Group #69578 — subgroups

Subgroups — #69578

Order 384.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₃×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×#808	C₂	1	2	yes	maximal
3	C₂×#807	C₂	1	2	yes	maximal
4	#4936	C₂	1	1	yes	maximal, characteristic
5	C₂×#803	C₂	1	1	yes	maximal, characteristic
6	#1847	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₄	—	2	2	no	—
8	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃⋊#95	—	2	2	no	—
14	#808	—	2	4	no	—
15	#807	—	2	4	no	—
16	#803	C₄	1	4	yes	complemented
17	C₂×#107	—	3	2	no	—
18	C₂×#106	—	3	2	no	—
19	C₂×#97	—	3	1	no	—
20	#384	D₃	1	1	yes	characteristic
21	#382	—	3	1	no	—
22	C₂²×C₃×C₄	—	2	2	no	—
23	C₂²×C₃×C₄	D₄	1	2	yes	—
24	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃	—	2	1	no	—
26	C₂²×Dic₃	—	2	1	no	—
27	C₃×C₄⋊C₄	—	2	2	no	—
28	C₃×C₄⋊C₄	—	2	2	no	—
29	C₃×C₄⋊C₄	—	4	4	no	—
30	C₃×C₄²	C₂×C₄	1	2	yes	—
31	C₃×C₄²	Q₈	1	2	yes	—
32	C₄⋊Dic₃	—	2	2	no	—
33	C₄⋊Dic₃	—	2	2	no	—
34	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
35	C₂×C₃⋊C₈	D₄	1	4	yes	—
36	C₂×C₃⋊C₈	Q₈	1	4	yes	—
37	C₂²×C₈	—	3	2	no	—
38	C₂×C₄⋊C₄	—	2	2	no	—
39	C₂×C₄⋊C₄	—	3	1	no	—
40	C₂×C₄⋊C₄	—	3	1	no	—
41	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
42	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
43	#107	—	6	4	no	—
44	#106	—	6	4	no	—
45	#97	—	3	4	no	—
46	#95	—	6	2	no	—
47	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₄	—	2	2	no	—
49	C₂×C₃×C₄	—	2	2	no	—
50	C₂×C₃×C₄	—	4	2	no	—
51	C₂×C₃×C₄	—	4	4	no	—
52	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
53	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
55	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
57	C₂×Dic₃	—	4	1	no	—
58	C₂×Dic₃	—	4	1	no	—
59	C₂×Dic₃	—	4	2	no	—
60	C₂×Dic₃	—	4	2	no	—
61	C₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	—	6	1	no	—
64	C₂²×C₄	—	6	1	no	—
65	C₂²×C₄	#67	1	2	yes	—
66	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₈	—	3	4	no	—
68	C₂×C₈	—	3	4	no	—
69	C₂×C₈	—	3	4	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	4	4	no	—
73	C₄⋊C₄	—	6	2	no	—
74	C₄⋊C₄	—	6	2	no	—
75	C₄²	Dic₆	1	2	yes	—
76	C₄²	C₄×D₃	1	2	yes	—
77	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₃	#127	1	1	yes	characteristic
79	C₂²×C₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
81	C₂²×C₃	#137	1	2	yes	—
82	C₃×C₄	—	2	4	no	—
83	C₃×C₄	—	8	2	no	—
84	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
85	C₃×C₄	C₄⋊Q₈	1	2	yes	—
86	Dic₃	—	8	1	no	—
87	Dic₃	—	8	1	no	—
88	C₂³	C₂×#67	1	1	yes	characteristic
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	4	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	12	2	no	—
94	C₂×C₄	—	12	1	no	—
95	C₂×C₄	—	12	1	no	—
96	C₂×C₄	—	12	2	no	—
97	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
98	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
99	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
100	C₂×C₄	D₃×D₄	1	2	yes	—
101	C₂×C₄	D₃×Q₈	1	2	yes	—
102	C₈	—	3	8	no	—
103	C₂×C₃	#384	1	1	yes	characteristic
104	C₂×C₃	#428	1	2	yes	—
105	C₂×C₃	#496	1	1	yes	characteristic
106	C₂×C₃	#497	1	1	yes	characteristic
107	C₂×C₃	#501	1	2	yes	—
108	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
109	C₂²	#932	1	2	yes	—
110	C₂²	#940	1	1	yes	characteristic
111	C₂²	#942	1	2	yes	—
112	C₂²	#944	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	8	2	no	—
115	C₄	—	24	1	no	—
116	C₄	—	24	1	no	—
117	C₄	#888	1	2	yes	—
118	C₄	D₃×C₄⋊C₄	1	2	yes	—
119	C₃	#1847	1	1	yes	characteristic, complemented, Sylow
120	C₂	#4964	1	2	yes	—
121	C₂	#4975	1	1	yes	characteristic
122	C₂	#5074	1	1	yes	characteristic
123	C₂	#5081	1	2	yes	—
124	C₂	#5086	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#384

C₂

yes

maximal, characteristic

C₂×#808

C₂

yes

maximal

C₂×#807

C₂

yes

maximal

#4936

C₂

yes

maximal, characteristic

C₂×#803

C₂

yes

maximal, characteristic

#1847

—

maximal, Sylow

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

D₃

yes

characteristic

#382

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

Q₈

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×C₃⋊C₈

Q₈

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

—

—

—

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₆

yes

—

C₄²

C₄×D₃

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

#127

yes

characteristic

C₂²×C₃

C₄⋊Q₈

yes

characteristic

C₂²×C₃

AGL(1,ℤ/8)

yes

—

C₂²×C₃

#137

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×C₄⋊C₄

yes

—

C₃×C₄

C₄⋊Q₈

yes

—

Dic₃

—

Dic₃

—

C₂³

C₂×#67

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₃⋊C₄

yes

—

C₂×C₄

C₂×Dic₆

yes

characteristic

C₂×C₄

C₂×C₄×D₃

yes

characteristic

100

C₂×C₄

D₃×D₄

yes

—

101

C₂×C₄

D₃×Q₈

yes

—

102

C₈

—

103

C₂×C₃

#384

yes

characteristic

104

C₂×C₃

#428

yes

—

105

C₂×C₃

#496

yes

characteristic

106

C₂×C₃

#497

yes

characteristic

107

C₂×C₃

#501

yes

—

108

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

109

C₂²

#932

yes

—

110

C₂²

#940

yes

characteristic

111

C₂²

#942

yes

—

112

C₂²

#944

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#888

yes

—

118

C₄

D₃×C₄⋊C₄

yes

—

119

C₃

#1847

yes

characteristic, complemented, Sylow

120

C₂

#4964

yes

—

121

C₂

#4975

yes

characteristic

122

C₂

#5074

yes

characteristic

123

C₂

#5081

yes

—

124

C₂

#5086

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#384	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#384	D₃
hypocenter	C₃	#1847
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#384

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
16. #803	C₄	semidirect
61. C₃⋊C₈	C₄⋊C₄	semidirect
119. C₃	#1847	semidirect

H ≅ G/N

type

16. #803

C₄

semidirect

61. C₃⋊C₈

C₄⋊C₄

semidirect

119. C₃

#1847

semidirect

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

subgroup	index
C₃×#384	2
C₂×#808	2
C₂×#807	2
#4936	2
C₂×#803	2
#1847	3

subgroup

index

Subgroups — #69578

Conjugacy classes of subgroups

Complements

Maximal subgroups