Group #135673 — subgroups

Subgroups — #135673

Order 512.

254 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 130 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	#12590	C₂	1	1	yes	maximal, characteristic
2	#8806	C₂	1	2	yes	maximal
3	#7154	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆	C₂²	1	2	yes	—
5	#1842	—	2	2	no	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₂×#335	—	2	2	no	—
8	C₂×#335	C₂²	1	4	yes	—
9	#1374	C₄	1	4	yes	—
10	C₂²×C₁₆	C₂×C₄	1	4	yes	—
11	C₂×C₄⋊C₈	—	2	4	no	—
12	C₂×C₄⋊C₈	—	4	2	no	—
13	C₂×C₄⋊C₈	—	4	2	no	—
14	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₈	D₄	1	1	yes	characteristic
16	C₂×C₄×C₈	Q₈	1	2	yes	—
17	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
18	C₄³	D₄	1	1	yes	characteristic
19	C₄×C₁₆	—	2	4	no	—
20	#335	—	2	8	no	—
21	#335	—	4	4	no	—
22	C₂²×C₈	—	4	4	no	—
23	C₂²×C₈	C₄⋊C₄	1	2	yes	—
24	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄²	C₄⋊C₄	1	2	yes	—
27	C₂×C₄²	D₈	1	1	yes	characteristic
28	C₂×C₄²	D₈	1	1	yes	characteristic
29	C₂×C₄²	Q₁₆	1	2	yes	—
30	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
31	C₂×C₁₆	—	2	4	no	—
32	C₂×C₁₆	—	2	8	no	—
33	C₄⋊C₈	—	4	4	no	—
34	C₄⋊C₈	—	4	4	no	—
35	C₄⋊C₈	—	8	4	no	—
36	C₄⋊C₈	—	8	4	no	—
37	C₄×C₈	—	2	4	no	—
38	C₄×C₈	—	4	2	no	—
39	C₄×C₈	SD₁₆	1	4	yes	—
40	C₄×C₈	#41	1	4	yes	—
41	C₂²×C₄	#107	1	2	yes	—
42	C₂²×C₄	#107	1	2	yes	—
43	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
45	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	8	4	no	—
50	C₂×C₈	—	8	4	no	—
51	C₂×C₈	—	8	4	no	—
52	C₂×C₈	#106	1	4	yes	—
53	C₂×C₈	#117	1	2	yes	—
54	C₂×C₈	#125	1	2	yes	—
55	C₂×C₈	#131	1	4	yes	—
56	C₄²	—	2	2	no	—
57	C₄²	—	2	2	no	—
58	C₄²	—	4	2	no	—
59	C₄²	D₁₆	1	2	yes	—
60	C₄²	SD₃₂	1	4	yes	—
61	C₄²	Q₃₂	1	2	yes	—
62	C₄²	#124	1	2	yes	—
63	C₄²	C₂×SD₁₆	1	2	yes	—
64	C₁₆	—	4	4	no	—
65	C₂³	C₂×#107	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	#366	1	4	yes	—
72	C₂×C₄	#367	1	4	yes	—
73	C₂×C₄	#382	1	1	yes	characteristic
74	C₂×C₄	#424	1	2	yes	—
75	C₂×C₄	C₂×#106	1	1	yes	characteristic
76	C₂×C₄	#430	1	2	yes	—
77	C₂×C₄	#486	1	2	yes	—
78	C₂×C₄	#487	1	2	yes	—
79	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
80	C₂×C₄	C₂×SD₃₂	1	2	yes	—
81	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
82	C₈	—	2	4	no	—
83	C₈	—	2	2	no	—
84	C₈	—	16	4	no	—
85	C₈	#433	1	4	yes	—
86	C₂²	#1571	1	2	yes	—
87	C₂²	#1834	1	1	yes	characteristic
88	C₂²	C₂×#366	1	1	yes	characteristic
89	C₂²	C₂×#367	1	1	yes	characteristic
90	C₂²	#2161	1	2	yes	—
91	C₄	—	2	4	no	—
92	C₄	—	2	4	no	—
93	C₄	—	2	2	no	—
94	C₄	#1570	1	2	yes	—
95	C₄	#1832	1	1	yes	characteristic
96	C₄	#1838	1	1	yes	characteristic
97	C₄	#2240	1	2	yes	—
98	C₄	#2241	1	2	yes	—
99	C₂	#8787	1	1	yes	characteristic
100	C₂	#12032	1	2	yes	—
101	C₂	#12033	1	2	yes	—
102	C₂	#12888	1	1	yes	characteristic
103	C₂	#12892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12590

C₂

yes

maximal, characteristic

#8806

C₂

yes

maximal

#7154

C₂

yes

maximal

C₂×C₄×C₁₆

C₂²

yes

—

#1842

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#335

C₂²

yes

—

#1374

C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

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

—

#335

—

#335

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

Q₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#125

yes

—

C₂×C₈

#131

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₁₆

yes

—

C₄²

SD₃₂

yes

—

C₄²

Q₃₂

yes

—

C₄²

#124

yes

—

C₄²

C₂×SD₁₆

yes

—

C₁₆

—

C₂³

C₂×#107

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#366

yes

—

C₂×C₄

#367

yes

—

C₂×C₄

#382

yes

characteristic

C₂×C₄

#424

yes

—

C₂×C₄

C₂×#106

yes

characteristic

C₂×C₄

#430

yes

—

C₂×C₄

#486

yes

—

C₂×C₄

#487

yes

—

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

C₂×SD₃₂

yes

—

C₂×C₄

C₂×Q₃₂

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#433

yes

—

C₂²

#1571

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

C₄

#1570

yes

—

C₄

yes

C₄

yes

C₄

yes

—

C₄

#2241

yes

—

C₂

#8787

yes

characteristic

100

C₂

#12032

yes

—

101

C₂

#12033

yes

—

102

C₂

#12888

yes

characteristic

103

C₂

#12892

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₈

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×#107

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ 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
#12590	2
#8806	2
#7154	2

subgroup

index

#12590

#8806

#7154

Subgroups — #135673

Conjugacy classes of subgroups

Maximal subgroups