

—

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2011
prime radical	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
63. C₂×C₁₆	Q₁₆	semidirect

H ≅ G/N

type

63. C₂×C₁₆

Q₁₆

semidirect

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

subgroup	index
#33728	2
#19618	2
#13286	2
#9861	2
#7137	2

subgroup

index

Subgroups — #144200

Conjugacy classes of subgroups

Complements

Maximal subgroups