Group #68606 — subgroups

Subgroups — #68606

Order 384.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×Dic₃⋊C₈	C₂	1	2	yes	maximal
2	#4932	C₂	1	1	yes	maximal, characteristic
3	C₃×#336	C₂	1	1	yes	maximal, characteristic
4	#4550	C₂	1	2	yes	maximal
5	#4524	C₂	1	1	yes	maximal, characteristic
6	#1832	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
12	C₃⋊#95	C₄	1	2	yes	—
13	C₃⋊#95	C₄	1	2	yes	—
14	Dic₃⋊C₈	—	2	4	no	—
15	C₂×C₄⋊C₈	—	3	2	no	—
16	#376	—	3	1	no	—
17	#336	—	3	2	no	—
18	#336	—	3	1	no	—
19	#336	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₃	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₈	—	2	2	no	—
26	C₂×C₃×C₈	—	2	2	no	—
27	C₂×C₃×C₈	—	2	2	no	—
28	C₃×C₄²	—	2	2	no	—
29	C₄×Dic₃	—	2	2	no	—
30	C₄×Dic₃	D₄	1	2	yes	—
31	C₄×Dic₃	Q₈	1	2	yes	—
32	C₂×C₃⋊C₈	—	2	2	no	—
33	C₂×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂²×C₈	—	3	2	no	—
36	C₂²×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₄²	—	3	1	no	—
38	C₂×C₄²	—	3	1	no	—
39	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₄⋊C₈	—	6	4	no	—
41	#95	—	3	2	no	—
42	#95	—	3	2	no	—
43	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄	—	2	2	no	—
45	C₂×C₃×C₄	—	2	2	no	—
46	C₂×C₃×C₄	—	2	2	no	—
47	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₃×C₄	#41	1	2	yes	—
50	C₂×C₃×C₄	#41	1	2	yes	—
51	C₂×Dic₃	—	2	2	no	—
52	C₂×Dic₃	—	2	2	no	—
53	C₂×Dic₃	—	2	1	no	—
54	C₂×Dic₃	—	2	1	no	—
55	C₂×Dic₃	—	2	2	no	—
56	C₂×Dic₃	—	2	2	no	—
57	C₂×Dic₃	C₄⋊C₄	1	4	yes	—
58	C₃×C₈	—	4	2	no	—
59	C₃⋊C₈	—	4	2	no	—
60	C₂²×C₄	—	3	2	no	—
61	C₂²×C₄	—	3	1	no	—
62	C₂²×C₄	—	3	1	no	—
63	C₂²×C₄	C₄×D₃	1	2	yes	—
64	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	6	2	no	—
69	C₂×C₈	—	6	2	no	—
70	C₂×C₈	—	6	2	no	—
71	C₄²	—	2	2	no	—
72	C₄²	—	3	2	no	—
73	C₄²	—	3	2	no	—
74	C₄²	—	6	2	no	—
75	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₃	#117	1	1	yes	characteristic
77	C₂²×C₃	#117	1	1	yes	characteristic
78	C₂²×C₃	#131	1	2	yes	—
79	C₂²×C₃	#131	1	2	yes	—
80	C₃×C₄	—	4	2	no	—
81	C₃×C₄	#124	1	2	yes	—
82	C₃×C₄	#125	1	2	yes	—
83	Dic₃	—	2	4	no	—
84	Dic₃	—	4	1	no	—
85	Dic₃	—	4	1	no	—
86	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	3	4	no	—
91	C₂×C₄	—	6	2	no	—
92	C₂×C₄	—	6	2	no	—
93	C₂×C₄	—	6	1	no	—
94	C₂×C₄	—	6	1	no	—
95	C₂×C₄	—	6	2	no	—
96	C₂×C₄	—	6	2	no	—
97	C₂×C₄	#235	1	2	yes	—
98	C₂×C₄	D₃×D₄	1	1	yes	characteristic
99	C₂×C₄	#237	1	2	yes	—
100	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
101	C₈	—	4	2	no	—
102	C₈	—	12	2	no	—
103	C₂×C₃	#382	1	1	yes	characteristic
104	C₂×C₃	#424	1	2	yes	—
105	C₂×C₃	#433	1	1	yes	characteristic
106	C₂×C₃	#433	1	1	yes	characteristic
107	C₂×C₃	#433	1	2	yes	—
108	C₂²	#872	1	1	yes	characteristic
109	C₂²	#877	1	1	yes	characteristic
110	C₂²	D₃×C₄⋊C₄	1	1	yes	characteristic
111	C₂²	#901	1	2	yes	—
112	C₂²	#907	1	2	yes	—
113	C₄	—	4	2	no	—
114	C₄	—	6	4	no	—
115	C₄	—	12	1	no	—
116	C₄	—	12	1	no	—
117	C₄	#885	1	2	yes	—
118	C₄	#889	1	2	yes	—
119	C₃	#1832	1	1	yes	characteristic, complemented, Sylow
120	C₂	#4650	1	1	yes	characteristic
121	C₂	#4690	1	1	yes	characteristic
122	C₂	#4709	1	1	yes	characteristic
123	C₂	#4720	1	2	yes	—
124	C₂	#4839	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₃⋊C₈

C₂

yes

maximal

#4932

C₂

yes

maximal, characteristic

C₃×#336

C₂

yes

maximal, characteristic

#4550

C₂

yes

maximal

#4524

C₂

yes

maximal, characteristic

#1832

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

—

C₃⋊#95

C₄

yes

—

C₃⋊#95

C₄

yes

—

Dic₃⋊C₈

—

C₂×C₄⋊C₈

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

D₄

yes

—

C₄×Dic₃

Q₈

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

#95

—

#95

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₄⋊C₄

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂²×C₃

C₂×C₄⋊C₄

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₂²×C₃

#131

yes

—

C₃×C₄

—

C₃×C₄

#124

yes

—

C₃×C₄

#125

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

C₂³

C₂×C₄×D₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#235

yes

—

C₂×C₄

D₃×D₄

yes

characteristic

C₂×C₄

#237

yes

—

100

C₂×C₄

D₃×Q₈

yes

characteristic

101

C₈

—

102

C₈

—

103

C₂×C₃

#382

yes

characteristic

104

C₂×C₃

#424

yes

—

105

C₂×C₃

#433

yes

characteristic

106

C₂×C₃

#433

yes

characteristic

107

C₂×C₃

#433

yes

—

108

C₂²

#872

yes

characteristic

109

C₂²

#877

yes

characteristic

110

C₂²

D₃×C₄⋊C₄

yes

characteristic

111

C₂²

#901

yes

—

112

C₂²

#907

yes

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#885

yes

—

118

C₄

#889

yes

—

119

C₃

#1832

yes

characteristic, complemented, Sylow

120

C₂

#4650

yes

characteristic

121

C₂

#4690

yes

characteristic

122

C₂

#4709

yes

characteristic

123

C₂

#4720

yes

—

124

C₂

#4839

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator, socle, prime radical	C₂³×C₃	C₂²×C₄
Fitting	C₃×#336	C₂
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#336	D₃
hypocenter	C₃	#1832
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃

commutator, socle, prime radical

C₂³×C₃

C₂²×C₄

Fitting

C₃×#336

C₂

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
119. C₃	#1832	semidirect

H ≅ G/N

type

119. C₃

#1832

semidirect

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

subgroup	index
C₂×Dic₃⋊C₈	2
#4932	2
C₃×#336	2
#4550	2
#4524	2
#1832	3

subgroup

index

C₂×Dic₃⋊C₈

Subgroups — #68606

Conjugacy classes of subgroups

Complements

Maximal subgroups