Group #135801 — subgroups

Subgroups — #135801

Order 512.

217 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 110 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	#12020	C₂	1	4	yes	maximal
2	#11402	C₂	1	1	yes	maximal, characteristic
3	#8824	C₂	1	1	yes	maximal, characteristic
4	#8822	C₂	1	1	yes	maximal, characteristic
5	C₄×M₃₂	C₂²	1	2	yes	—
6	#1848	—	2	1	no	—
7	#1837	—	2	1	no	—
8	#1567	—	2	1	no	—
9	#1566	—	2	1	no	—
10	#1566	C₂²	1	2	yes	—
11	#1566	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1368	—	2	8	no	—
14	#1312	C₄	1	2	yes	—
15	#1312	C₄	1	2	yes	—
16	C₂×M₃₂	Q₈	1	4	yes	—
17	C₂×#108	—	4	2	no	—
18	#423	—	2	2	no	—
19	#423	—	2	2	no	—
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
23	#346	D₄	1	4	yes	—
24	C₄×C₁₆	C₂×C₄	1	4	yes	—
25	#341	—	4	2	no	—
26	#335	—	4	2	no	—
27	#334	—	2	4	no	—
28	#334	—	2	4	no	—
29	#334	—	4	2	no	—
30	C₈²	C₂×C₄	1	1	yes	characteristic
31	C₈²	C₂×C₄	1	1	yes	characteristic
32	C₈²	Q₈	1	1	yes	characteristic
33	C₈²	Q₈	1	1	yes	characteristic
34	C₂×M₁₆	—	4	2	no	—
35	C₂×M₁₆	—	4	2	no	—
36	C₂²×C₈	SD₁₆	1	2	yes	—
37	C₂²×C₈	SD₁₆	1	2	yes	—
38	C₂²×C₈	C₂×Q₈	1	2	yes	—
39	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
40	M₃₂	—	4	4	no	—
41	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
42	#108	—	8	4	no	—
43	C₄⋊C₈	—	4	2	no	—
44	C₄⋊C₈	—	4	2	no	—
45	C₄⋊C₈	—	4	2	no	—
46	C₄⋊C₈	—	4	2	no	—
47	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₈	SD₁₆	1	2	yes	—
52	C₄×C₈	SD₁₆	1	2	yes	—
53	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₄×C₈	C₂×D₄	1	2	yes	—
55	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
59	M₁₆	—	8	4	no	—
60	M₁₆	—	8	4	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	8	2	no	—
65	C₂×C₈	—	8	2	no	—
66	C₂×C₈	#106	1	2	yes	—
67	C₂×C₈	#106	1	2	yes	—
68	C₂×C₈	#106	1	2	yes	—
69	C₂×C₈	#106	1	2	yes	—
70	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₈	C₄⋊Q₈	1	2	yes	—
72	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₄²	#127	1	1	yes	characteristic
74	C₄²	C₂×SD₁₆	1	1	yes	characteristic
75	C₄²	C₂×SD₁₆	1	1	yes	characteristic
76	C₁₆	—	2	8	no	—
77	C₂³	#498	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	#384	1	1	yes	characteristic
82	C₂×C₄	C₂×#106	1	1	yes	characteristic
83	C₂×C₄	C₂×#106	1	1	yes	characteristic
84	C₂×C₄	#492	1	1	yes	characteristic
85	C₂×C₄	#492	1	1	yes	characteristic
86	C₂×C₄	#498	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	—	2	2	no	—
89	C₈	—	2	2	no	—
90	C₈	—	2	2	no	—
91	C₈	—	16	2	no	—
92	C₈	—	16	2	no	—
93	C₈	#365	1	4	yes	—
94	C₈	#430	1	4	yes	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#1844	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	#1849	1	2	yes	—
103	C₄	C₂×#365	1	2	yes	—
104	C₄	#2155	1	2	yes	—
105	C₂	—	4	1	no	—
106	C₂	#8825	1	1	yes	characteristic
107	C₂	#12903	1	1	yes	characteristic
108	C₂	#12904	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12020

C₂

yes

maximal

#11402

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal, characteristic

#8822

C₂

yes

maximal, characteristic

C₄×M₃₂

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#1566

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1368

—

#1312

C₄

yes

—

#1312

C₄

yes

—

C₂×M₃₂

Q₈

yes

—

C₂×#108

—

#423

—

#423

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#346

D₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×Q₈

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

C₄⋊C₄

yes

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#127

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₁₆

—

C₂³

#498

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

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#365

yes

—

C₈

#430

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#1844

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#1849

yes

—

103

C₄

C₂×#365

yes

—

104

C₄

#2155

yes

—

105

C₂

—

106

C₂

#8825

yes

characteristic

107

C₂

#12903

yes

characteristic

108

C₂

#12904

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1844
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#498
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1844

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#498

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ 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
#12020	2
#11402	2
#8824	2
#8822	2

subgroup

index

Subgroups — #135801

Conjugacy classes of subgroups

Maximal subgroups