Group #135806 — subgroups

Subgroups — #135806

Order 512.

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

normal subgroups only

This group has 216 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	#12023	C₂	1	2	yes	maximal
2	#12022	C₂	1	2	yes	maximal
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	#8824	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	#1837	—	2	2	no	—
8	#1567	C₂²	1	4	yes	—
9	#1566	—	2	2	no	—
10	C₂×C₈²	C₂²	1	1	yes	characteristic
11	#1372	—	2	4	no	—
12	#1371	—	2	4	no	—
13	#1312	C₄	1	4	yes	—
14	C₂²×C₁₆	Q₈	1	2	yes	—
15	C₂²×C₁₆	Q₈	1	2	yes	—
16	#423	—	2	4	no	—
17	C₂×C₄×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
20	C₄×C₁₆	C₂×C₄	1	2	yes	—
21	C₄×C₁₆	C₂×C₄	1	2	yes	—
22	C₄×C₁₆	D₄	1	2	yes	—
23	C₄×C₁₆	D₄	1	2	yes	—
24	#341	—	4	4	no	—
25	#335	—	2	4	no	—
26	#335	—	2	4	no	—
27	#334	—	4	4	no	—
28	C₈²	C₂×C₄	1	2	yes	—
29	C₈²	Q₈	1	2	yes	—
30	C₂×M₁₆	—	4	4	no	—
31	C₂²×C₈	D₈	1	2	yes	—
32	C₂²×C₈	Q₁₆	1	2	yes	—
33	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
36	C₂×C₁₆	—	4	2	no	—
37	C₂×C₁₆	—	4	2	no	—
38	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
39	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
40	C₄⋊C₈	—	4	4	no	—
41	C₄⋊C₈	—	4	4	no	—
42	C₄×C₈	C₄⋊C₄	1	2	yes	—
43	C₄×C₈	C₄⋊C₄	1	2	yes	—
44	C₄×C₈	D₈	1	2	yes	—
45	C₄×C₈	Q₁₆	1	2	yes	—
46	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₈	C₂×D₄	1	1	yes	characteristic
48	C₄×C₈	C₂×D₄	1	1	yes	characteristic
49	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
53	M₁₆	—	8	8	no	—
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	—	8	4	no	—
59	C₂×C₈	#107	1	4	yes	—
60	C₂×C₈	#107	1	4	yes	—
61	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
65	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄²	#127	1	1	yes	characteristic
67	C₄²	C₂×D₈	1	1	yes	characteristic
68	C₄²	C₂×Q₁₆	1	1	yes	characteristic
69	C₁₆	—	2	4	no	—
70	C₁₆	—	2	4	no	—
71	C₂³	#500	1	1	yes	characteristic
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	#384	1	1	yes	characteristic
76	C₂×C₄	C₂×#107	1	1	yes	characteristic
77	C₂×C₄	C₂×#107	1	1	yes	characteristic
78	C₂×C₄	#493	1	1	yes	characteristic
79	C₂×C₄	#494	1	1	yes	characteristic
80	C₂×C₄	#500	1	1	yes	characteristic
81	C₂×C₄	#508	1	4	yes	—
82	C₈	—	2	1	no	—
83	C₈	—	2	1	no	—
84	C₈	—	2	4	no	—
85	C₈	—	2	4	no	—
86	C₈	—	16	4	no	—
87	C₈	#430	1	2	yes	—
88	C₈	#430	1	2	yes	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#1845	1	1	yes	characteristic
92	C₂²	#2253	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	#1849	1	1	yes	characteristic
96	C₄	#1849	1	1	yes	characteristic
97	C₄	#2158	1	2	yes	—
98	C₄	#2161	1	2	yes	—
99	C₄	#2249	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	#8826	1	1	yes	characteristic
102	C₂	#12907	1	1	yes	characteristic
103	C₂	#12910	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12023

C₂

yes

maximal

#12022

C₂

yes

maximal

C₂×#1312

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1837

—

#1567

C₂²

yes

—

#1566

—

C₂×C₈²

C₂²

yes

—

—

C₄

yes

—

C₂²×C₁₆

Q₈

yes

—

C₂²×C₁₆

Q₈

yes

—

#423

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

—

—

—

—

C₈²

C₂×C₄

yes

—

C₈²

Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

Q₁₆

yes

—

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#127

yes

characteristic

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

#500

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#430

yes

—

C₈

#430

yes

—

C₂²

—

C₂²

—

C₂²

#1845

yes

characteristic

C₂²

#2253

yes

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₄

#2161

yes

—

C₄

#2249

yes

—

100

C₂

—

101

C₂

#8826

yes

characteristic

102

C₂

#12907

yes

characteristic

103

C₂

#12910

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1845
prime radical	C₂³	#500
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#493

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#12023	2
#12022	2
C₂×#1312	2
#8824	2

subgroup

index

Subgroups — #135806

Conjugacy classes of subgroups

Maximal subgroups