Group #72004 — subgroups

Subgroups — #72004

Order 384.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#4549	C₂	1	4	yes	maximal, complemented
5	#2113	—	3	1	no	maximal, Sylow
6	C₂×C₃×#41	C₄	1	1	yes	characteristic
7	C₃×#131	—	2	2	no	—
8	C₃×#131	—	2	2	no	—
9	C₂×C₃×M₁₆	C₄	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₄	1	2	yes	—
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
13	#812	—	2	2	no	—
14	C₂×C₃⋊C₁₆	—	2	2	no	—
15	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
17	C₂×#131	D₃	1	1	yes	characteristic
18	C₂×M₃₂	—	3	1	no	—
19	C₂²×C₁₆	—	3	1	no	—
20	C₂²⋊C₁₆	—	3	4	no	—
21	C₃×#41	—	2	2	no	—
22	C₃×#41	—	2	2	no	—
23	C₂×C₃×Q₈	C₈	1	1	yes	characteristic
24	C₂×C₃×D₄	C₈	1	2	yes	—
25	C₂×C₃×D₄	C₈	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
28	C₃×M₁₆	—	2	2	no	—
29	C₃×M₁₆	—	2	4	no	—
30	C₂×C₃×C₈	—	2	2	no	—
31	C₂×C₃×C₈	—	2	2	no	—
32	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
37	C₃⋊C₁₆	—	2	2	no	—
38	C₃⋊C₁₆	—	2	2	no	—
39	C₂×#41	Dic₃	1	1	yes	characteristic
40	#131	—	2	2	no	—
41	#131	—	2	2	no	—
42	C₂×M₁₆	Dic₃	1	1	yes	characteristic
43	C₂×M₁₆	Dic₃	1	2	yes	—
44	C₂²×C₈	C₂×D₃	1	2	yes	—
45	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
46	M₃₂	—	6	2	no	—
47	C₂×C₁₆	—	3	2	no	—
48	C₂×C₁₆	—	3	2	no	—
49	C₂×C₁₆	—	6	2	no	—
50	C₂³×C₃	C₂×C₈	1	2	yes	—
51	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
52	C₃×Q₈	—	2	2	no	—
53	C₃×D₄	—	2	4	no	—
54	C₃×D₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
59	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃×C₄	M₁₆	1	2	yes	—
61	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₈	—	2	2	no	—
63	C₃×C₈	C₂²⋊C₄	1	2	yes	—
64	C₃×C₈	C₂×D₄	1	1	yes	characteristic
65	C₃×C₈	C₂×D₄	1	1	yes	characteristic
66	#41	—	2	2	no	—
67	#41	—	2	2	no	—
68	C₂×Q₈	C₃⋊C₈	1	1	yes	characteristic
69	C₂×D₄	C₃⋊C₈	1	2	yes	—
70	C₂×D₄	C₃⋊C₈	1	1	yes	characteristic
71	C₂²×C₄	C₂×Dic₃	1	2	yes	—
72	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
73	M₁₆	—	2	2	no	—
74	M₁₆	—	2	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	C₂×Dic₃	1	2	yes	—
78	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
79	C₂×C₈	#67	1	2	yes	—
80	C₂×C₈	#67	1	2	yes	—
81	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
82	C₁₆	—	6	2	no	—
83	C₁₆	—	6	2	no	—
84	C₂²×C₃	—	2	2	no	—
85	C₂²×C₃	—	2	2	no	—
86	C₂²×C₃	—	2	4	no	—
87	C₂²×C₃	C₂²⋊C₈	1	2	yes	—
88	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
89	C₃×C₄	—	2	2	no	—
90	C₃×C₄	C₂²⋊C₈	1	2	yes	—
91	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
93	C₂³	C₂×C₃⋊C₈	1	2	yes	—
94	C₂³	C₂×C₃⋊C₈	1	1	yes	characteristic
95	Q₈	—	2	2	no	—
96	D₄	—	2	4	no	—
97	D₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
101	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
102	C₂×C₄	#208	1	2	yes	—
103	C₂×C₄	#217	1	2	yes	—
104	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	#217	1	2	yes	—
107	C₈	C₂×#67	1	1	yes	characteristic
108	C₈	C₂×#67	1	1	yes	characteristic
109	C₂×C₃	—	2	2	no	—
110	C₂×C₃	—	4	2	no	—
111	C₂×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₃	#504	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#830	1	2	yes	—
117	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	#830	1	2	yes	—
120	C₄	C₂×#208	1	1	yes	characteristic
121	C₄	C₂×#217	1	1	yes	characteristic
122	C₃	#2113	1	1	yes	characteristic, complemented, Sylow
123	C₂	—	2	2	no	—
124	C₂	—	4	2	no	—
125	C₂	#5140	1	2	yes	—
126	C₂	C₂×#830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#131

C₂

yes

maximal, characteristic

C₂×#812

C₂

yes

maximal, characteristic, complemented

C₂²×C₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#4549

C₂

yes

maximal, complemented

#2113

—

maximal, Sylow

C₂×C₃×#41

C₄

yes

characteristic

C₃×#131

—

C₃×#131

—

C₂×C₃×M₁₆

C₄

yes

characteristic

C₂×C₃×M₁₆

C₄

yes

—

C₂²×C₃×C₈

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

characteristic

#812

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

C₂²

yes

—

C₂×C₃⋊C₁₆

C₂²

yes

—

C₂×#131

D₃

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

C₈

yes

characteristic

C₂×C₃×D₄

C₈

yes

—

C₂×C₃×D₄

C₈

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

—

C₂×C₃×C₈

D₄

yes

—

C₂×C₃×C₈

C₂³

yes

characteristic

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂×#41

Dic₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₃

yes

characteristic

C₂×M₁₆

Dic₃

yes

—

C₂²×C₈

C₂×D₃

yes

—

C₂²×C₈

C₂×D₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂³×C₃

C₂×C₈

yes

—

C₂³×C₃

C₂×C₈

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

C₂×C₈

yes

characteristic

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₃×C₈

—

C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₈

C₂×D₄

yes

characteristic

C₃×C₈

C₂×D₄

yes

characteristic

#41

—

#41

—

C₂×Q₈

C₃⋊C₈

yes

characteristic

C₂×D₄

C₃⋊C₈

yes

—

C₂×D₄

C₃⋊C₈

yes

characteristic

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂×Dic₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃

yes

—

C₂×C₈

C₂×Dic₃

yes

characteristic

C₂×C₈

#67

yes

—

C₂×C₈

#67

yes

—

C₂×C₈

C₂²×D₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²⋊C₈

yes

—

C₂²×C₃

C₂²×C₈

yes

characteristic

C₃×C₄

—

C₃×C₄

C₂²⋊C₈

yes

—

C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₃×C₄

C₂×M₁₆

yes

characteristic

C₂³

C₂×C₃⋊C₈

yes

—

C₂³

C₂×C₃⋊C₈

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×C₃⋊C₈

yes

—

101

C₂×C₄

C₂×C₃⋊C₈

yes

characteristic

102

C₂×C₄

#208

yes

—

103

C₂×C₄

#217

yes

—

104

C₂×C₄

C₂²×Dic₃

yes

characteristic

105

C₈

—

106

C₈

#217

yes

—

107

C₈

C₂×#67

yes

characteristic

108

C₈

C₂×#67

yes

characteristic

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

C₂×C₂²⋊C₈

yes

characteristic

112

C₂×C₃

#504

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#830

yes

—

117

C₂²

C₂²×C₃⋊C₈

yes

characteristic

118

C₄

—

119

C₄

#830

yes

—

120

C₄

C₂×#208

yes

characteristic

121

C₄

C₂×#217

yes

characteristic

122

C₃

#2113

yes

characteristic, complemented, Sylow

123

C₂

—

124

C₂

—

125

C₂

#5140

yes

—

126

C₂

C₂×#830

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₈
Fitting	C₂×C₃×#131	C₂
prime radical	C₂×C₃×#41	C₄
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×#131	D₃
hypocenter	C₃	#2113
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃

commutator, socle

C₂²×C₃

C₂²×C₈

Fitting

C₂×C₃×#131

C₂

prime radical

C₂×C₃×#41

C₄

cosocle

C₂×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂×C₈ ◁ C₂×#131

N	H ≅ G/N	type
2. C₂×#812	C₂	semidirect
3. C₂²×C₃⋊C₁₆	C₂	semidirect
4. #4549	C₂	semidirect
122. C₃	#2113	semidirect

H ≅ G/N

type

2. C₂×#812

C₂

semidirect

3. C₂²×C₃⋊C₁₆

C₂

semidirect

4. #4549

C₂

semidirect

122. C₃

#2113

semidirect

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

subgroup	index
C₂×C₃×#131	2
C₂×#812	2
C₂²×C₃⋊C₁₆	2
#4549	2
#2113	3

subgroup

index

C₂×C₃×#131

C₂×#812

C₂²×C₃⋊C₁₆

#4549

#2113

Subgroups — #72004

Conjugacy classes of subgroups

Complements

Maximal subgroups