Group #67745 — subgroups

Subgroups — #67745

Order 384.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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×#431	C₂	1	1	yes	maximal, characteristic
2	#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#1301	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
6	C₂×#208	—	2	1	no	—
7	C₂²×C₃⋊C₈	—	2	1	no	—
8	C₃×C₄⋊C₈	—	2	1	no	—
9	C₃×C₂²⋊C₈	C₄	1	2	yes	—
10	C₃×C₄×C₈	—	2	1	no	—
11	C₄⋊(C₃⋊C₈)	—	2	1	no	—
12	C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
14	C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
15	#431	D₃	1	1	yes	characteristic
16	#423	—	3	1	no	—
17	C₂×C₄⋊C₈	—	3	1	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
21	C₂×C₃×C₈	—	2	2	no	—
22	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄²	D₄	1	1	yes	characteristic
25	C₃×C₄²	D₄	1	1	yes	characteristic
26	#208	—	4	1	no	—
27	#208	—	4	1	no	—
28	C₂×C₃⋊C₈	—	2	2	no	—
29	C₂×C₃⋊C₈	—	2	2	no	—
30	C₂×C₃⋊C₈	—	4	1	no	—
31	C₂×C₃⋊C₈	—	4	1	no	—
32	C₂×M₁₆	—	6	1	no	—
33	C₂²×C₈	—	6	1	no	—
34	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	3	2	no	—
37	C₄⋊C₈	—	3	2	no	—
38	C₄⋊C₈	—	6	1	no	—
39	C₄⋊C₈	—	6	1	no	—
40	C₂²⋊C₈	Dic₃	1	2	yes	—
41	C₄×C₈	—	2	1	no	—
42	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	—	4	1	no	—
45	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃×C₄	D₈	1	1	yes	characteristic
52	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
53	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
54	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
55	C₃×C₈	—	4	2	no	—
56	C₃⋊C₈	—	4	2	no	—
57	C₃⋊C₈	—	4	2	no	—
58	C₂²×C₄	Dic₆	1	1	yes	characteristic
59	C₂²×C₄	D₁₂	1	1	yes	characteristic
60	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
61	M₁₆	—	12	1	no	—
62	M₁₆	—	12	1	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	6	2	no	—
65	C₂×C₈	—	6	2	no	—
66	C₂×C₈	—	12	1	no	—
67	C₂×C₈	—	12	1	no	—
68	C₄²	C₄×D₃	1	1	yes	characteristic
69	C₄²	C₄×D₃	1	1	yes	characteristic
70	C₄²	#67	1	1	yes	characteristic
71	C₄²	#67	1	1	yes	characteristic
72	C₂²×C₃	—	2	1	no	—
73	C₂²×C₃	—	2	1	no	—
74	C₂²×C₃	#95	1	1	yes	characteristic
75	C₂²×C₃	#106	1	1	yes	characteristic
76	C₂²×C₃	#107	1	1	yes	characteristic
77	C₃×C₄	—	2	1	no	—
78	C₃×C₄	—	2	1	no	—
79	C₃×C₄	—	2	1	no	—
80	C₃×C₄	#102	1	1	yes	characteristic
81	C₃×C₄	#102	1	1	yes	characteristic
82	C₃×C₄	#103	1	1	yes	characteristic
83	C₃×C₄	#103	1	1	yes	characteristic
84	C₃×C₄	#108	1	2	yes	—
85	C₂³	C₄⋊Dic₃	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	#204	1	1	yes	characteristic
89	C₂×C₄	#204	1	1	yes	characteristic
90	C₂×C₄	D₂₄	1	1	yes	characteristic
91	C₂×C₄	Dic₁₂	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
93	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
94	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
95	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
96	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#217	1	1	yes	characteristic
98	C₈	—	4	2	no	—
99	C₈	—	12	2	no	—
100	C₈	—	12	2	no	—
101	C₂×C₃	—	2	1	no	—
102	C₂×C₃	—	2	1	no	—
103	C₂×C₃	#340	1	1	yes	characteristic
104	C₂×C₃	#341	1	1	yes	characteristic
105	C₂×C₃	#344	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#817	1	1	yes	characteristic
109	C₂²	#818	1	1	yes	characteristic
110	C₂²	C₃⋊#95	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	#816	1	1	yes	characteristic
115	C₄	#816	1	1	yes	characteristic
116	C₄	#821	1	1	yes	characteristic
117	C₄	#821	1	1	yes	characteristic
118	C₄	#822	1	2	yes	—
119	C₃	#1301	1	1	yes	characteristic, complemented, Sylow
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#4533	1	1	yes	characteristic
123	C₂	#4551	1	1	yes	characteristic
124	C₂	#4558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#431

C₂

yes

maximal, characteristic

#4924

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#1301

—

maximal, Sylow

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

—

C₂²×C₃⋊C₈

—

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

C₄

yes

—

C₃×C₄×C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

C₄

yes

—

C₄⋊(C₃⋊C₈)

C₄

yes

—

#431

D₃

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

Q₈

yes

characteristic

C₂×C₃×C₈

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

Dic₃

yes

—

C₄×C₈

—

C₂³×C₃

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

D₈

yes

characteristic

C₂×C₃×C₄

SD₁₆

yes

characteristic

C₂×C₃×C₄

SD₁₆

yes

characteristic

C₂×C₃×C₄

Q₁₆

yes

characteristic

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

Dic₆

yes

characteristic

C₂²×C₄

D₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₃

yes

characteristic

C₄²

C₄×D₃

yes

C₄²

yes

C₄²

yes

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

C₂³

C₄⋊Dic₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

D₂₄

yes

characteristic

C₂×C₄

Dic₁₂

yes

characteristic

C₂×C₄

C₄×Dic₃

yes

characteristic

C₂×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₄

#217

yes

characteristic

C₈

—

C₈

—

100

C₈

—

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

#340

yes

characteristic

104

C₂×C₃

#341

yes

characteristic

105

C₂×C₃

#344

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

#817

yes

characteristic

109

C₂²

#818

yes

characteristic

110

C₂²

C₃⋊#95

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#816

yes

characteristic

115

C₄

#816

yes

characteristic

116

C₄

#821

yes

characteristic

117

C₄

#821

yes

characteristic

118

C₄

#822

yes

—

119

C₃

#1301

yes

characteristic, complemented, Sylow

120

C₂

—

121

C₂

—

122

C₂

#4533

yes

characteristic

123

C₂

#4551

yes

characteristic

124

C₂

#4558

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₄²
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₃×#431	C₂
socle	C₂²×C₃	#95
prime radical	C₂³×C₃	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	#431	D₃
hypocenter	C₃	#1301
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₃)⋊C₄

commutator

C₂×C₃×C₄

C₄²

Frattini

C₂×C₄²

C₂×D₃

C₂

C₂²×C₃

C₂³×C₃

C₄⋊C₄

cosocle

C₂×C₃×C₄²

C₂²

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
119. C₃	#1301	semidirect

H ≅ G/N

type

119. C₃

#1301

semidirect

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

subgroup	index
C₃×#431	2
#4924	2
C₂×C₄⋊(C₃⋊C₈)	2
#1301	3

subgroup

index

C₃×#431

#4924

C₂×C₄⋊(C₃⋊C₈)

#1301

Subgroups — #67745

Conjugacy classes of subgroups

Complements

Maximal subgroups