Group #135752 — subgroups

Subgroups — #135752

Order 512.

208 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 98 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	#12884	C₂	1	1	yes	maximal, characteristic
2	#12880	C₂	1	1	yes	maximal, characteristic
3	#12005	C₂	1	2	yes	maximal
4	#11997	C₂	1	2	yes	maximal
5	#8137	C₂	1	1	yes	maximal, characteristic
6	#2150	C₂²	1	2	yes	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	#1726	C₂²	1	1	yes	characteristic
9	#1450	C₂²	1	1	yes	characteristic
10	#1450	C₂²	1	1	yes	characteristic
11	#1376	—	2	2	no	—
12	#1374	—	2	2	no	—
13	#1370	—	2	4	no	—
14	#1367	—	2	4	no	—
15	#1275	C₄	1	2	yes	—
16	#1274	C₄	1	2	yes	—
17	C₂×M₃₂	—	2	2	no	—
18	C₂²×C₁₆	—	2	2	no	—
19	#431	C₂×C₄	1	2	yes	—
20	#431	C₂×C₄	1	2	yes	—
21	C₂×#97	C₂×C₄	1	1	yes	characteristic
22	C₂×#97	Q₈	1	2	yes	—
23	C₂×C₄×C₈	D₄	1	2	yes	—
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	#376	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₁₆	—	2	2	no	—
27	C₄⋊C₁₆	—	2	2	no	—
28	C₄⋊C₁₆	—	2	2	no	—
29	C₄⋊C₁₆	—	2	2	no	—
30	#322	Q₈	1	2	yes	—
31	#322	Q₈	1	2	yes	—
32	C₈²	D₄	1	2	yes	—
33	C₈²	D₄	1	2	yes	—
34	C₂²×C₈	C₄⋊C₄	1	2	yes	—
35	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄²	C₄⋊C₄	1	2	yes	—
38	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
39	M₃₂	—	4	4	no	—
40	C₂×C₁₆	—	4	2	no	—
41	C₂×C₁₆	—	4	2	no	—
42	C₂×C₁₆	—	4	4	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	—	2	2	no	—
45	C₂²⋊C₈	—	2	2	no	—
46	C₂²⋊C₈	—	2	2	no	—
47	#97	—	2	2	no	—
48	#97	—	4	2	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	C₄⋊C₄	1	4	yes	—
51	C₄×C₈	C₄⋊C₄	1	4	yes	—
52	C₄×C₈	C₂×D₄	1	1	yes	characteristic
53	C₄×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
56	#95	—	2	2	no	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄	#131	1	2	yes	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	#127	1	1	yes	characteristic
67	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₈	#135	1	4	yes	—
71	C₄²	—	2	2	no	—
72	C₄²	—	2	2	no	—
73	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₄²	#131	1	2	yes	—
76	C₁₆	—	8	2	no	—
77	C₁₆	—	8	2	no	—
78	C₂³	#424	1	1	yes	characteristic
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	4	4	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	#384	1	1	yes	characteristic
83	C₂×C₄	#424	1	2	yes	—
84	C₂×C₄	#424	1	2	yes	—
85	C₂×C₄	#424	1	1	yes	characteristic
86	C₂×C₄	#427	1	2	yes	—
87	C₂×C₄	#430	1	2	yes	—
88	C₈	—	2	1	no	—
89	C₈	—	2	1	no	—
90	C₈	—	4	4	no	—
91	C₈	—	4	4	no	—
92	C₈	—	4	2	no	—
93	C₈	#495	1	2	yes	—
94	C₈	#499	1	2	yes	—
95	C₂²	—	2	2	no	—
96	C₂²	#1572	1	2	yes	—
97	C₂²	#1843	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	8	2	no	—
101	C₄	#1572	1	2	yes	—
102	C₄	#1846	1	1	yes	characteristic
103	C₄	#1849	1	1	yes	characteristic
104	C₂	—	2	2	no	—
105	C₂	#8811	1	1	yes	characteristic
106	C₂	#12902	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12884

C₂

yes

maximal, characteristic

#12880

C₂

yes

maximal, characteristic

#12005

C₂

yes

maximal

#11997

C₂

yes

maximal

#8137

C₂

yes

maximal, characteristic

#2150

C₂²

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1726

C₂²

yes

characteristic

#1450

C₂²

yes

characteristic

#1450

C₂²

yes

—

—

—

—

C₄

yes

—

#1274

C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#431

C₂×C₄

yes

—

#431

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

Q₈

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#376

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#322

Q₈

yes

—

#322

Q₈

yes

—

C₈²

D₄

yes

—

C₈²

D₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#131

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

#135

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#131

yes

—

C₁₆

—

C₁₆

—

C₂³

#424

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#384

yes

characteristic

C₂×C₄

#424

yes

—

C₂×C₄

#424

yes

—

C₂×C₄

#424

yes

characteristic

C₂×C₄

#427

yes

—

C₂×C₄

#430

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#495

yes

—

C₈

#499

yes

—

C₂²

—

C₂²

#1572

yes

—

C₂²

#1843

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

#1572

yes

—

102

C₄

#1846

yes

characteristic

103

C₄

#1849

yes

characteristic

104

C₂

—

105

C₂

#8811

yes

characteristic

106

C₂

#12902

yes

—

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1843
prime radical	C₂³	#424
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#127

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×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
#12884	2
#12880	2
#12005	2
#11997	2
#8137	2

subgroup

index

Subgroups — #135752

Conjugacy classes of subgroups

Maximal subgroups