Group #94204 — subgroups

Subgroups — #94204

Order 512.

133 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 133 conjugacy classes of subgroups — too many to draw. Only its 62 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	#13639	C₂	1	1	yes	maximal, characteristic
2	#12891	C₂	1	1	yes	maximal, characteristic
3	#12023	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
5	#1848	—	2	1	no	—
6	#1567	—	2	1	no	—
7	C₄⋊C₃₂	—	2	1	no	—
8	C₂²⋊C₃₂	—	2	1	no	—
9	C₄×C₃₂	C₄	1	2	yes	—
10	#1381	—	2	1	no	—
11	#1381	C₄	1	2	yes	—
12	#1372	—	2	1	no	—
13	#1372	C₄	1	2	yes	—
14	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₁₆	D₄	1	1	yes	characteristic
16	C₂×#108	—	2	2	no	—
17	C₂×#108	—	4	1	no	—
18	#423	—	4	1	no	—
19	#423	—	4	1	no	—
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₃₂	—	2	2	no	—
22	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₄×C₁₆	D₄	1	1	yes	characteristic
25	C₄×C₁₆	Q₈	1	1	yes	characteristic
26	#335	—	2	2	no	—
27	#335	—	4	1	no	—
28	C₂×M₁₆	—	4	2	no	—
29	C₂×M₁₆	—	8	1	no	—
30	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₈	SD₁₆	1	1	yes	characteristic
32	C₂×C₄²	D₈	1	1	yes	characteristic
33	C₂×C₁₆	—	4	1	no	—
34	C₂×C₁₆	—	4	1	no	—
35	C₂×C₁₆	C₄²	1	1	yes	characteristic
36	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
39	#108	—	4	2	no	—
40	#108	—	4	2	no	—
41	#108	—	8	2	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄⋊C₈	—	8	1	no	—
44	C₄⋊C₈	—	8	1	no	—
45	C₄⋊C₈	—	8	1	no	—
46	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₈	SD₁₆	1	1	yes	characteristic
49	C₄×C₈	Q₁₆	1	1	yes	characteristic
50	C₃₂	—	4	2	no	—
51	C₂²×C₄	#102	1	1	yes	characteristic
52	C₂²×C₄	SD₃₂	1	1	yes	characteristic
53	C₂²×C₄	Q₃₂	1	1	yes	characteristic
54	M₁₆	—	8	2	no	—
55	M₁₆	—	8	2	no	—
56	M₁₆	—	16	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	8	2	no	—
61	C₂×C₈	—	8	2	no	—
62	C₂×C₈	#95	1	1	yes	characteristic
63	C₂×C₈	#103	1	1	yes	characteristic
64	C₂×C₈	#103	1	1	yes	characteristic
65	C₂×C₈	#106	1	1	yes	characteristic
66	C₄²	#102	1	1	yes	characteristic
67	C₄²	#107	1	1	yes	characteristic
68	C₄²	D₁₆	1	1	yes	characteristic
69	C₄²	SD₃₂	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	—	2	1	no	—
72	C₁₆	—	2	1	no	—
73	C₁₆	—	2	1	no	—
74	C₂³	#358	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	#340	1	1	yes	characteristic
80	C₂×C₄	#357	1	1	yes	characteristic
81	C₂×C₄	#357	1	1	yes	characteristic
82	C₂×C₄	#358	1	1	yes	characteristic
83	C₂×C₄	#366	1	1	yes	characteristic
84	C₂×C₄	#367	1	1	yes	characteristic
85	C₈	—	2	1	no	—
86	C₈	—	2	1	no	—
87	C₈	—	16	2	no	—
88	C₈	—	16	2	no	—
89	C₈	H(3,ℤ/4)	1	1	yes	characteristic
90	C₈	#344	1	1	yes	characteristic
91	C₈	#365	1	2	yes	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	#1380	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	—	2	1	no	—
99	C₄	#1286	1	1	yes	characteristic
100	C₄	#1383	1	1	yes	characteristic
101	C₄	#1425	1	2	yes	—
102	C₄	#1426	1	2	yes	—
103	C₂	—	4	1	no	—
104	C₂	#7217	1	1	yes	characteristic
105	C₂	#7472	1	1	yes	characteristic
106	C₂	#7474	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13639

C₂

yes

maximal, characteristic

#12891

C₂

yes

maximal, characteristic

#12023

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

—

#1567

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₄×C₃₂

C₄

yes

—

#1381

—

#1381

C₄

yes

—

#1372

—

#1372

C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₃₂

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

Q₈

yes

characteristic

#335

—

#335

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₃₂

—

C₂²×C₄

#102

yes

characteristic

C₂²×C₄

SD₃₂

yes

characteristic

C₂²×C₄

Q₃₂

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₄²

yes

C₄²

D₁₆

yes

characteristic

C₄²

SD₃₂

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#358

yes

characteristic

C₂×C₄

—

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

yes

C₈

yes

C₈

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#1380

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#1286

yes

characteristic

100

C₄

#1383

yes

characteristic

101

C₄

#1425

yes

—

102

C₄

#1426

yes

—

103

C₂

—

104

C₂

#7217

yes

characteristic

105

C₂

#7472

yes

characteristic

106

C₂

#7474

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
prime radical	C₂³	#358
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1380

commutator

C₂×C₁₆

C₄²

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

prime radical

C₂³

#358

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13639	2
#12891	2
#12023	2

subgroup

index

#13639

#12891

#12023

Subgroups — #94204

Conjugacy classes of subgroups

Maximal subgroups