Group #72234412 — subgroups

Subgroups — #72234412

Order 4096.

1129 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1129 conjugacy classes of subgroups — too many to draw. Only its 651 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	#72234325	C₂	1	3	yes	maximal
2	#72234310	—	2	6	no	—
3	C₄×C₁₆²	C₂²	1	1	yes	characteristic
4	C₄×C₈×C₃₂	C₄	1	6	yes	—
5	C₄²×C₃₂	C₈	1	12	yes	—
6	C₂×C₈×C₃₂	—	2	6	no	—
7	C₂×C₈×C₃₂	—	2	6	no	—
8	C₄×C₈×C₁₆	C₂×C₄	1	3	yes	—
9	C₂×C₁₆²	D₄	1	3	yes	—
10	C₂×C₁₆²	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₃₂	—	2	12	no	—
12	C₂×C₄×C₃₂	—	2	12	no	—
13	C₂×C₄×C₃₂	C₁₆	1	24	yes	—
14	C₄²×C₁₆	C₂×C₈	1	6	yes	—
15	C₂×C₈×C₁₆	C₂²⋊C₄	1	3	yes	—
16	C₂×C₈×C₁₆	C₄⋊C₄	1	3	yes	—
17	C₂×C₈×C₁₆	M₁₆	1	6	yes	—
18	C₄×C₈²	C₄²	1	1	yes	characteristic
19	C₈×C₃₂	—	4	12	no	—
20	C₈×C₃₂	—	4	12	no	—
21	C₁₆²	—	4	3	no	—
22	C₁₆²	—	4	1	no	—
23	C₂²×C₃₂	—	2	24	no	—
24	C₂×C₄×C₁₆	C₂²⋊C₈	1	6	yes	—
25	C₂×C₄×C₁₆	C₄⋊C₈	1	6	yes	—
26	C₂×C₄×C₁₆	C₂×C₁₆	1	12	yes	—
27	C₂×C₄×C₁₆	M₃₂	1	12	yes	—
28	C₄²×C₈	C₄×C₈	1	3	yes	—
29	C₂×C₈²	#95	1	1	yes	characteristic
30	C₂×C₈²	#97	1	3	yes	—
31	C₄×C₃₂	—	2	24	no	—
32	C₄×C₃₂	—	4	12	no	—
33	C₄×C₃₂	—	4	12	no	—
34	C₄×C₃₂	—	4	24	no	—
35	C₄×C₃₂	C₃₂	1	48	yes	complemented
36	C₈×C₁₆	—	2	6	no	—
37	C₈×C₁₆	—	2	3	no	—
38	C₈×C₁₆	—	2	3	no	—
39	C₈×C₁₆	—	2	12	no	—
40	C₂²×C₁₆	C₂²⋊C₁₆	1	12	yes	—
41	C₂²×C₁₆	C₄⋊C₁₆	1	12	yes	—
42	C₂×C₄×C₈	#322	1	3	yes	—
43	C₂×C₄×C₈	#336	1	3	yes	—
44	C₂×C₄×C₈	C₄×C₁₆	1	6	yes	—
45	C₂×C₄×C₈	#346	1	6	yes	—
46	C₄³	C₈²	1	1	yes	characteristic
47	C₂×C₃₂	—	2	48	no	—
48	C₂×C₃₂	—	4	24	no	—
49	C₂×C₃₂	—	4	24	no	—
50	C₄×C₁₆	—	2	6	no	—
51	C₄×C₁₆	—	2	6	no	—
52	C₄×C₁₆	—	2	24	no	—
53	C₄×C₁₆	—	2	6	no	—
54	C₄×C₁₆	—	2	6	no	—
55	C₄×C₁₆	C₂×C₃₂	1	24	yes	—
56	C₄×C₁₆	M₆₄	1	24	yes	—
57	C₈²	H(3,ℤ/4)	1	3	yes	—
58	C₈²	#338	1	1	yes	characteristic
59	C₈²	#347	1	12	yes	—
60	C₂²×C₈	#1312	1	6	yes	—
61	C₂²×C₈	#1374	1	6	yes	—
62	C₂×C₄²	#1273	1	1	yes	characteristic
63	C₂×C₄²	C₈×C₁₆	1	3	yes	—
64	C₂×C₄²	#1311	1	3	yes	—
65	C₂×C₁₆	—	2	12	no	—
66	C₂×C₁₆	—	2	12	no	—
67	C₂×C₁₆	—	2	12	no	—
68	C₂×C₁₆	—	2	12	no	—
69	C₂×C₁₆	C₂²⋊C₃₂	1	24	yes	—
70	C₂×C₁₆	C₄⋊C₃₂	1	24	yes	—
71	C₄×C₈	#1274	1	3	yes	—
72	C₄×C₈	#1275	1	3	yes	—
73	C₄×C₈	#1313	1	12	yes	—
74	C₄×C₈	#1375	1	6	yes	—
75	C₄×C₈	C₄×C₃₂	1	12	yes	—
76	C₄×C₈	#1397	1	12	yes	—
77	C₄×C₈	#1398	1	24	yes	—
78	C₃₂	—	4	48	no	—
79	C₂²×C₄	C₁₆²	1	1	yes	characteristic
80	C₂²×C₄	#7052	1	3	yes	—
81	C₂²×C₄	#7136	1	3	yes	—
82	C₂×C₈	#7053	1	12	yes	—
83	C₂×C₈	#7142	1	6	yes	—
84	C₂×C₈	#7145	1	6	yes	—
85	C₂×C₈	#7331	1	12	yes	—
86	C₂×C₈	#7332	1	12	yes	—
87	C₂×C₈	#7465	1	12	yes	—
88	C₂×C₈	#7466	1	12	yes	—
89	C₄²	#7014	1	1	yes	characteristic
90	C₄²	#7137	1	3	yes	—
91	C₄²	C₈×C₃₂	1	6	yes	—
92	C₄²	#7329	1	6	yes	—
93	C₄²	#7330	1	12	yes	—
94	C₁₆	—	2	24	no	—
95	C₁₆	—	2	24	no	—
96	C₂³	#92895	1	1	yes	characteristic
97	C₂×C₄	#92896	1	3	yes	—
98	C₂×C₄	#93249	1	3	yes	—
99	C₂×C₄	C₁₆×C₃₂	1	3	yes	—
100	C₂×C₄	#93664	1	3	yes	—
101	C₂×C₄	#93665	1	6	yes	—
102	C₂×C₄	#93666	1	12	yes	—
103	C₂×C₄	#93989	1	6	yes	—
104	C₂×C₄	#93990	1	6	yes	—
105	C₈	#93667	1	24	yes	—
106	C₈	#93998	1	12	yes	—
107	C₈	#94005	1	12	yes	—
108	C₂²	#72210714	1	1	yes	characteristic
109	C₂²	#72234310	1	3	yes	—
110	C₂²	#72234311	1	3	yes	—
111	C₄	C₃₂²	1	1	yes	characteristic
112	C₄	#72225038	1	3	yes	—
113	C₄	#72225071	1	12	yes	—
114	C₄	#72234282	1	6	yes	—
115	C₄	#72234299	1	6	yes	—
116	C₂	#72234312	1	3	yes	—
117	C₂	#72234410	1	1	yes	characteristic
118	C₂	#72234411	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#72234325

C₂

yes

maximal

#72234310

—

C₄×C₁₆²

C₂²

yes

characteristic

C₄×C₈×C₃₂

C₄

yes

—

C₄²×C₃₂

C₈

yes

—

C₂×C₈×C₃₂

—

C₂×C₈×C₃₂

—

C₄×C₈×C₁₆

C₂×C₄

yes

—

C₂×C₁₆²

D₄

yes

—

C₂×C₁₆²

Q₈

yes

characteristic

C₂×C₄×C₃₂

—

C₂×C₄×C₃₂

—

C₂×C₄×C₃₂

C₁₆

yes

—

C₄²×C₁₆

C₂×C₈

yes

—

C₂×C₈×C₁₆

C₂²⋊C₄

yes

—

C₂×C₈×C₁₆

C₄⋊C₄

yes

—

C₂×C₈×C₁₆

M₁₆

yes

—

C₄×C₈²

C₄²

yes

characteristic

C₈×C₃₂

—

C₈×C₃₂

—

C₁₆²

—

C₁₆²

—

C₂²×C₃₂

—

C₂×C₄×C₁₆

C₂²⋊C₈

yes

—

C₂×C₄×C₁₆

C₄⋊C₈

yes

—

C₂×C₄×C₁₆

C₂×C₁₆

yes

—

C₂×C₄×C₁₆

M₃₂

yes

—

C₄²×C₈

C₄×C₈

yes

—

C₂×C₈²

#95

yes

characteristic

C₂×C₈²

#97

yes

—

C₄×C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

C₃₂

yes

complemented

C₈×C₁₆

—

C₈×C₁₆

—

C₈×C₁₆

—

C₈×C₁₆

—

C₂²×C₁₆

C₂²⋊C₁₆

yes

—

C₂²×C₁₆

C₄⋊C₁₆

yes

—

C₂×C₄×C₈

#322

yes

—

C₂×C₄×C₈

#336

yes

—

C₂×C₄×C₈

C₄×C₁₆

yes

—

C₂×C₄×C₈

#346

yes

—

C₄³

C₈²

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₃₂

yes

—

C₄×C₁₆

M₆₄

yes

—

C₈²

H(3,ℤ/4)

yes

—

C₈²

#338

yes

characteristic

C₈²

#347

yes

—

C₂²×C₈

#1312

yes

—

C₂²×C₈

#1374

yes

—

C₂×C₄²

#1273

yes

characteristic

C₂×C₄²

C₈×C₁₆

yes

—

C₂×C₄²

#1311

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₃₂

yes

—

C₂×C₁₆

C₄⋊C₃₂

yes

—

C₄×C₈

#1274

yes

—

C₄×C₈

#1275

yes

—

C₄×C₈

#1313

yes

—

C₄×C₈

#1375

yes

—

C₄×C₈

C₄×C₃₂

yes

—

C₄×C₈

#1397

yes

—

C₄×C₈

#1398

yes

—

C₃₂

—

C₂²×C₄

C₁₆²

yes

characteristic

C₂²×C₄

#7052

yes

—

C₂²×C₄

#7136

yes

—

C₂×C₈

#7053

yes

—

C₂×C₈

#7142

yes

—

C₂×C₈

#7145

yes

—

C₂×C₈

#7331

yes

—

C₂×C₈

#7332

yes

—

C₂×C₈

#7465

yes

—

C₂×C₈

#7466

yes

—

C₄²

#7014

yes

characteristic

C₄²

#7137

yes

—

C₄²

C₈×C₃₂

yes

—

C₄²

#7329

yes

—

C₄²

#7330

yes

—

C₁₆

—

C₁₆

—

C₂³

#92895

yes

characteristic

C₂×C₄

#92896

yes

—

C₂×C₄

#93249

yes

—

C₂×C₄

C₁₆×C₃₂

yes

—

100

C₂×C₄

#93664

yes

—

101

C₂×C₄

#93665

yes

—

102

C₂×C₄

#93666

yes

—

103

C₂×C₄

#93989

yes

—

104

C₂×C₄

#93990

yes

—

105

C₈

#93667

yes

—

106

C₈

#93998

yes

—

107

C₈

#94005

yes

—

108

C₂²

#72210714

yes

characteristic

109

C₂²

#72234310

yes

—

110

C₂²

#72234311

yes

—

111

C₄

C₃₂²

yes

characteristic

112

C₄

#72225038

yes

—

113

C₄

#72225071

yes

—

114

C₄

#72234282

yes

—

115

C₄

#72234299

yes

—

116

C₂

#72234312

yes

—

117

C₂

#72234410

yes

characteristic

118

C₂

#72234411

yes

—

	subgroup	G/H
center	C₄×C₈²	C₄²
commutator	C₄	C₃₂²
Frattini, cosocle	C₄×C₁₆²	C₂²
socle, prime radical	C₂³	#92895
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄×C₁₆² ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₄×C₈²

C₄²

commutator

C₄

C₃₂²

Frattini, cosocle

C₄×C₁₆²

C₂²

C₂³

G ▷ C₄ ▷ C₁

C₁ ◁ C₄×C₈² ◁ G

G ▷ C₄×C₁₆² ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
35. C₄×C₃₂	C₃₂	semidirect

H ≅ G/N

type

35. C₄×C₃₂

C₃₂

semidirect

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

subgroup	index
#72234325	2

subgroup

index

#72234325

Subgroups — #72234412

Conjugacy classes of subgroups

Complements

Maximal subgroups