Group #93308 — subgroups

Subgroups — #93308

Order 512.

220 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 106 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	#8784	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆	C₄	1	2	yes	—
4	#1842	—	2	2	no	—
5	C₄²×C₈	C₂²	1	1	yes	characteristic
6	#1374	—	2	2	no	—
7	#1277	—	2	2	no	—
8	#1277	C₄	1	4	yes	—
9	C₂²×C₁₆	—	2	2	no	—
10	C₂×C₄⋊C₈	—	2	4	no	—
11	C₂×C₄⋊C₈	—	4	2	no	—
12	C₂×C₄⋊C₈	—	4	2	no	—
13	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₈	D₄	1	2	yes	—
16	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
17	C₄³	D₄	1	1	yes	characteristic
18	C₄×C₁₆	—	2	2	no	—
19	C₄×C₁₆	C₈	1	4	yes	complemented
20	C₂²×C₈	—	4	4	no	—
21	C₂²×C₈	C₄²	1	1	yes	characteristic
22	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₈	C₄⋊C₄	1	2	yes	—
24	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄²	D₈	1	1	yes	characteristic
27	C₂×C₄²	SD₁₆	1	2	yes	—
28	C₂×C₄²	Q₁₆	1	1	yes	characteristic
29	C₂×C₁₆	—	2	4	no	—
30	C₂×C₁₆	—	4	4	no	—
31	C₄⋊C₈	—	4	4	no	—
32	C₄⋊C₈	—	4	4	no	—
33	C₄⋊C₈	—	8	4	no	—
34	C₄⋊C₈	—	8	2	no	—
35	C₄⋊C₈	—	8	2	no	—
36	C₄×C₈	—	2	2	no	—
37	C₄×C₈	—	4	2	no	—
38	C₄×C₈	—	4	2	no	—
39	C₄×C₈	C₂×C₈	1	2	yes	—
40	C₄×C₈	M₁₆	1	2	yes	—
41	C₂²×C₄	#95	1	1	yes	characteristic
42	C₂²×C₄	#102	1	2	yes	—
43	C₂²×C₄	#103	1	2	yes	—
44	C₂²×C₄	#106	1	1	yes	characteristic
45	C₂²×C₄	#107	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	—	2	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	8	4	no	—
52	C₂×C₈	—	8	4	no	—
53	C₂×C₈	—	8	4	no	—
54	C₂×C₈	C₄×C₈	1	1	yes	characteristic
55	C₂×C₈	#97	1	1	yes	characteristic
56	C₂×C₈	C₂²⋊C₈	1	2	yes	—
57	C₄²	—	2	2	no	—
58	C₄²	—	2	2	no	—
59	C₄²	—	2	4	no	—
60	C₄²	—	2	2	no	—
61	C₄²	C₄⋊C₈	1	2	yes	—
62	C₄²	#108	1	2	yes	—
63	C₄²	D₁₆	1	1	yes	characteristic
64	C₄²	SD₃₂	1	2	yes	—
65	C₄²	Q₃₂	1	1	yes	characteristic
66	C₁₆	—	4	4	no	—
67	C₂³	#340	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	#334	1	2	yes	—
74	C₂×C₄	#335	1	2	yes	—
75	C₂×C₄	#336	1	1	yes	characteristic
76	C₂×C₄	H(3,ℤ/4)	1	2	yes	—
77	C₂×C₄	#341	1	1	yes	characteristic
78	C₂×C₄	#344	1	2	yes	—
79	C₂×C₄	#357	1	2	yes	—
80	C₂×C₄	#358	1	2	yes	—
81	C₂×C₄	#365	1	4	yes	—
82	C₂×C₄	#366	1	1	yes	characteristic
83	C₂×C₄	#367	1	1	yes	characteristic
84	C₂×C₄	#368	1	2	yes	—
85	C₈	—	2	1	no	—
86	C₈	—	2	1	no	—
87	C₈	—	2	2	no	—
88	C₈	—	4	2	no	—
89	C₈	—	16	4	no	—
90	C₂²	#1277	1	1	yes	characteristic
91	C₂²	#1286	1	2	yes	—
92	C₂²	#1380	1	1	yes	characteristic
93	C₂²	#1381	1	1	yes	characteristic
94	C₂²	#1383	1	2	yes	—
95	C₄	—	2	2	no	—
96	C₄	—	2	2	no	—
97	C₄	—	4	2	no	—
98	C₄	#1274	1	1	yes	characteristic
99	C₄	#1281	1	1	yes	characteristic
100	C₄	#1287	1	2	yes	—
101	C₄	#1368	1	4	yes	—
102	C₄	#1371	1	2	yes	—
103	C₄	#1372	1	2	yes	—
104	C₂	#7018	1	1	yes	characteristic
105	C₂	#7154	1	1	yes	characteristic
106	C₂	#7156	1	1	yes	characteristic
107	C₂	#7217	1	2	yes	—
108	C₂	#7219	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12590

C₂

yes

maximal, characteristic

#8784

C₂

yes

maximal

C₂×C₄×C₁₆

C₄

yes

—

#1842

—

C₄²×C₈

C₂²

yes

—

—

C₄

yes

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

D₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

Q₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₂²×C₄

#95

yes

characteristic

C₂²×C₄

#102

yes

—

C₂²×C₄

#103

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

#97

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

—

C₄²

#108

yes

—

C₄²

D₁₆

yes

characteristic

C₄²

SD₃₂

yes

—

C₄²

Q₃₂

yes

characteristic

C₁₆

—

C₂³

#340

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#334

yes

—

C₂×C₄

#335

yes

—

C₂×C₄

#336

yes

characteristic

C₂×C₄

H(3,ℤ/4)

yes

—

C₂×C₄

#341

yes

characteristic

C₂×C₄

#344

yes

—

C₂×C₄

#357

yes

—

C₂×C₄

#358

yes

—

C₂×C₄

#365

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

#1277

yes

characteristic

C₂²

#1286

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#1287

yes

—

101

C₄

#1368

yes

—

102

C₄

#1371

yes

—

103

C₄

#1372

yes

—

104

C₂

#7018

yes

characteristic

105

C₂

#7154

yes

characteristic

106

C₂

#7156

yes

characteristic

107

C₂

#7217

yes

—

108

C₂

#7219

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#340

commutator

C₂×C₈

C₄×C₈

Frattini, cosocle

C₄²×C₈

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
19. C₄×C₁₆	C₈	semidirect

H ≅ G/N

type

19. C₄×C₁₆

C₈

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#12590	2
#8784	2

subgroup

index

#12590

#8784

Subgroups — #93308

Conjugacy classes of subgroups

Complements

Maximal subgroups