Group #144199 — subgroups

Subgroups — #144199

Order 512.

368 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 100 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	#33731	C₂	1	2	yes	maximal
2	C₂×#1710	C₂	1	1	yes	maximal, characteristic
3	#13290	C₂	1	2	yes	maximal
4	#9855	C₂	1	1	yes	maximal, characteristic
5	#7138	C₂	1	1	yes	maximal, characteristic
6	#3160	C₂²	1	1	yes	characteristic
7	C₂×#500	C₂²	1	2	yes	—
8	C₂×#487	—	2	2	no	—
9	#2255	—	2	4	no	—
10	#2254	—	2	2	no	—
11	#2252	—	2	2	no	—
12	#2159	—	2	4	no	—
13	C₄×M₃₂	C₂²	1	2	yes	—
14	#1848	C₂²	1	1	yes	characteristic
15	#1710	—	2	4	no	—
16	#1493	—	2	2	no	—
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1387	—	2	4	no	—
19	#533	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×M₃₂	D₄	1	4	yes	—
22	#500	—	2	2	no	—
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	D₄	1	4	yes	—
26	#500	D₄	1	2	yes	—
27	#499	—	2	2	no	—
28	#498	—	2	1	no	—
29	#487	—	2	8	no	—
30	#487	—	4	4	no	—
31	C₂×#108	—	2	2	no	—
32	#427	—	2	2	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	2	2	no	—
35	#423	—	2	1	no	—
36	C₂×#103	—	4	4	no	—
37	C₂×C₄×C₈	D₄	1	2	yes	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#367	—	2	8	no	—
40	#366	—	2	8	no	—
41	#346	—	2	2	no	—
42	C₄×C₁₆	—	2	2	no	—
43	Q₈⋊C₈	—	4	4	no	—
44	C₈²	—	2	2	no	—
45	C₂²×Q₈	—	4	2	no	—
46	C₂×M₁₆	—	4	1	no	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	C₂×D₄	1	2	yes	—
49	C₄⋊Q₈	—	2	4	no	—
50	C₄⋊Q₈	—	2	2	no	—
51	C₄⋊Q₈	—	4	2	no	—
52	C₄⋊Q₈	—	4	2	no	—
53	C₄⋊Q₈	—	4	2	no	—
54	#125	—	4	1	no	—
55	C₂²⋊Q₈	—	4	2	no	—
56	C₄×Q₈	—	4	2	no	—
57	#117	—	4	1	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
61	M₃₂	—	2	4	no	—
62	M₃₂	Q₁₆	1	8	yes	—
63	C₂×C₁₆	—	2	4	no	—
64	#108	—	4	4	no	—
65	#107	—	2	2	no	—
66	#107	—	2	4	no	—
67	#107	—	2	2	no	—
68	#107	—	2	4	no	—
69	#107	—	4	2	no	—
70	#107	—	4	2	no	—
71	#106	—	4	2	no	—
72	C₄⋊C₈	—	4	2	no	—
73	#103	—	4	16	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	C₂×Q₈	—	8	8	no	—
82	C₂×Q₈	—	8	2	no	—
83	C₂²×C₄	—	8	2	no	—
84	C₂²×C₄	#127	1	1	yes	characteristic
85	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
86	M₁₆	—	8	2	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	4	4	no	—
89	C₂×C₈	—	8	1	no	—
90	C₂×C₈	#120	1	2	yes	—
91	C₂×C₈	#121	1	2	yes	—
92	C₂×C₈	C₂×Q₁₆	1	4	yes	—
93	C₂×C₈	#137	1	4	yes	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	4	4	no	—
96	C₄⋊C₄	—	4	4	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	8	2	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₄⋊C₄	—	8	2	no	—
101	C₂²⋊C₄	—	8	1	no	—
102	C₄²	—	8	1	no	—
103	C₄²	#120	1	1	yes	characteristic
104	C₄²	#121	1	1	yes	characteristic
105	C₄²	C₂×D₈	1	2	yes	—
106	C₁₆	—	2	8	no	—
107	C₂³	#496	1	1	yes	characteristic
108	Q₈	—	8	8	no	—
109	Q₈	—	8	4	no	—
110	Q₈	—	16	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	8	4	no	—
114	C₂×C₄	—	16	4	no	—
115	C₂×C₄	#392	1	1	yes	characteristic
116	C₂×C₄	#447	1	2	yes	—
117	C₂×C₄	#459	1	2	yes	—
118	C₂×C₄	#493	1	1	yes	characteristic
119	C₂×C₄	#494	1	2	yes	—
120	C₂×C₄	#497	1	2	yes	—
121	C₈	—	4	4	no	—
122	C₈	—	16	1	no	—
123	C₈	#451	1	4	yes	—
124	C₈	#462	1	4	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	#1723	1	2	yes	—
127	C₂²	#2011	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	16	4	no	—
130	C₄	—	16	2	no	—
131	C₄	#1713	1	1	yes	characteristic
132	C₄	#1722	1	1	yes	characteristic
133	C₄	#2016	1	2	yes	—
134	C₄	#2212	1	4	yes	—
135	C₂	—	2	2	no	—
136	C₂	#10309	1	1	yes	characteristic
137	C₂	#13332	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33731

C₂

yes

maximal

C₂×#1710

C₂

yes

maximal, characteristic

#13290

C₂

yes

maximal

#9855

C₂

yes

maximal, characteristic

#7138

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

C₂×#500

C₂²

yes

—

—

—

—

—

—

C₄×M₃₂

C₂²

yes

—

#1848

C₂²

yes

characteristic

#1710

—

#1493

—

C₂×C₈²

C₂²

yes

characteristic

#1387

—

#533

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

—

—

—

D₄

yes

—

#500

D₄

yes

—

—

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄×C₁₆

—

Q₈⋊C₈

—

C₈²

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

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

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

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

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

#120

yes

characteristic

104

C₄²

#121

yes

characteristic

105

C₄²

C₂×D₈

yes

—

106

C₁₆

—

107

C₂³

#496

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#392

yes

characteristic

116

C₂×C₄

#447

yes

—

117

C₂×C₄

#459

yes

—

118

C₂×C₄

#493

yes

characteristic

119

C₂×C₄

#494

yes

—

120

C₂×C₄

#497

yes

—

121

C₈

—

122

C₈

—

123

C₈

#451

yes

—

124

C₈

#462

yes

—

125

C₂²

—

126

C₂²

#1723

yes

—

127

C₂²

#2011

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#1713

yes

characteristic

132

C₄

#1722

yes

characteristic

133

C₄

#2016

yes

—

134

C₄

#2212

yes

—

135

C₂

—

136

C₂

#10309

yes

characteristic

137

C₂

#13332

yes

—

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
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

#2011

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#496

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

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

subgroup	index
#33731	2
C₂×#1710	2
#13290	2
#9855	2
#7138	2

subgroup

index

Subgroups — #144199

Conjugacy classes of subgroups

Maximal subgroups