Group #74221 — subgroups

Subgroups — #74221

Order 384.

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

normal subgroups only

This group has 504 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×#888	C₂	1	2	yes	maximal
2	C₃×#382	C₂	1	1	yes	maximal, characteristic
3	#4977	C₂	1	2	yes	maximal
4	#4936	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
6	#4646	C₂	1	8	yes	maximal
7	#2862	—	3	1	no	maximal, Sylow
8	C₂²×Dic₆	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×Dic₃⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄×Dic₃	C₂²	1	4	yes	—
14	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
16	#888	—	2	8	no	—
17	C₃×#95	C₂²	1	4	yes	—
18	C₃⋊#95	C₂²	1	4	yes	—
19	C₃⋊#95	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈	—	3	2	no	—
21	#391	—	3	8	no	—
22	#382	—	3	2	no	—
23	#382	—	3	1	no	—
24	#382	D₃	1	1	yes	characteristic
25	C₄³	—	3	1	no	—
26	C₂²×C₃×C₄	C₂³	1	4	yes	—
27	C₂²×C₃×C₄	C₂³	1	2	yes	—
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃	C₂³	1	4	yes	—
30	C₂²×Dic₃	C₂³	1	2	yes	—
31	C₂²×Dic₃	C₂³	1	2	yes	—
32	C₂×Dic₆	—	2	8	no	—
33	C₂×Dic₆	—	2	16	no	—
34	C₃×C₄⋊C₄	—	2	4	no	—
35	C₃×C₄²	Q₈	1	4	yes	complemented
36	C₄⋊Dic₃	—	2	4	no	—
37	Dic₃⋊C₄	—	2	8	no	—
38	C₄×Dic₃	—	2	8	no	—
39	C₄×Dic₃	D₄	1	4	yes	—
40	C₄×Dic₃	Q₈	1	4	yes	complemented
41	C₂²×Q₈	—	3	4	no	—
42	C₄⋊Q₈	—	6	8	no	—
43	C₂×C₄⋊C₄	—	3	2	no	—
44	C₂×C₄⋊C₄	—	3	4	no	—
45	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
46	C₂×C₄²	—	3	1	no	—
47	C₂×C₄²	—	3	1	no	—
48	C₂×C₄²	—	3	4	no	—
49	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
50	#95	—	3	8	no	—
51	#95	—	3	4	no	—
52	#95	C₂×D₃	1	4	yes	—
53	C₂³×C₃	C₂⁴	1	1	yes	characteristic
54	C₂×C₃×C₄	—	2	4	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	—	2	8	no	—
57	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
59	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
60	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
61	C₂×Dic₃	—	2	4	no	—
62	C₂×Dic₃	—	2	4	no	—
63	C₂×Dic₃	—	2	8	no	—
64	C₂×Dic₃	C₂×D₄	1	4	yes	—
65	C₂×Dic₃	C₂×Q₈	1	4	yes	—
66	C₂×Dic₃	#41	1	8	yes	—
67	Dic₆	—	4	16	no	—
68	C₂×Q₈	—	6	8	no	—
69	C₂×Q₈	—	6	16	no	—
70	C₂²×C₄	—	3	2	no	—
71	C₂²×C₄	—	3	2	no	—
72	C₂²×C₄	—	3	4	no	—
73	C₂²×C₄	C₂²×D₃	1	4	yes	—
74	C₂²×C₄	C₂²×D₃	1	2	yes	—
75	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	6	8	no	—
78	C₄⋊C₄	—	6	4	no	—
79	C₄²	—	3	4	no	—
80	C₄²	—	3	4	no	—
81	C₄²	—	6	8	no	—
82	C₄²	Dic₆	1	4	yes	complemented
83	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₃	C₂×#41	1	4	yes	—
87	C₃×C₄	—	2	4	no	—
88	C₃×C₄	—	4	4	no	—
89	C₃×C₄	C₄⋊Q₈	1	4	yes	—
90	Dic₃	—	2	4	no	—
91	Dic₃	—	4	4	no	—
92	Dic₃	#124	1	8	yes	—
93	C₂³	C₂³×D₃	1	1	yes	characteristic
94	Q₈	—	12	16	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	3	4	no	—
99	C₂×C₄	—	3	8	no	—
100	C₂×C₄	—	3	4	no	—
101	C₂×C₄	—	6	4	no	—
102	C₂×C₄	—	6	4	no	—
103	C₂×C₄	—	6	8	no	—
104	C₂×C₄	C₂×Dic₆	1	4	yes	—
105	C₂×C₄	C₂×Dic₆	1	2	yes	—
106	C₂×C₄	D₃×D₄	1	2	yes	—
107	C₂×C₄	D₃×Q₈	1	2	yes	—
108	C₂×C₃	C₂×#124	1	2	yes	—
109	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₃	#533	1	2	yes	—
111	C₂×C₃	#533	1	2	yes	—
112	C₂²	C₂²×Dic₆	1	1	yes	characteristic
113	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
114	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
115	C₂²	D₃×#41	1	4	yes	—
116	C₄	—	2	4	no	—
117	C₄	—	3	8	no	—
118	C₄	—	4	4	no	—
119	C₄	—	6	4	no	—
120	C₄	—	12	4	no	—
121	C₄	#888	1	4	yes	—
122	C₃	#2862	1	1	yes	characteristic, complemented, Sylow
123	C₂	C₂×#888	1	1	yes	characteristic
124	C₂	#5517	1	2	yes	—
125	C₂	D₃×#124	1	2	yes	—
126	C₂	#5688	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#888

C₂

yes

maximal

C₃×#382

C₂

yes

maximal, characteristic

#4977

C₂

yes

maximal

#4936

C₂

yes

maximal, characteristic

C₄²×Dic₃

C₂

yes

maximal, characteristic

#4646

C₂

yes

maximal

#2862

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

#888

—

C₃×#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₂×C₄⋊Q₈

—

—

—

—

D₃

yes

characteristic

C₄³

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄²

Q₈

yes

complemented

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

D₄

yes

—

C₄×Dic₃

Q₈

yes

complemented

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

—

—

C₂×D₃

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

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×D₄

yes

—

C₂×Dic₃

C₂×Q₈

yes

—

C₂×Dic₃

#41

yes

—

Dic₆

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₆

yes

complemented

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂×#41

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄⋊Q₈

yes

—

Dic₃

—

Dic₃

—

Dic₃

#124

yes

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×Dic₆

yes

—

105

C₂×C₄

C₂×Dic₆

yes

—

106

C₂×C₄

D₃×D₄

yes

—

107

C₂×C₄

D₃×Q₈

yes

—

108

C₂×C₃

C₂×#124

yes

—

109

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

110

C₂×C₃

#533

yes

—

111

C₂×C₃

#533

yes

—

112

C₂²

C₂²×Dic₆

yes

characteristic

113

C₂²

C₂×D₃×D₄

yes

characteristic

114

C₂²

C₂×D₃×Q₈

yes

characteristic

115

C₂²

D₃×#41

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#888

yes

—

122

C₃

#2862

yes

characteristic, complemented, Sylow

123

C₂

C₂×#888

yes

characteristic

124

C₂

#5517

yes

—

125

C₂

D₃×#124

yes

—

126

C₂

#5688

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#382	C₂
hypercenter	#382	D₃
hypocenter	C₃	#2862
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #382

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 ◁ #382

N	H ≅ G/N	type
35. C₃×C₄²	Q₈	semidirect
40. C₄×Dic₃	Q₈	semidirect
82. C₄²	Dic₆	semidirect
122. C₃	#2862	semidirect

H ≅ G/N

type

35. C₃×C₄²

Q₈

semidirect

40. C₄×Dic₃

Q₈

semidirect

82. C₄²

Dic₆

semidirect

122. C₃

#2862

semidirect

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

subgroup	index
C₂×#888	2
C₃×#382	2
#4977	2
#4936	2
C₄²×Dic₃	2
#4646	2
#2862	3

subgroup

index

C₄²×Dic₃

Subgroups — #74221

Conjugacy classes of subgroups

Complements

Maximal subgroups