Group #151996 — subgroups

Subgroups — #151996

Order 512.

198 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1425	C₂	1	1	yes	maximal, characteristic
2	#13633	C₂	1	1	yes	maximal, characteristic
3	C₂×#1397	C₂	1	1	yes	maximal, characteristic
4	#12670	C₂	1	2	yes	maximal
5	#7473	C₂	1	2	yes	maximal
6	C₂²×C₃₂	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	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1772	—	2	2	no	—
12	#1425	C₄	1	4	yes	complemented
13	#1424	C₄	1	2	yes	complemented
14	#1423	C₄	1	2	yes	complemented
15	#1397	C₄	1	4	yes	complemented
16	#1382	—	2	2	no	—
17	C₂²×C₁₆	D₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂³	1	1	yes	characteristic
19	C₂×#108	—	2	2	no	—
20	#427	—	2	2	no	—
21	#405	—	4	2	no	—
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₃₂	—	2	1	no	—
24	C₂×C₃₂	—	2	1	no	—
25	C₂×C₃₂	C₂×C₄	1	2	yes	—
26	C₂×C₃₂	C₂×C₄	1	2	yes	—
27	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃₂	C₂×C₄	1	2	yes	—
30	#368	—	2	2	no	—
31	#368	C₂×C₄	1	4	yes	—
32	#367	—	2	2	no	—
33	#366	C₂×C₄	1	4	yes	—
34	C₄×C₁₆	C₂×C₄	1	2	yes	—
35	C₄×C₁₆	Q₈	1	2	yes	—
36	#339	—	4	2	no	—
37	C₂×M₁₆	—	4	2	no	—
38	C₂²×C₈	D₈	1	1	yes	characteristic
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	#117	—	4	2	no	—
41	C₂×C₄²	D₈	1	1	yes	characteristic
42	C₂×C₁₆	—	2	1	no	—
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
45	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₁₆	#41	1	1	yes	characteristic
50	C₂×C₁₆	#41	1	1	yes	characteristic
51	#108	—	2	4	no	—
52	#108	—	4	2	no	—
53	#107	—	2	4	no	—
54	#106	—	4	2	no	—
55	#97	—	8	2	no	—
56	C₄×C₈	—	8	2	no	—
57	C₄×C₈	C₄⋊C₄	1	2	yes	—
58	C₄×C₈	Q₁₆	1	2	yes	—
59	C₃₂	—	2	2	no	—
60	C₃₂	C₄²	1	2	yes	—
61	C₃₂	C₄²	1	2	yes	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
64	M₁₆	—	4	4	no	—
65	M₁₆	—	8	2	no	—
66	C₂×C₈	—	8	2	no	—
67	C₂×C₈	#107	1	2	yes	—
68	C₂×C₈	D₁₆	1	2	yes	—
69	C₂×C₈	Q₃₂	1	2	yes	—
70	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈	C₄×D₄	1	1	yes	characteristic
72	C₂×C₈	C₄×D₄	1	1	yes	characteristic
73	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₈	#135	1	1	yes	characteristic
75	C₂×C₈	#135	1	1	yes	characteristic
76	C₄⋊C₄	—	4	4	no	—
77	C₂²⋊C₄	—	8	2	no	—
78	C₄²	—	2	1	no	—
79	C₄²	—	8	2	no	—
80	C₄²	#107	1	2	yes	—
81	C₁₆	—	2	2	no	—
82	C₁₆	C₂×C₄²	1	1	yes	characteristic
83	C₁₆	#117	1	1	yes	characteristic
84	C₁₆	C₄×Q₈	1	1	yes	characteristic
85	C₁₆	C₄×Q₈	1	1	yes	characteristic
86	C₂³	#508	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	8	4	no	—
90	C₂×C₄	C₂×#107	1	1	yes	characteristic
91	C₂×C₄	C₄×D₈	1	1	yes	characteristic
92	C₂×C₄	C₄×D₈	1	1	yes	characteristic
93	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
95	C₂×C₄	#508	1	1	yes	characteristic
96	C₈	—	8	4	no	—
97	C₈	#366	1	4	yes	—
98	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₈	#427	1	1	yes	characteristic
100	C₈	C₄×Q₁₆	1	1	yes	characteristic
101	C₈	C₄×Q₁₆	1	1	yes	characteristic
102	C₂²	—	2	2	no	—
103	C₂²	#2158	1	1	yes	characteristic
104	C₂²	C₄×D₁₆	1	1	yes	characteristic
105	C₂²	C₄×Q₃₂	1	1	yes	characteristic
106	C₄	—	4	2	no	—
107	C₄	—	16	4	no	—
108	C₄	C₄×#107	1	1	yes	characteristic
109	C₄	C₂×#366	1	1	yes	characteristic
110	C₄	C₄×D₁₆	1	1	yes	characteristic
111	C₄	C₄×Q₃₂	1	1	yes	characteristic
112	C₂	—	2	1	no	—
113	C₂	—	2	1	no	—
114	C₂	C₄×#366	1	1	yes	characteristic
115	C₂	#13656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1425

C₂

yes

maximal, characteristic

#13633

C₂

yes

maximal, characteristic

C₂×#1397

C₂

yes

maximal, characteristic

#12670

C₂

yes

maximal

#7473

C₂

yes

maximal

C₂²×C₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

#2158

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1772

—

#1425

C₄

yes

complemented

#1424

C₄

yes

complemented

#1423

C₄

yes

complemented

#1397

C₄

yes

complemented

#1382

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

C₂×C₄

yes

characteristic

C₂×C₃₂

C₂×C₄

yes

characteristic

C₂×C₃₂

C₂×C₄

yes

—

#368

—

#368

C₂×C₄

yes

—

#367

—

#366

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

Q₈

yes

—

#339

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂²×C₄

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

—

C₃₂

C₄²

yes

—

C₃₂

C₄²

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#107

yes

—

C₁₆

—

C₁₆

C₂×C₄²

yes

characteristic

C₁₆

#117

yes

characteristic

C₁₆

C₄×Q₈

yes

characteristic

C₁₆

C₄×Q₈

yes

characteristic

C₂³

#508

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#107

yes

characteristic

C₂×C₄

C₄×D₈

yes

characteristic

C₂×C₄

C₄×D₈

yes

characteristic

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

C₂×Q₃₂

yes

characteristic

C₂×C₄

#508

yes

characteristic

C₈

—

C₈

#366

yes

—

C₈

C₄×C₄⋊C₄

yes

characteristic

C₈

#427

yes

characteristic

100

C₈

C₄×Q₁₆

yes

characteristic

101

C₈

C₄×Q₁₆

yes

characteristic

102

C₂²

—

103

C₂²

#2158

yes

characteristic

104

C₂²

C₄×D₁₆

yes

characteristic

105

C₂²

C₄×Q₃₂

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₄×#107

yes

characteristic

109

C₄

C₂×#366

yes

characteristic

110

C₄

C₄×D₁₆

yes

characteristic

111

C₄

C₄×Q₃₂

yes

characteristic

112

C₂

—

113

C₂

—

114

C₂

C₄×#366

yes

characteristic

115

C₂

#13656

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₄×D₁₆
prime radical	C₂³	#508
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂×D₁₆

commutator

C₁₆

C₂×C₄²

Frattini, cosocle

C₂²×C₁₆

C₂³

socle

C₂²

C₄×D₁₆

prime radical

C₂³

#508

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
12. #1425	C₄	semidirect
13. #1424	C₄	semidirect
14. #1423	C₄	semidirect
15. #1397	C₄	semidirect

H ≅ G/N

type

12. #1425

C₄

semidirect

13. #1424

C₄

semidirect

14. #1423

C₄

semidirect

15. #1397

C₄

semidirect

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

subgroup	index
C₂×#1425	2
#13633	2
C₂×#1397	2
#12670	2
#7473	2

subgroup

index

Subgroups — #151996

Conjugacy classes of subgroups

Complements

Maximal subgroups