Group #74335 — subgroups

Subgroups — #74335

Order 384.

556 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 197 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×#888	C₂	1	4	yes	maximal
2	C₃×#384	C₂	1	1	yes	maximal, characteristic
3	#4979	C₂	1	4	yes	maximal
4	#4972	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₃	C₂	1	2	yes	maximal
6	#2551	—	3	1	no	maximal, Sylow
7	C₂²×Dic₆	C₂²	1	4	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	2	yes	—
13	C₂×Dic₃⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₃	C₂²	1	4	yes	—
15	#888	—	2	16	no	—
16	C₃⋊#95	C₂²	1	4	yes	—
17	C₃⋊#95	C₂²	1	4	yes	—
18	C₂×C₄⋊Q₈	—	3	4	no	—
19	#395	—	3	4	no	—
20	#389	—	3	4	no	—
21	#384	D₃	1	1	yes	characteristic
22	C₄×C₄⋊C₄	—	3	2	no	—
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	16	no	—
29	C₂×Dic₆	—	4	4	no	—
30	C₃×C₄⋊C₄	—	2	8	no	—
31	C₃×C₄⋊C₄	—	4	2	no	—
32	C₃×C₄²	Q₈	1	4	yes	complemented
33	C₄⋊Dic₃	D₄	1	8	yes	—
34	C₄⋊Dic₃	Q₈	1	8	yes	complemented
35	Dic₃⋊C₄	—	2	16	no	—
36	C₄×Dic₃	—	2	8	no	—
37	C₂²×Q₈	—	3	4	no	—
38	C₄⋊Q₈	—	6	16	no	—
39	C₂×C₄⋊C₄	—	3	2	no	—
40	C₂×C₄⋊C₄	—	3	2	no	—
41	C₂×C₄⋊C₄	—	3	8	no	—
42	C₂×C₄⋊C₄	C₂×D₃	1	4	yes	—
43	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
44	C₂×C₄²	—	3	4	no	—
45	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
46	#95	—	3	4	no	—
47	#95	—	3	4	no	—
48	C₂³×C₃	C₂⁴	1	1	yes	characteristic
49	C₂×C₃×C₄	—	2	1	no	—
50	C₂×C₃×C₄	—	2	1	no	—
51	C₂×C₃×C₄	—	2	8	no	—
52	C₂×C₃×C₄	—	2	4	no	—
53	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
54	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
55	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
56	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
57	C₂×Dic₃	—	2	4	no	—
58	C₂×Dic₃	—	2	4	no	—
59	C₂×Dic₃	—	2	4	no	—
60	C₂×Dic₃	—	2	4	no	—
61	C₂×Dic₃	C₂×D₄	1	8	yes	—
62	C₂×Dic₃	C₂×Q₈	1	8	yes	—
63	Dic₆	—	4	16	no	—
64	C₂×Q₈	—	6	16	no	—
65	C₂×Q₈	—	12	4	no	—
66	C₂²×C₄	—	3	4	no	—
67	C₂²×C₄	—	3	4	no	—
68	C₂²×C₄	C₂²×D₃	1	4	yes	—
69	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₃	1	2	yes	—
71	C₄⋊C₄	—	2	8	no	—
72	C₄⋊C₄	—	3	8	no	—
73	C₄⋊C₄	—	3	8	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	6	16	no	—
76	C₄²	—	6	8	no	—
77	C₄²	Dic₆	1	4	yes	complemented
78	C₂²×C₃	C₂²×D₄	1	2	yes	—
79	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₃	C₂²×Q₈	1	2	yes	—
81	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
83	C₃×C₄	—	2	2	no	—
84	C₃×C₄	—	4	4	no	—
85	C₃×C₄	C₄⋊Q₈	1	8	yes	complemented
86	Dic₃	—	2	8	no	—
87	Dic₃	—	2	8	no	—
88	C₂³	C₂³×D₃	1	1	yes	characteristic
89	Q₈	—	12	16	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	3	8	no	—
95	C₂×C₄	—	3	8	no	—
96	C₂×C₄	—	6	4	no	—
97	C₂×C₄	—	6	4	no	—
98	C₂×C₄	—	6	4	no	—
99	C₂×C₄	—	6	4	no	—
100	C₂×C₄	C₂×Dic₆	1	2	yes	—
101	C₂×C₄	C₂×Dic₆	1	4	yes	—
102	C₂×C₄	D₃×D₄	1	4	yes	—
103	C₂×C₄	D₃×Q₈	1	4	yes	—
104	C₂×C₃	C₂×C₄⋊Q₈	1	2	yes	—
105	C₂×C₃	#544	1	1	yes	characteristic
106	C₂×C₃	D₄²	1	1	yes	characteristic
107	C₂×C₃	D₄×Q₈	1	2	yes	—
108	C₂×C₃	Q₈²	1	1	yes	characteristic
109	C₂²	C₂²×Dic₆	1	1	yes	characteristic
110	C₂²	C₂×D₃×D₄	1	2	yes	—
111	C₂²	#1004	1	1	yes	characteristic
112	C₂²	C₂×D₃×Q₈	1	2	yes	—
113	C₂²	#1007	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	6	8	no	—
117	C₄	—	6	8	no	—
118	C₄	#888	1	8	yes	complemented
119	C₃	#2551	1	1	yes	characteristic, complemented, Sylow
120	C₂	C₂×#888	1	2	yes	—
121	C₂	#5500	1	1	yes	characteristic
122	C₂	#5717	1	1	yes	characteristic
123	C₂	#5722	1	1	yes	characteristic
124	C₂	#5727	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#888

C₂

yes

maximal

C₃×#384

C₂

yes

maximal, characteristic

#4979

C₂

yes

maximal

#4972

C₂

yes

maximal

C₄×C₄⋊Dic₃

C₂

yes

maximal

#2551

—

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

—

#888

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₂×C₄⋊Q₈

—

#395

—

#389

—

#384

D₃

yes

characteristic

C₄×C₄⋊C₄

—

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

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

Q₈

yes

complemented

C₄⋊Dic₃

D₄

yes

—

C₄⋊Dic₃

Q₈

yes

complemented

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×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×D₄

yes

—

C₂×Dic₃

C₂×Q₈

yes

—

Dic₆

—

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

Dic₆

yes

complemented

C₂²×C₃

C₂²×D₄

yes

—

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

—

C₂²×C₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₃

2₋¹⁺⁴

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄⋊Q₈

yes

complemented

Dic₃

—

Dic₃

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×Dic₆

yes

—

101

C₂×C₄

C₂×Dic₆

yes

—

102

C₂×C₄

D₃×D₄

yes

—

103

C₂×C₄

D₃×Q₈

yes

—

104

C₂×C₃

C₂×C₄⋊Q₈

yes

—

105

C₂×C₃

#544

yes

characteristic

106

C₂×C₃

D₄²

yes

characteristic

107

C₂×C₃

D₄×Q₈

yes

—

108

C₂×C₃

Q₈²

yes

characteristic

109

C₂²

C₂²×Dic₆

yes

characteristic

110

C₂²

C₂×D₃×D₄

yes

—

111

C₂²

#1004

yes

characteristic

112

C₂²

C₂×D₃×Q₈

yes

—

113

C₂²

#1007

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#888

yes

complemented

119

C₃

#2551

yes

characteristic, complemented, Sylow

120

C₂

C₂×#888

yes

—

121

C₂

#5500

yes

characteristic

122

C₂

#5717

yes

characteristic

123

C₂

#5722

yes

characteristic

124

C₂

#5727

yes

—

	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	#2551
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
32. C₃×C₄²	Q₈	semidirect
34. C₄⋊Dic₃	Q₈	semidirect
77. C₄²	Dic₆	semidirect
85. C₃×C₄	C₄⋊Q₈	semidirect
118. C₄	#888	semidirect
119. C₃	#2551	semidirect

H ≅ G/N

type

32. C₃×C₄²

Q₈

semidirect

34. C₄⋊Dic₃

Q₈

semidirect

77. C₄²

Dic₆

semidirect

85. C₃×C₄

C₄⋊Q₈

semidirect

118. C₄

#888

semidirect

119. C₃

#2551

semidirect

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

subgroup	index
C₂×#888	2
C₃×#384	2
#4979	2
#4972	2
C₄×C₄⋊Dic₃	2
#2551	3

subgroup

index

C₄×C₄⋊Dic₃

Subgroups — #74335

Conjugacy classes of subgroups

Complements

Maximal subgroups