Group #7041 — subgroups

Subgroups — #7041

Order 256.

139 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 139 conjugacy classes of subgroups — too many to draw. Only its 55 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	#1839	C₂	1	1	yes	maximal, characteristic
2	#1831	C₂	1	1	yes	maximal, characteristic
3	#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈	—	2	1	no	—
5	C₂×C₄⋊C₈	—	2	1	no	—
6	C₂×#97	—	2	1	no	—
7	C₄³	C₂²	1	1	yes	characteristic
8	#336	—	2	1	no	—
9	#336	—	2	1	no	—
10	#336	—	2	1	no	—
11	#336	C₄	1	2	yes	—
12	#336	C₄	1	2	yes	—
13	#336	C₄	1	2	yes	—
14	C₂²×C₈	—	2	2	no	—
15	C₂²×C₈	—	2	2	no	—
16	C₂²×C₈	—	2	2	no	—
17	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²	D₄	1	1	yes	characteristic
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	C₂×C₄²	D₄	1	1	yes	characteristic
23	C₂×C₄²	Q₈	1	1	yes	characteristic
24	C₄⋊C₈	—	2	4	no	—
25	C₄⋊C₈	—	4	1	no	—
26	C₄⋊C₈	—	4	1	no	—
27	#97	—	4	1	no	—
28	#97	—	4	1	no	—
29	C₂²×C₄	C₄²	1	1	yes	characteristic
30	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈	—	2	4	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	—	4	2	no	—
39	C₂×C₈	—	4	2	no	—
40	C₂×C₈	—	4	2	no	—
41	C₂×C₈	—	4	2	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	—	4	2	no	—
44	C₂×C₈	—	4	2	no	—
45	C₄²	—	2	2	no	—
46	C₄²	—	2	2	no	—
47	C₄²	—	4	1	no	—
48	C₄²	—	4	1	no	—
49	C₄²	—	4	1	no	—
50	C₄²	—	4	1	no	—
51	C₄²	D₈	1	1	yes	characteristic
52	C₄²	SD₁₆	1	1	yes	characteristic
53	C₄²	SD₁₆	1	1	yes	characteristic
54	C₄²	Q₁₆	1	1	yes	characteristic
55	C₂³	#95	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	#102	1	1	yes	characteristic
73	C₂×C₄	#102	1	1	yes	characteristic
74	C₂×C₄	#103	1	1	yes	characteristic
75	C₂×C₄	#103	1	1	yes	characteristic
76	C₂×C₄	#104	1	2	yes	—
77	C₂×C₄	#106	1	1	yes	characteristic
78	C₂×C₄	#107	1	1	yes	characteristic
79	C₂×C₄	#108	1	2	yes	—
80	C₈	—	4	4	no	—
81	C₈	—	8	2	no	—
82	C₈	—	8	2	no	—
83	C₂²	H(3,ℤ/4)	1	1	yes	characteristic
84	C₂²	#339	1	1	yes	characteristic
85	C₂²	#340	1	1	yes	characteristic
86	C₂²	#341	1	1	yes	characteristic
87	C₂²	#344	1	1	yes	characteristic
88	C₂²	#344	1	1	yes	characteristic
89	C₂²	#344	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	2	1	no	—
92	C₄	—	2	2	no	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	—	4	1	no	—
99	C₄	—	4	1	no	—
100	C₂	#1286	1	1	yes	characteristic
101	C₂	#1287	1	1	yes	characteristic
102	C₂	#1288	1	1	yes	characteristic
103	C₂	#1301	1	1	yes	characteristic
104	C₂	#1302	1	1	yes	characteristic
105	C₂	#1303	1	1	yes	characteristic
106	C₂	#1308	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1839

C₂

yes

maximal, characteristic

#1831

C₂

yes

maximal, characteristic

#1767

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

C₂²

yes

—

—

—

C₄

yes

—

#336

C₄

yes

—

#336

C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

Q₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

C₂²×C₄

C₄²

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₈

yes

characteristic

C₄²

SD₁₆

yes

characteristic

C₄²

SD₁₆

yes

characteristic

C₄²

Q₁₆

yes

characteristic

C₂³

#95

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

100

C₂

#1286

yes

characteristic

101

C₂

#1287

yes

characteristic

102

C₂

#1288

yes

characteristic

103

C₂

#1301

yes

characteristic

104

C₂

#1302

yes

characteristic

105

C₂

#1303

yes

characteristic

106

C₂

#1308

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#95
commutator	C₂²×C₄	C₄²
Frattini, cosocle	C₄³	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#95

commutator

C₂²×C₄

C₄²

Frattini, cosocle

C₄³

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

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

subgroup	index
#1839	2
#1831	2
#1767	2

subgroup

index

#1839

#1831

#1767

Subgroups — #7041

Conjugacy classes of subgroups

Maximal subgroups