Group #152869 — subgroups

Subgroups — #152869

Order 512.

198 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₂×#1426	C₂	1	2	yes	maximal
2	#13627	C₂	1	1	yes	maximal, characteristic
3	#12906	C₂	1	1	yes	maximal, characteristic
4	C₄×#368	C₂	1	1	yes	maximal, characteristic
5	#7474	C₂	1	2	yes	maximal
6	C₂×M₆₄	C₂²	1	2	yes	—
7	C₂×#368	C₂²	1	1	yes	characteristic
8	C₂×#368	C₂²	1	1	yes	characteristic
9	C₂×#367	—	2	1	no	—
10	C₂×#367	C₂²	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1845	—	2	1	no	—
13	C₄×#108	—	2	1	no	—
14	#1426	—	2	4	no	—
15	C₄⋊C₃₂	C₄	1	4	yes	complemented
16	#1381	—	2	1	no	—
17	C₂²×C₁₆	D₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂³	1	1	yes	characteristic
19	C₂×#108	—	2	1	no	—
20	C₂×#108	—	2	1	no	—
21	C₂×#107	—	2	2	no	—
22	C₂×#107	—	4	1	no	—
23	C₄×M₁₆	—	4	1	no	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	#384	—	4	1	no	—
26	M₆₄	—	2	2	no	—
27	M₆₄	—	2	2	no	—
28	C₂×C₃₂	C₂×C₄	1	4	yes	—
29	#368	—	2	1	no	—
30	#368	—	2	1	no	—
31	#368	D₄	1	2	yes	—
32	#368	Q₈	1	2	yes	—
33	#367	—	2	2	no	—
34	#367	—	2	2	no	—
35	#367	—	4	2	no	—
36	C₄×C₁₆	C₂×C₄	1	2	yes	—
37	C₄×C₁₆	Q₈	1	2	yes	—
38	#341	—	4	1	no	—
39	C₂×M₁₆	—	4	1	no	—
40	C₂×M₁₆	—	4	1	no	—
41	C₂²×C₈	D₈	1	1	yes	characteristic
42	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄	—	4	2	no	—
44	C₂×C₄⋊C₄	—	8	2	no	—
45	C₂×C₄²	D₈	1	1	yes	characteristic
46	C₂×C₁₆	—	2	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
49	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
51	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₁₆	#41	1	1	yes	characteristic
54	C₂×C₁₆	#41	1	1	yes	characteristic
55	#108	—	2	2	no	—
56	#108	—	2	2	no	—
57	#108	—	4	1	no	—
58	#108	—	4	1	no	—
59	#107	—	4	2	no	—
60	#107	—	4	2	no	—
61	#107	—	8	2	no	—
62	#97	—	8	1	no	—
63	C₄×C₈	—	8	1	no	—
64	C₄×C₈	C₄⋊C₄	1	2	yes	—
65	C₄×C₈	Q₁₆	1	2	yes	—
66	C₃₂	—	2	4	no	—
67	C₂²×C₄	—	8	2	no	—
68	C₂²×C₄	SD₃₂	1	2	yes	—
69	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
70	M₁₆	—	4	2	no	—
71	M₁₆	—	4	2	no	—
72	M₁₆	—	8	1	no	—
73	M₁₆	—	8	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	8	1	no	—
77	C₂×C₈	—	8	1	no	—
78	C₂×C₈	#107	1	2	yes	—
79	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₈	#121	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₈	#137	1	1	yes	characteristic
85	C₄⋊C₄	—	8	2	no	—
86	C₄⋊C₄	—	8	2	no	—
87	C₄⋊C₄	—	16	4	no	—
88	C₄²	#107	1	2	yes	—
89	C₄²	SD₃₂	1	1	yes	characteristic
90	C₄²	SD₃₂	1	1	yes	characteristic
91	C₁₆	—	2	2	no	—
92	C₁₆	C₄×D₄	1	1	yes	characteristic
93	C₁₆	C₄×Q₈	1	1	yes	characteristic
94	C₁₆	#125	1	1	yes	characteristic
95	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
96	C₂³	C₂×SD₃₂	1	1	yes	characteristic
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	16	4	no	—
99	C₂×C₄	—	16	2	no	—
100	C₂×C₄	#367	1	2	yes	—
101	C₂×C₄	#367	1	2	yes	—
102	C₂×C₄	C₂×#107	1	1	yes	characteristic
103	C₂×C₄	#459	1	1	yes	characteristic
104	C₂×C₄	#474	1	1	yes	characteristic
105	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
106	C₂×C₄	#509	1	1	yes	characteristic
107	C₂×C₄	#510	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	8	2	no	—
110	C₈	—	8	2	no	—
111	C₈	#389	1	1	yes	characteristic
112	C₈	#428	1	1	yes	characteristic
113	C₈	#462	1	1	yes	characteristic
114	C₈	#477	1	1	yes	characteristic
115	C₂²	—	2	2	no	—
116	C₂²	C₂×#367	1	1	yes	characteristic
117	C₂²	#2207	1	1	yes	characteristic
118	C₂²	#2225	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	32	2	no	—
121	C₄	#1426	1	4	yes	complemented
122	C₄	#1947	1	1	yes	characteristic
123	C₄	#2159	1	1	yes	characteristic
124	C₄	#2208	1	1	yes	characteristic
125	C₄	#2226	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	#13177	1	1	yes	characteristic
129	C₂	C₂×#1426	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1426

C₂

yes

maximal

#13627

C₂

yes

maximal, characteristic

#12906

C₂

yes

maximal, characteristic

C₄×#368

C₂

yes

maximal, characteristic

#7474

C₂

yes

maximal

C₂×M₆₄

C₂²

yes

—

C₂×#368

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

C₂×#367

—

C₂×#367

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

—

—

C₄⋊C₃₂

C₄

yes

complemented

#1381

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

#384

—

M₆₄

—

M₆₄

—

C₂×C₃₂

C₂×C₄

yes

—

#368

—

#368

—

#368

D₄

yes

—

#368

Q₈

yes

—

#367

—

#367

—

#367

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

Q₈

yes

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

Q₁₆

yes

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

SD₃₂

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#107

yes

—

C₄²

SD₃₂

yes

characteristic

C₄²

SD₃₂

yes

characteristic

C₁₆

—

C₁₆

C₄×D₄

yes

characteristic

C₁₆

C₄×Q₈

yes

characteristic

C₁₆

#125

yes

characteristic

C₁₆

C₄⋊Q₈

yes

characteristic

C₂³

C₂×SD₃₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#367

yes

—

101

C₂×C₄

#367

yes

—

102

C₂×C₄

C₂×#107

yes

characteristic

103

C₂×C₄

#459

yes

characteristic

104

C₂×C₄

#474

yes

characteristic

105

C₂×C₄

C₂×SD₃₂

yes

characteristic

106

C₂×C₄

#509

yes

characteristic

107

C₂×C₄

#510

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

#389

yes

characteristic

112

C₈

#428

yes

characteristic

113

C₈

#462

yes

characteristic

114

C₈

#477

yes

characteristic

115

C₂²

—

116

C₂²

C₂×#367

yes

characteristic

117

C₂²

#2207

yes

characteristic

118

C₂²

#2225

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#1426

yes

complemented

122

C₄

#1947

yes

characteristic

123

C₄

#2159

yes

characteristic

124

C₄

#2208

yes

characteristic

125

C₄

#2226

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#13177

yes

characteristic

129

C₂

C₂×#1426

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#367
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#367

commutator

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

C₂×SD₃₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
15. C₄⋊C₃₂	C₄	semidirect
121. C₄	#1426	semidirect

H ≅ G/N

type

15. C₄⋊C₃₂

C₄

semidirect

121. C₄

#1426

semidirect

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

subgroup	index
C₂×#1426	2
#13627	2
#12906	2
C₄×#368	2
#7474	2

subgroup

index

Subgroups — #152869

Conjugacy classes of subgroups

Complements

Maximal subgroups