Group #152409 — subgroups

Subgroups — #152409

Order 512.

180 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 104 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	#13638	C₂	1	2	yes	maximal
2	C₂×#1425	C₂	1	1	yes	maximal, characteristic
3	#13633	C₂	1	1	yes	maximal, characteristic
4	#13549	C₂	1	1	yes	maximal, characteristic
5	#12911	C₂	1	2	yes	maximal
6	C₂×M₆₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×#368	C₂²	1	2	yes	—
9	#2158	C₂²	1	2	yes	—
10	#2155	—	2	2	no	—
11	#2108	C₂²	1	1	yes	characteristic
12	#1833	—	2	2	no	—
13	#1426	—	2	4	no	—
14	#1425	—	2	1	no	—
15	#1425	—	2	1	no	—
16	#1424	—	2	1	no	—
17	#1423	—	2	1	no	—
18	#1397	C₄	1	2	yes	complemented
19	C₄×C₃₂	C₄	1	2	yes	complemented
20	C₂×M₃₂	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	C₂³	1	1	yes	characteristic
22	C₂×#108	—	2	2	no	—
23	#427	—	2	2	no	—
24	#405	D₄	1	1	yes	characteristic
25	M₆₄	D₄	1	2	yes	—
26	M₆₄	Q₈	1	2	yes	—
27	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃₂	C₂×C₄	1	2	yes	—
29	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃₂	D₄	1	1	yes	characteristic
31	C₂×C₃₂	D₄	1	1	yes	characteristic
32	C₂×C₃₂	Q₈	1	1	yes	characteristic
33	C₂×C₃₂	Q₈	1	1	yes	characteristic
34	#368	—	2	2	no	—
35	#368	—	2	2	no	—
36	#367	—	2	2	no	—
37	#366	—	2	2	no	—
38	#365	—	4	4	no	—
39	#346	Q₈	1	2	yes	—
40	C₄×C₁₆	C₂×C₄	1	2	yes	—
41	#344	—	4	2	no	—
42	H(3,ℤ/4)	—	4	2	no	—
43	C₂×M₁₆	—	4	2	no	—
44	C₂×M₁₆	D₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	#117	—	4	2	no	—
47	#117	D₈	1	1	yes	characteristic
48	M₃₂	D₈	1	2	yes	—
49	M₃₂	Q₁₆	1	2	yes	—
50	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
51	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
54	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
57	#108	—	4	2	no	—
58	#108	—	4	2	no	—
59	#107	—	4	2	no	—
60	#106	—	4	2	no	—
61	#97	Q₁₆	1	2	yes	—
62	C₄×C₈	C₄⋊C₄	1	2	yes	—
63	C₃₂	C₄⋊C₄	1	1	yes	characteristic
64	C₃₂	C₄⋊C₄	1	1	yes	characteristic
65	C₃₂	C₄⋊C₄	1	1	yes	characteristic
66	C₃₂	C₄⋊C₄	1	1	yes	characteristic
67	C₃₂	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
69	M₁₆	—	2	1	no	—
70	M₁₆	—	2	1	no	—
71	M₁₆	—	8	2	no	—
72	M₁₆	—	8	2	no	—
73	C₂×C₈	—	8	2	no	—
74	C₂×C₈	#107	1	2	yes	—
75	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈	#127	1	1	yes	characteristic
77	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₈	C₂×D₈	1	1	yes	characteristic
79	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	8	2	no	—
83	C₂²⋊C₄	—	2	1	no	—
84	C₂²⋊C₄	—	8	2	no	—
85	C₄²	—	8	2	no	—
86	C₄²	#107	1	2	yes	—
87	C₁₆	#107	1	4	yes	—
88	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
91	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
92	C₂³	#508	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	16	2	no	—
95	C₂×C₄	C₂×#107	1	1	yes	characteristic
96	C₂×C₄	#493	1	1	yes	characteristic
97	C₂×C₄	#500	1	1	yes	characteristic
98	C₂×C₄	#508	1	1	yes	characteristic
99	C₂×C₄	#508	1	1	yes	characteristic
100	C₂×C₄	#508	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	16	2	no	—
103	C₈	#384	1	1	yes	characteristic
104	C₈	C₂×#107	1	1	yes	characteristic
105	C₈	#494	1	1	yes	characteristic
106	C₈	#500	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	#2158	1	1	yes	characteristic
109	C₂²	#2249	1	1	yes	characteristic
110	C₂²	#2253	1	1	yes	characteristic
111	C₄	—	4	2	no	—
112	C₄	—	32	2	no	—
113	C₄	#1845	1	1	yes	characteristic
114	C₄	#2158	1	1	yes	characteristic
115	C₄	#2249	1	1	yes	characteristic
116	C₄	#2253	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#12907	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13638

C₂

yes

maximal

C₂×#1425

C₂

yes

maximal, characteristic

#13633

C₂

yes

maximal, characteristic

#13549

C₂

yes

maximal, characteristic

#12911

C₂

yes

maximal

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

#2158

C₂²

yes

—

#2155

—

#2108

C₂²

yes

—

—

—

—

—

—

C₄

yes

complemented

C₄×C₃₂

C₄

yes

complemented

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

D₄

yes

characteristic

M₆₄

D₄

yes

—

M₆₄

Q₈

yes

—

C₂×C₃₂

C₂×C₄

yes

characteristic

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

C₂×C₄

yes

characteristic

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

Q₈

yes

characteristic

C₂×C₃₂

Q₈

yes

—

—

—

—

—

Q₈

yes

—

C₄×C₁₆

C₂×C₄

yes

—

#344

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

#117

D₈

yes

characteristic

M₃₂

D₈

yes

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

—

—

—

Q₁₆

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₃₂

C₄⋊C₄

yes

characteristic

C₃₂

C₄⋊C₄

yes

characteristic

C₃₂

C₄⋊C₄

yes

characteristic

C₃₂

C₄⋊C₄

yes

characteristic

C₃₂

C₄⋊C₄

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

#107

yes

—

C₁₆

#107

yes

—

C₁₆

C₂×C₄⋊C₄

yes

characteristic

C₁₆

C₂×C₄⋊C₄

yes

characteristic

C₁₆

C₄⋊Q₈

yes

characteristic

C₁₆

C₄⋊Q₈

yes

characteristic

C₂³

#508

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#508

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

#384

yes

characteristic

104

C₈

C₂×#107

yes

characteristic

105

C₈

#494

yes

characteristic

106

C₈

#500

yes

characteristic

107

C₂²

—

108

C₂²

#2158

yes

characteristic

109

C₂²

#2249

yes

characteristic

110

C₂²

#2253

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#1845

yes

characteristic

114

C₄

#2158

yes

characteristic

115

C₄

#2249

yes

characteristic

116

C₄

#2253

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#12907

yes

characteristic

	subgroup	G/H
center	C₄	#2158
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12907
prime radical	C₂³	#508
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#2158

commutator

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

C₂

C₂³

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
18. #1397	C₄	semidirect
19. C₄×C₃₂	C₄	semidirect

H ≅ G/N

type

18. #1397

C₄

semidirect

19. C₄×C₃₂

C₄

semidirect

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

subgroup	index
#13638	2
C₂×#1425	2
#13633	2
#13549	2
#12911	2

subgroup

index

Subgroups — #152409

Conjugacy classes of subgroups

Complements

Maximal subgroups