Group #94213 — subgroups

Subgroups — #94213

Order 512.

178 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 60 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	C₂×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#12837	C₂	1	2	yes	maximal
3	C₂³×C₁₆	C₂²	1	1	yes	characteristic
4	C₂²×C₃₂	C₄	1	2	yes	—
5	C₂×#368	—	2	2	no	—
6	#1810	—	2	2	no	—
7	C₂²⋊C₃₂	—	2	2	no	—
8	#1381	C₄	1	4	yes	—
9	C₂³×C₈	D₄	1	1	yes	characteristic
10	C₂²×C₁₆	—	4	2	no	—
11	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₁₆	C₂×C₄	1	2	yes	—
13	C₂²×C₁₆	D₄	1	2	yes	—
14	C₂²×C₁₆	Q₈	1	1	yes	characteristic
15	C₂×#108	—	2	4	no	—
16	#407	—	4	2	no	—
17	C₂×C₃₂	—	2	2	no	—
18	C₂×C₃₂	—	4	2	no	—
19	#368	—	4	4	no	—
20	#341	—	4	2	no	—
21	C₂³×C₄	D₈	1	1	yes	characteristic
22	C₂×M₁₆	—	4	4	no	—
23	C₂²×C₈	—	4	1	no	—
24	C₂²×C₈	—	4	1	no	—
25	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₈	SD₁₆	1	2	yes	—
28	C₂²×C₈	Q₁₆	1	1	yes	characteristic
29	C₂×C₁₆	—	2	4	no	—
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	—	4	2	no	—
32	C₂×C₁₆	—	4	1	no	—
33	C₂×C₁₆	C₄²	1	1	yes	characteristic
34	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
36	#108	—	4	4	no	—
37	#108	—	4	4	no	—
38	C₂²⋊C₈	—	8	2	no	—
39	C₂²⋊C₈	—	8	2	no	—
40	C₃₂	—	4	2	no	—
41	C₂⁴	Q₃₂	1	1	yes	characteristic
42	C₂²×C₄	—	4	1	no	—
43	C₂²×C₄	—	4	1	no	—
44	C₂²×C₄	#102	1	2	yes	—
45	C₂²×C₄	#107	1	1	yes	characteristic
46	C₂²×C₄	D₁₆	1	1	yes	characteristic
47	C₂²×C₄	SD₃₂	1	2	yes	—
48	M₁₆	—	8	4	no	—
49	M₁₆	—	8	4	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	—	4	1	no	—
56	C₂×C₈	—	8	4	no	—
57	C₂×C₈	#95	1	1	yes	characteristic
58	C₂×C₈	#103	1	2	yes	—
59	C₂×C₈	#106	1	1	yes	characteristic
60	C₁₆	—	2	1	no	—
61	C₁₆	—	2	2	no	—
62	C₁₆	#99	1	2	yes	—
63	C₂³	—	2	2	no	—
64	C₂³	—	4	1	no	—
65	C₂³	—	4	1	no	—
66	C₂³	#358	1	2	yes	—
67	C₂³	#366	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	#340	1	1	yes	characteristic
75	C₂×C₄	#357	1	2	yes	—
76	C₂×C₄	#367	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	2	1	no	—
79	C₈	—	16	4	no	—
80	C₈	H(3,ℤ/4)	1	1	yes	characteristic
81	C₈	#342	1	1	yes	characteristic
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	2	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	4	4	no	—
88	C₂²	#1380	1	1	yes	characteristic
89	C₂²	#1425	1	2	yes	—
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	#1289	1	1	yes	characteristic
93	C₄	#1384	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	4	2	no	—
96	C₂	#7230	1	1	yes	characteristic
97	C₂	#7472	1	1	yes	characteristic
98	C₂	#7476	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊C₃₂

C₂

yes

maximal, characteristic

#12837

C₂

yes

maximal

C₂³×C₁₆

C₂²

yes

characteristic

C₂²×C₃₂

C₄

yes

—

C₂×#368

—

#1810

—

C₂²⋊C₃₂

—

#1381

C₄

yes

—

C₂³×C₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

D₄

yes

—

C₂²×C₁₆

Q₈

yes

characteristic

C₂×#108

—

#407

—

C₂×C₃₂

—

C₂×C₃₂

—

#368

—

#341

—

C₂³×C₄

D₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

Q₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

—

#108

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₃₂

—

C₂⁴

Q₃₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#102

yes

—

C₂²×C₄

#107

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

SD₃₂

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#95

yes

characteristic

C₂×C₈

#103

yes

—

C₂×C₈

#106

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#99

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#358

yes

—

C₂³

#366

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#340

yes

characteristic

C₂×C₄

#357

yes

—

C₂×C₄

#367

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#1380

yes

characteristic

C₂²

#1425

yes

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂³×C₁₆	C₂²
prime radical	C₂⁴	Q₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1380

commutator

C₂×C₁₆

C₄²

Frattini, cosocle

C₂³×C₁₆

C₂²

prime radical

C₂⁴

Q₃₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G

LECS

G ▷ C₂³×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
C₂×C₂²⋊C₃₂	2
#12837	2

subgroup

index

C₂×C₂²⋊C₃₂

#12837

Subgroups — #94213

Conjugacy classes of subgroups

Maximal subgroups