Group #129095 — subgroups

Subgroups — #129095

Order 512.

284 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#12008	C₂	1	2	yes	maximal
3	#8266	C₂	1	2	yes	maximal
4	#7155	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆	C₄	1	2	yes	—
6	C₂×C₄×C₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	#1564	C₂²	1	4	yes	—
11	#1450	—	2	2	no	—
12	#1372	C₄	1	4	yes	complemented
13	#1371	C₄	1	4	yes	complemented
14	#1276	—	2	2	no	—
15	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂×C₄	1	2	yes	—
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	#431	—	2	4	no	—
19	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	Q₈	1	2	yes	—
22	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
23	C₄³	D₄	1	1	yes	characteristic
24	C₄×C₁₆	—	2	4	no	—
25	C₄×C₁₆	—	2	2	no	—
26	C₄×C₁₆	C₈	1	8	yes	complemented
27	C₄×C₁₆	C₂×C₄	1	2	yes	—
28	C₄×C₁₆	C₂×C₄	1	2	yes	—
29	#335	C₂×C₄	1	8	yes	—
30	#334	—	2	4	no	—
31	#322	—	4	2	no	—
32	C₈²	—	4	2	no	—
33	C₂²×C₈	C₄⋊C₄	1	2	yes	—
34	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄²	—	2	2	no	—
37	C₂×C₄²	C₄⋊C₄	1	2	yes	—
38	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	C₄²	1	2	yes	—
43	C₂×C₁₆	C₄²	1	1	yes	characteristic
44	C₂×C₁₆	C₄²	1	1	yes	characteristic
45	C₂×C₁₆	C₂×C₈	1	4	yes	—
46	C₂×C₁₆	C₂×C₈	1	4	yes	—
47	C₄⋊C₈	—	2	8	no	—
48	C₂²⋊C₈	—	4	4	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	—	4	4	no	—
51	C₄×C₈	C₂×C₈	1	4	yes	—
52	C₄×C₈	M₁₆	1	4	yes	—
53	C₄×C₈	D₈	1	1	yes	characteristic
54	C₄×C₈	D₈	1	1	yes	characteristic
55	C₄×C₈	Q₁₆	1	2	yes	—
56	C₄×C₈	C₂²×C₄	1	2	yes	—
57	C₄×C₈	#41	1	2	yes	—
58	C₂²×C₄	—	2	2	no	—
59	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄	#135	1	2	yes	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	4	8	no	—
63	C₂×C₈	C₄⋊C₈	1	4	yes	—
64	C₂×C₈	#107	1	2	yes	—
65	C₂×C₈	#107	1	2	yes	—
66	C₂×C₈	#108	1	4	yes	—
67	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₈	#117	1	1	yes	characteristic
69	C₂×C₈	C₄×Q₈	1	2	yes	—
70	C₂×C₈	C₂²×C₈	1	2	yes	—
71	C₂×C₈	C₂×M₁₆	1	2	yes	—
72	C₄²	—	2	8	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₄⋊C₈	1	4	yes	—
75	C₄²	C₄×D₄	1	2	yes	—
76	C₄²	C₂×D₈	1	1	yes	characteristic
77	C₄²	C₂×Q₁₆	1	1	yes	characteristic
78	C₁₆	—	2	2	no	—
79	C₁₆	—	2	4	no	—
80	C₁₆	C₄×C₈	1	2	yes	—
81	C₁₆	C₄×C₈	1	2	yes	—
82	C₂³	#427	1	1	yes	characteristic
83	C₂×C₄	—	2	8	no	—
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
88	C₂×C₄	C₂×#107	1	1	yes	characteristic
89	C₂×C₄	C₂×#108	1	2	yes	—
90	C₂×C₄	C₄×D₈	1	2	yes	—
91	C₂×C₄	C₄×Q₁₆	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	8	8	no	—
94	C₈	#335	1	4	yes	—
95	C₈	#335	1	4	yes	—
96	C₈	C₂×C₄×C₈	1	1	yes	characteristic
97	C₈	C₄×M₁₆	1	1	yes	characteristic
98	C₈	#431	1	2	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	#1564	1	2	yes	—
101	C₂²	C₄×#107	1	1	yes	characteristic
102	C₄	—	2	8	no	—
103	C₄	—	2	2	no	—
104	C₄	C₂×#335	1	2	yes	—
105	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
106	C₄	C₄×#108	1	1	yes	characteristic
107	C₄	#2178	1	4	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#335	1	1	yes	characteristic
110	C₂	#12670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₁₆

C₂

yes

maximal, characteristic

#12008

C₂

yes

maximal

#8266

C₂

yes

maximal

#7155

C₂

yes

maximal

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

—

#1450

—

#1372

C₄

yes

complemented

#1371

C₄

yes

complemented

#1276

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

complemented

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

#335

C₂×C₄

yes

—

#334

—

#322

—

C₈²

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#135

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

—

C₄²

C₄×D₄

yes

—

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₄×C₈

yes

—

C₁₆

C₄×C₈

yes

—

C₂³

#427

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

—

C₂×C₄

C₂×#107

yes

characteristic

C₂×C₄

C₂×#108

yes

—

C₂×C₄

C₄×D₈

yes

—

C₂×C₄

C₄×Q₁₆

yes

—

C₈

—

C₈

—

C₈

#335

yes

—

C₈

#335

yes

—

C₈

C₂×C₄×C₈

yes

characteristic

C₈

C₄×M₁₆

yes

characteristic

C₈

#431

yes

—

C₂²

—

100

C₂²

#1564

yes

—

101

C₂²

C₄×#107

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂×#335

yes

—

105

C₄

C₄×C₄⋊C₈

yes

characteristic

106

C₄

C₄×#108

yes

characteristic

107

C₄

#2178

yes

—

108

C₂

—

109

C₂

C₄×#335

yes

characteristic

110

C₂

#12670

yes

—

	subgroup	G/H
center	C₄×C₈	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#107
prime radical	C₂³	#427
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄×C₈

D₈

commutator

C₈

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
12. #1372	C₄	semidirect
13. #1371	C₄	semidirect
26. C₄×C₁₆	C₈	semidirect

H ≅ G/N

type

12. #1372

C₄

semidirect

13. #1371

C₄

semidirect

26. C₄×C₁₆

C₈

semidirect

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

subgroup	index
C₄²×C₁₆	2
#12008	2
#8266	2
#7155	2

subgroup

index

C₄²×C₁₆

#12008

#8266

#7155

Subgroups — #129095

Conjugacy classes of subgroups

Complements

Maximal subgroups