Group #10618 — subgroups

Subgroups — #10618

Order 256.

183 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 71 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	#1984	C₂	1	1	yes	maximal, characteristic
2	#1845	C₂	1	1	yes	maximal, characteristic
3	#1803	C₂	1	2	yes	maximal
4	#1296	C₂	1	2	yes	maximal
5	#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	C₂×#107	—	2	1	no	—
8	C₂×#107	—	2	1	no	—
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#103	—	2	2	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	C₂²	1	2	yes	—
14	#340	—	2	2	no	—
15	#336	—	2	1	no	—
16	#336	—	2	1	no	—
17	C₂²×Q₈	D₄	1	1	yes	characteristic
18	C₂²×C₈	—	2	1	no	—
19	C₂²×C₈	—	2	2	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₄×Q₈	—	2	2	no	—
23	C₄×Q₈	—	2	2	no	—
24	C₂×C₄⋊C₄	—	2	1	no	—
25	C₂×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
29	C₂×C₄²	—	2	1	no	—
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	#107	—	4	2	no	—
32	#107	—	4	2	no	—
33	C₄⋊C₈	—	2	2	no	—
34	C₄⋊C₈	—	2	2	no	—
35	C₄⋊C₈	—	2	2	no	—
36	#103	—	4	4	no	—
37	C₄×C₈	—	2	2	no	—
38	C₂×Q₈	—	2	1	no	—
39	C₂×Q₈	—	2	2	no	—
40	C₂×Q₈	—	2	2	no	—
41	C₂×Q₈	Q₁₆	1	2	yes	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	4	2	no	—
45	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄	#41	1	1	yes	characteristic
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	4	2	no	—
53	C₂×C₈	—	4	1	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	4	1	no	—
57	C₂×C₈	—	4	1	no	—
58	C₂×C₈	SD₁₆	1	2	yes	—
59	C₂×C₈	Q₁₆	1	2	yes	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	—	8	2	no	—
67	C₄²	—	4	1	no	—
68	C₄²	—	4	1	no	—
69	C₄²	#41	1	2	yes	—
70	C₄²	#41	1	2	yes	—
71	C₂³	#121	1	1	yes	characteristic
72	Q₈	—	2	4	no	—
73	Q₈	—	4	2	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	8	2	no	—
84	C₂×C₄	—	8	2	no	—
85	C₂×C₄	#123	1	2	yes	—
86	C₂×C₄	#123	1	1	yes	characteristic
87	C₂×C₄	#124	1	1	yes	characteristic
88	C₂×C₄	#126	1	2	yes	—
89	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄	#135	1	1	yes	characteristic
93	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
94	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄	#137	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₈	—	2	2	no	—
98	C₈	—	8	2	no	—
99	C₈	—	8	1	no	—
100	C₂²	#397	1	1	yes	characteristic
101	C₂²	#460	1	1	yes	characteristic
102	C₂²	#462	1	1	yes	characteristic
103	C₂²	#467	1	1	yes	characteristic
104	C₂²	#469	1	1	yes	characteristic
105	C₂²	#472	1	1	yes	characteristic
106	C₂²	#473	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	4	2	no	—
109	C₄	—	8	1	no	—
110	C₄	—	16	2	no	—
111	C₄	#482	1	2	yes	—
112	C₄	#483	1	1	yes	characteristic
113	C₄	#484	1	1	yes	characteristic
114	C₄	#484	1	2	yes	—
115	C₄	#487	1	1	yes	characteristic
116	C₄	#489	1	1	yes	characteristic
117	C₂	#1670	1	1	yes	characteristic
118	C₂	#1676	1	1	yes	characteristic
119	C₂	#1688	1	1	yes	characteristic
120	C₂	#1694	1	1	yes	characteristic
121	C₂	#2072	1	1	yes	characteristic
122	C₂	#2076	1	1	yes	characteristic
123	C₂	#2078	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1984

C₂

yes

maximal, characteristic

#1845

C₂

yes

maximal, characteristic

#1803

C₂

yes

maximal

#1296

C₂

yes

maximal

#1277

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂²

yes

characteristic

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

#340

—

#336

—

#336

—

C₂²×Q₈

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

#41

yes

—

C₄²

#41

yes

—

C₂³

#121

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#123

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

100

C₂²

#397

yes

characteristic

101

C₂²

#460

yes

characteristic

102

C₂²

#462

yes

characteristic

103

C₂²

#467

yes

characteristic

104

C₂²

#469

yes

characteristic

105

C₂²

#472

yes

characteristic

106

C₂²

#473

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#482

yes

—

112

C₄

#483

yes

characteristic

113

C₄

#484

yes

characteristic

114

C₄

#484

yes

—

115

C₄

#487

yes

characteristic

116

C₄

#489

yes

characteristic

117

C₂

#1670

yes

characteristic

118

C₂

#1676

yes

characteristic

119

C₂

#1688

yes

characteristic

120

C₂

#1694

yes

characteristic

121

C₂

#2072

yes

characteristic

122

C₂

#2076

yes

characteristic

123

C₂

#2078

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#121

commutator, Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#1984	2
#1845	2
#1803	2
#1296	2
#1277	2

subgroup

index

Subgroups — #10618

Conjugacy classes of subgroups

Maximal subgroups