Group #146108 — subgroups

Subgroups — #146108

Order 512.

170 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 52 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	#11973	C₂	1	2	yes	maximal
2	#9759	C₂	1	1	yes	maximal, characteristic
3	#8798	C₂	1	1	yes	maximal, characteristic
4	#7218	C₂	1	2	yes	maximal
5	#7198	C₂	1	1	yes	maximal, characteristic
6	#3160	C₂²	1	1	yes	characteristic
7	#2162	C₂²	1	2	yes	—
8	#1835	C₂²	1	1	yes	characteristic
9	#1568	C₂²	1	2	yes	—
10	#1518	—	2	2	no	—
11	#1452	C₂²	1	1	yes	characteristic
12	#1365	—	2	2	no	—
13	#1364	—	2	2	no	—
14	#1287	—	2	2	no	—
15	#533	—	2	1	no	—
16	#500	D₄	1	2	yes	—
17	#499	—	2	2	no	—
18	#498	—	2	1	no	—
19	#432	—	2	2	no	—
20	#431	D₄	1	2	yes	—
21	#427	—	2	2	no	—
22	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
23	#382	—	2	1	no	—
24	C₄⋊C₁₆	—	2	2	no	—
25	C₂²⋊C₁₆	—	2	2	no	—
26	C₄×C₁₆	—	2	2	no	—
27	#340	—	2	2	no	—
28	#335	—	2	2	no	—
29	#334	—	2	2	no	—
30	#328	—	4	4	no	—
31	#322	—	2	2	no	—
32	C₂²×C₈	#41	1	2	yes	—
33	C₄⋊Q₈	—	2	2	no	—
34	#125	—	4	1	no	—
35	C₂²⋊Q₈	—	4	2	no	—
36	C₄×Q₈	—	4	2	no	—
37	#117	—	4	1	no	—
38	C₂×C₄⋊C₄	—	4	1	no	—
39	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
40	C₂×C₁₆	—	4	2	no	—
41	#107	—	2	4	no	—
42	#106	—	4	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	—	4	2	no	—
45	C₂²⋊C₈	—	2	2	no	—
46	C₂²⋊C₈	—	4	2	no	—
47	#97	—	4	2	no	—
48	C₄×C₈	—	2	2	no	—
49	C₄×C₈	C₂×D₄	1	1	yes	characteristic
50	C₄×C₈	#41	1	1	yes	characteristic
51	C₄×C₈	#41	1	2	yes	—
52	#95	—	4	2	no	—
53	C₂×Q₈	—	4	2	no	—
54	C₂²×C₄	—	8	1	no	—
55	C₂²×C₄	#124	1	1	yes	characteristic
56	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	—	4	2	no	—
59	C₂×C₈	—	8	2	no	—
60	C₂×C₈	#123	1	2	yes	—
61	C₂×C₈	#126	1	2	yes	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	8	2	no	—
65	C₄⋊C₄	—	8	2	no	—
66	C₂²⋊C₄	—	8	1	no	—
67	C₄²	—	8	1	no	—
68	C₄²	#121	1	1	yes	characteristic
69	C₄²	#123	1	1	yes	characteristic
70	C₄²	#135	1	2	yes	—
71	C₁₆	—	8	2	no	—
72	C₂³	#490	1	1	yes	characteristic
73	Q₈	—	16	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	8	2	no	—
77	C₂×C₄	—	16	1	no	—
78	C₂×C₄	—	16	2	no	—
79	C₂×C₄	#397	1	1	yes	characteristic
80	C₂×C₄	#463	1	2	yes	—
81	C₂×C₄	#482	1	2	yes	—
82	C₂×C₄	#488	1	1	yes	characteristic
83	C₈	—	2	2	no	—
84	C₈	—	2	2	no	—
85	C₈	—	8	2	no	—
86	C₈	—	16	2	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	2	no	—
89	C₂²	#2073	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	16	2	no	—
94	C₄	—	32	1	no	—
95	C₄	#2082	1	2	yes	—
96	C₂	—	4	1	no	—
97	C₂	#10635	1	1	yes	characteristic
98	C₂	#13489	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11973

C₂

yes

maximal

#9759

C₂

yes

maximal, characteristic

#8798

C₂

yes

maximal, characteristic

#7218

C₂

yes

maximal

#7198

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

#2162

C₂²

yes

—

#1835

C₂²

yes

characteristic

#1568

C₂²

yes

—

#1518

—

#1452

C₂²

yes

—

—

—

—

D₄

yes

—

—

—

—

D₄

yes

—

#427

—

C₂×C₄×C₈

C₂³

yes

characteristic

#382

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

—

—

—

—

—

C₂²×C₈

#41

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#124

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#123

yes

—

C₂×C₈

#126

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

yes

C₄²

yes

C₄²

yes

—

C₁₆

—

C₂³

#490

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#397

yes

characteristic

C₂×C₄

#463

yes

—

C₂×C₄

#482

yes

—

C₂×C₄

#488

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#2073

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#2082

yes

—

C₂

—

C₂

#10635

yes

characteristic

C₂

#13489

yes

—

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#490
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2073

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#490

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ 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
#11973	2
#9759	2
#8798	2
#7218	2
#7198	2

subgroup

index

Subgroups — #146108

Conjugacy classes of subgroups

Maximal subgroups