Group #74343 — subgroups

Subgroups — #74343

Order 384.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×C₄×Dic₆	C₂	1	2	yes	maximal
2	C₃×#384	C₂	1	1	yes	maximal, characteristic
3	#4984	C₂	1	4	yes	maximal
4	#4981	C₂	1	4	yes	maximal
5	#4973	C₂	1	4	yes	maximal
6	#2582	—	3	1	no	maximal, Sylow
7	C₂²×Dic₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
13	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄×Dic₃	C₂²	1	4	yes	—
15	C₄×Dic₆	—	2	8	no	—
16	C₃⋊#95	C₂²	1	4	yes	—
17	C₃⋊#95	C₂²	1	8	yes	—
18	C₂×C₄×Q₈	—	3	2	no	—
19	#398	—	3	4	no	—
20	#391	—	3	4	no	—
21	#389	—	3	4	no	—
22	#384	D₃	1	1	yes	characteristic
23	C₂²×C₃×C₄	C₂³	1	4	yes	—
24	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂³	1	2	yes	—
26	C₂²×Dic₃	C₂³	1	4	yes	—
27	C₂²×Dic₃	C₂³	1	4	yes	—
28	C₂×Dic₆	—	2	4	no	—
29	C₂×Dic₆	—	2	4	no	—
30	C₂×Dic₆	Q₈	1	8	yes	complemented
31	C₃×C₄⋊C₄	—	2	8	no	—
32	C₃×C₄⋊C₄	—	4	2	no	—
33	C₃×C₄²	D₄	1	4	yes	—
34	C₄⋊Dic₃	—	2	4	no	—
35	C₄⋊Dic₃	—	4	4	no	—
36	Dic₃⋊C₄	—	2	8	no	—
37	C₄×Dic₃	—	2	8	no	—
38	C₂²×Q₈	—	3	2	no	—
39	C₄×Q₈	—	6	8	no	—
40	C₂×C₄⋊C₄	—	3	2	no	—
41	C₂×C₄⋊C₄	—	3	4	no	—
42	C₂×C₄⋊C₄	—	3	4	no	—
43	C₂×C₄⋊C₄	C₂×D₃	1	4	yes	—
44	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
45	C₂×C₄²	—	3	4	no	—
46	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
47	#95	—	3	8	no	—
48	#95	—	3	4	no	—
49	C₂³×C₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₃×C₄	—	2	1	no	—
51	C₂×C₃×C₄	—	2	1	no	—
52	C₂×C₃×C₄	—	2	8	no	—
53	C₂×C₃×C₄	—	2	4	no	—
54	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
55	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
56	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
57	C₂×C₃×C₄	#41	1	4	yes	—
58	C₂×Dic₃	—	2	4	no	—
59	C₂×Dic₃	—	2	4	no	—
60	C₂×Dic₃	—	2	4	no	—
61	C₂×Dic₃	—	2	4	no	—
62	C₂×Dic₃	—	2	4	no	—
63	C₂×Dic₃	C₂×Q₈	1	8	yes	—
64	Dic₆	—	2	16	no	—
65	C₂×Q₈	—	3	8	no	—
66	C₂×Q₈	—	6	4	no	—
67	C₂×Q₈	—	6	4	no	—
68	C₂²×C₄	—	3	4	no	—
69	C₂²×C₄	—	3	4	no	—
70	C₂²×C₄	C₂²×D₃	1	4	yes	—
71	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₃	1	2	yes	—
73	C₄⋊C₄	—	2	8	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	6	8	no	—
76	C₄⋊C₄	—	6	4	no	—
77	C₄⋊C₄	—	12	4	no	—
78	C₄²	—	6	8	no	—
79	C₄²	D₁₂	1	4	yes	—
80	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₃	C₂²×Q₈	1	2	yes	—
82	C₂²×C₃	C₂×#41	1	2	yes	—
83	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
85	C₃×C₄	—	2	2	no	—
86	C₃×C₄	—	4	4	no	—
87	C₃×C₄	C₂²⋊Q₈	1	8	yes	—
88	Dic₃	—	2	8	no	—
89	Dic₃	—	4	4	no	—
90	C₂³	C₂³×D₃	1	1	yes	characteristic
91	Q₈	—	6	16	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	3	8	no	—
97	C₂×C₄	—	6	4	no	—
98	C₂×C₄	—	6	4	no	—
99	C₂×C₄	—	6	4	no	—
100	C₂×C₄	—	6	4	no	—
101	C₂×C₄	—	6	4	no	—
102	C₂×C₄	C₂×D₁₂	1	2	yes	—
103	C₂×C₄	C₂×D₁₂	1	4	yes	—
104	C₂×C₄	#237	1	4	yes	—
105	C₂×C₄	D₃×Q₈	1	4	yes	—
106	C₂×C₃	C₂×C₂²⋊Q₈	1	2	yes	—
107	C₂×C₃	#537	1	1	yes	characteristic
108	C₂×C₃	#557	1	2	yes	—
109	C₂×C₃	Q₈²	1	1	yes	characteristic
110	C₂×C₃	#559	1	1	yes	characteristic
111	C₂²	C₂²×D₁₂	1	1	yes	characteristic
112	C₂²	C₂×#237	1	2	yes	—
113	C₂²	#1004	1	1	yes	characteristic
114	C₂²	C₂×D₃×Q₈	1	2	yes	—
115	C₂²	#1007	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	4	no	—
118	C₄	—	6	8	no	—
119	C₄	—	12	4	no	—
120	C₄	#897	1	8	yes	—
121	C₃	#2582	1	1	yes	characteristic, complemented, Sylow
122	C₂	C₂×#897	1	2	yes	—
123	C₂	#5510	1	1	yes	characteristic
124	C₂	#5713	1	1	yes	characteristic
125	C₂	#5721	1	2	yes	—
126	C₂	#5722	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₆

C₂

yes

maximal

C₃×#384

C₂

yes

maximal, characteristic

#4984

C₂

yes

maximal

#4981

C₂

yes

maximal

#4973

C₂

yes

maximal

#2582

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₄×Dic₆

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₂×C₄×Q₈

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

Q₈

yes

complemented

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

#95

—

#95

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×Q₈

yes

—

Dic₆

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

—

C₂²×C₃

C₂×#41

yes

—

C₂²×C₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₃

2₋¹⁺⁴

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²⋊Q₈

yes

—

Dic₃

—

Dic₃

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂×D₁₂

yes

—

103

C₂×C₄

C₂×D₁₂

yes

—

104

C₂×C₄

#237

yes

—

105

C₂×C₄

D₃×Q₈

yes

—

106

C₂×C₃

C₂×C₂²⋊Q₈

yes

—

107

C₂×C₃

#537

yes

characteristic

108

C₂×C₃

#557

yes

—

109

C₂×C₃

Q₈²

yes

characteristic

110

C₂×C₃

#559

yes

characteristic

111

C₂²

C₂²×D₁₂

yes

characteristic

112

C₂²

C₂×#237

yes

—

113

C₂²

#1004

yes

characteristic

114

C₂²

C₂×D₃×Q₈

yes

—

115

C₂²

#1007

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#897

yes

—

121

C₃

#2582

yes

characteristic, complemented, Sylow

122

C₂

C₂×#897

yes

—

123

C₂

#5510

yes

characteristic

124

C₂

#5713

yes

characteristic

125

C₂

#5721

yes

—

126

C₂

#5722

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#384	C₂
hypercenter	#384	D₃
hypocenter	C₃	#2582
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

C₂

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
30. C₂×Dic₆	Q₈	semidirect
121. C₃	#2582	semidirect

H ≅ G/N

type

30. C₂×Dic₆

Q₈

semidirect

121. C₃

#2582

semidirect

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

subgroup	index
C₂×C₄×Dic₆	2
C₃×#384	2
#4984	2
#4981	2
#4973	2
#2582	3

subgroup

index

C₂×C₄×Dic₆

Subgroups — #74343

Conjugacy classes of subgroups

Complements

Maximal subgroups