Group #7031 — subgroups

Subgroups — #7031

Order 256.

176 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 93 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	#1842	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈	—	2	1	no	—
5	C₂×C₄⋊C₈	—	2	1	no	—
6	C₂×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄⋊C₈	C₄	1	2	yes	—
8	C₂×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₈	—	2	1	no	—
10	C₂×C₄×C₈	—	2	1	no	—
11	C₄³	C₂²	1	1	yes	characteristic
12	#336	—	2	1	no	—
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	—	4	2	no	—
27	C₄⋊C₈	—	4	1	no	—
28	C₄⋊C₈	—	4	1	no	—
29	C₄⋊C₈	—	4	1	no	—
30	C₄⋊C₈	—	4	1	no	—
31	C₄⋊C₈	C₈	1	4	yes	complemented
32	C₄×C₈	—	4	1	no	—
33	C₄×C₈	—	4	1	no	—
34	C₄×C₈	—	4	1	no	—
35	C₄×C₈	—	4	1	no	—
36	C₂²×C₄	C₄²	1	1	yes	characteristic
37	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈	—	2	4	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	—	4	2	no	—
52	C₄²	—	2	1	no	—
53	C₄²	—	2	1	no	—
54	C₄²	—	2	2	no	—
55	C₄²	—	2	2	no	—
56	C₄²	—	4	1	no	—
57	C₄²	C₂×C₈	1	2	yes	—
58	C₄²	M₁₆	1	2	yes	—
59	C₄²	D₈	1	1	yes	characteristic
60	C₄²	D₈	1	1	yes	characteristic
61	C₄²	SD₁₆	1	1	yes	characteristic
62	C₄²	SD₁₆	1	1	yes	characteristic
63	C₄²	SD₁₆	1	1	yes	characteristic
64	C₄²	SD₁₆	1	1	yes	characteristic
65	C₄²	Q₁₆	1	1	yes	characteristic
66	C₄²	Q₁₆	1	1	yes	characteristic
67	C₂³	#95	1	1	yes	characteristic
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	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄	#97	1	1	yes	characteristic
79	C₂×C₄	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄	#100	1	1	yes	characteristic
81	C₂×C₄	#101	1	1	yes	characteristic
82	C₂×C₄	#102	1	1	yes	characteristic
83	C₂×C₄	#102	1	1	yes	characteristic
84	C₂×C₄	#102	1	1	yes	characteristic
85	C₂×C₄	#102	1	1	yes	characteristic
86	C₂×C₄	#103	1	1	yes	characteristic
87	C₂×C₄	#103	1	1	yes	characteristic
88	C₂×C₄	#103	1	1	yes	characteristic
89	C₂×C₄	#103	1	1	yes	characteristic
90	C₂×C₄	#104	1	2	yes	—
91	C₂×C₄	C₄⋊C₈	1	2	yes	—
92	C₂×C₄	#106	1	1	yes	characteristic
93	C₂×C₄	#106	1	1	yes	characteristic
94	C₂×C₄	#107	1	1	yes	characteristic
95	C₂×C₄	#107	1	1	yes	characteristic
96	C₂×C₄	#108	1	2	yes	—
97	C₈	—	4	4	no	—
98	C₈	—	4	4	no	—
99	C₈	—	8	2	no	—
100	C₂²	#336	1	1	yes	characteristic
101	C₂²	#339	1	1	yes	characteristic
102	C₂²	#340	1	1	yes	characteristic
103	C₂²	#340	1	1	yes	characteristic
104	C₂²	#341	1	1	yes	characteristic
105	C₂²	#343	1	1	yes	characteristic
106	C₂²	#344	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	4	1	no	—
114	C₄	D₄⋊C₈	1	2	yes	—
115	C₄	Q₈⋊C₈	1	2	yes	—
116	C₄	#331	1	1	yes	characteristic
117	C₄	#332	1	1	yes	characteristic
118	C₄	#332	1	1	yes	characteristic
119	C₄	#333	1	1	yes	characteristic
120	C₄	#334	1	2	yes	—
121	C₄	#335	1	2	yes	—
122	C₂	#1277	1	1	yes	characteristic
123	C₂	#1279	1	1	yes	characteristic
124	C₂	#1283	1	1	yes	characteristic
125	C₂	#1296	1	1	yes	characteristic
126	C₂	#1299	1	1	yes	characteristic
127	C₂	#1301	1	1	yes	characteristic
128	C₂	#1303	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1842

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂²

yes

characteristic

#336

—

#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

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₈

yes

complemented

C₄×C₈

—

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

D₈

yes

characteristic

C₄²

SD₁₆

yes

characteristic

C₄²

SD₁₆

yes

characteristic

C₄²

SD₁₆

yes

characteristic

C₄²

SD₁₆

yes

characteristic

C₄²

Q₁₆

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

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

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

100

C₂²

#336

yes

characteristic

101

C₂²

#339

yes

characteristic

102

C₂²

#340

yes

characteristic

103

C₂²

#340

yes

characteristic

104

C₂²

#341

yes

characteristic

105

C₂²

#343

yes

characteristic

106

C₂²

#344

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

D₄⋊C₈

yes

—

115

C₄

Q₈⋊C₈

yes

—

116

C₄

#331

yes

characteristic

117

C₄

#332

yes

characteristic

118

C₄

#332

yes

characteristic

119

C₄

#333

yes

characteristic

120

C₄

#334

yes

—

121

C₄

#335

yes

—

122

C₂

#1277

yes

characteristic

123

C₂

#1279

yes

characteristic

124

C₂

#1283

yes

characteristic

125

C₂

#1296

yes

characteristic

126

C₂

#1299

yes

characteristic

127

C₂

#1301

yes

characteristic

128

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
31. C₄⋊C₈	C₈	semidirect

H ≅ G/N

type

31. C₄⋊C₈

C₈

semidirect

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

subgroup	index
#1842	2
C₄×C₄⋊C₈	2

subgroup

index

#1842

C₄×C₄⋊C₈

Subgroups — #7031

Conjugacy classes of subgroups

Complements

Maximal subgroups