Group #7027 — subgroups

Subgroups — #7027

Order 256.

169 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 75 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	#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈	—	2	1	no	—
5	C₂×C₄⋊C₈	C₄	1	2	yes	—
6	C₂×#97	—	2	1	no	—
7	C₂×C₄×C₈	—	2	1	no	—
8	C₂×C₄×C₈	—	2	1	no	—
9	C₄³	C₂²	1	1	yes	characteristic
10	#336	—	2	1	no	—
11	#336	—	2	1	no	—
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	—	2	2	no	—
26	C₄⋊C₈	C₈	1	4	yes	complemented
27	#97	—	4	1	no	—
28	#97	—	4	1	no	—
29	C₄×C₈	—	2	4	no	—
30	C₄×C₈	—	4	1	no	—
31	C₄×C₈	—	4	1	no	—
32	C₂²×C₄	C₄²	1	1	yes	characteristic
33	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈	—	2	4	no	—
40	C₂×C₈	—	2	4	no	—
41	C₂×C₈	—	2	4	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	—	4	2	no	—
44	C₂×C₈	—	4	2	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₄²	—	2	2	no	—
49	C₄²	—	2	2	no	—
50	C₄²	—	2	2	no	—
51	C₄²	—	4	1	no	—
52	C₄²	—	4	1	no	—
53	C₄²	C₂×C₈	1	2	yes	—
54	C₄²	M₁₆	1	2	yes	—
55	C₄²	D₈	1	1	yes	characteristic
56	C₄²	SD₁₆	1	1	yes	characteristic
57	C₄²	SD₁₆	1	1	yes	characteristic
58	C₄²	Q₁₆	1	1	yes	characteristic
59	C₂³	#95	1	1	yes	characteristic
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	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄	#97	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄	#102	1	1	yes	characteristic
76	C₂×C₄	#102	1	1	yes	characteristic
77	C₂×C₄	#103	1	1	yes	characteristic
78	C₂×C₄	#103	1	1	yes	characteristic
79	C₂×C₄	#104	1	2	yes	—
80	C₂×C₄	#104	1	2	yes	—
81	C₂×C₄	C₄⋊C₈	1	2	yes	—
82	C₂×C₄	#106	1	1	yes	characteristic
83	C₂×C₄	#107	1	1	yes	characteristic
84	C₂×C₄	#108	1	2	yes	—
85	C₈	—	4	4	no	—
86	C₈	—	4	4	no	—
87	C₈	—	4	4	no	—
88	C₂²	#336	1	1	yes	characteristic
89	C₂²	#339	1	1	yes	characteristic
90	C₂²	#339	1	1	yes	characteristic
91	C₂²	#340	1	1	yes	characteristic
92	C₂²	#341	1	1	yes	characteristic
93	C₂²	#344	1	1	yes	characteristic
94	C₂²	#344	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	2	1	no	—
97	C₄	—	2	2	no	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	2	2	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	2	no	—
104	C₄	D₄⋊C₈	1	2	yes	—
105	C₄	Q₈⋊C₈	1	2	yes	—
106	C₂	#1278	1	1	yes	characteristic
107	C₂	#1279	1	1	yes	characteristic
108	C₂	#1281	1	1	yes	characteristic
109	C₂	#1300	1	1	yes	characteristic
110	C₂	#1301	1	1	yes	characteristic
111	C₂	#1302	1	1	yes	characteristic
112	C₂	#1303	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1767

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂²

yes

—

—

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

C₈

yes

complemented

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

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

C₂×C₈

yes

—

C₄²

M₁₆

yes

—

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

C₄×C₈

yes

characteristic

C₂×C₄

#97

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#104

yes

—

C₂×C₄

C₄⋊C₈

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

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

D₄⋊C₈

yes

—

105

C₄

Q₈⋊C₈

yes

—

106

C₂

#1278

yes

characteristic

107

C₂

#1279

yes

characteristic

108

C₂

#1281

yes

characteristic

109

C₂

#1300

yes

characteristic

110

C₂

#1301

yes

characteristic

111

C₂

#1302

yes

characteristic

112

C₂

#1303

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈
Frattini, cosocle	C₄³	C₂²
socle, prime radical	C₂³	#95
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂²⋊C₄

commutator

C₂×C₄

C₄×C₈

Frattini, cosocle

C₄³

C₂²

C₂³

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
26. C₄⋊C₈	C₈	semidirect

H ≅ G/N

type

26. C₄⋊C₈

C₈

semidirect

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

subgroup	index
#1767	2
C₄×C₄⋊C₈	2
#1744	2

subgroup

index

#1767

C₄×C₄⋊C₈

#1744

Subgroups — #7027

Conjugacy classes of subgroups

Complements

Maximal subgroups