Group #68922 — subgroups

Subgroups — #68922

Order 384.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	#5133	C₂	1	2	yes	maximal
2	#5110	C₂	1	1	yes	maximal, characteristic
3	C₂×#817	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
5	#4554	C₂	1	2	yes	maximal, complemented
6	#2079	—	3	1	no	maximal, Sylow
7	C₂×#235	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×#208	C₂²	1	1	yes	characteristic
12	#821	—	2	2	no	—
13	(C₂×D₃)⋊C₈	—	2	2	no	—
14	#817	—	2	2	no	—
15	#816	—	2	2	no	—
16	C₂×#106	—	3	1	no	—
17	#417	—	3	2	no	—
18	#408	—	3	1	no	—
19	#343	—	3	2	no	—
20	#341	D₃	1	1	yes	characteristic, complemented
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂×#67	—	2	1	no	—
23	C₂²×Dic₃	—	2	2	no	—
24	#235	—	2	2	no	—
25	#235	—	2	2	no	—
26	C₂×D₁₂	D₄	1	1	yes	characteristic
27	C₂×C₄×D₃	—	2	1	no	—
28	C₂×Dic₆	D₄	1	1	yes	characteristic
29	C₃×M₁₆	—	2	2	no	—
30	C₃×M₁₆	—	2	2	no	—
31	C₂×C₃×C₈	—	2	2	no	—
32	C₂×C₃×C₈	—	2	2	no	—
33	C₂×C₃×C₈	D₄	1	2	yes	—
34	C₄⋊Dic₃	—	2	2	no	—
35	C₄⋊Dic₃	—	2	2	no	—
36	#208	—	2	2	no	—
37	C₂×C₃⋊C₈	—	2	1	no	—
38	C₂×#41	—	3	1	no	—
39	C₂×M₁₆	—	3	1	no	—
40	C₂×M₁₆	C₂×D₃	1	2	yes	—
41	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄⋊C₄	—	3	2	no	—
43	#106	—	6	2	no	—
44	#103	—	6	2	no	—
45	#102	—	6	2	no	—
46	C₂²⋊C₈	—	6	2	no	—
47	C₂³×C₃	#41	1	1	yes	characteristic
48	C₂²×D₃	—	2	1	no	—
49	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄	#41	1	2	yes	—
52	C₂×C₃×C₄	#41	1	2	yes	—
53	#67	—	4	2	no	—
54	C₂×Dic₃	—	2	1	no	—
55	C₂×Dic₃	—	4	2	no	—
56	C₂×Dic₃	—	8	2	no	—
57	D₁₂	—	2	2	no	—
58	C₄×D₃	—	4	1	no	—
59	C₄×D₃	—	4	1	no	—
60	Dic₆	—	2	2	no	—
61	C₃×C₈	—	2	2	no	—
62	C₃×C₈	—	4	2	no	—
63	C₃⋊C₈	—	4	1	no	—
64	#41	—	6	2	no	—
65	#41	—	6	2	no	—
66	C₂×Q₈	—	3	1	no	—
67	C₂×D₄	—	3	1	no	—
68	C₂×D₄	—	6	1	no	—
69	C₂²×C₄	—	6	1	no	—
70	C₂²×C₄	—	6	2	no	—
71	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
72	M₁₆	—	2	2	no	—
73	M₁₆	—	2	2	no	—
74	M₁₆	—	6	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	6	1	no	—
78	C₂×C₈	D₁₂	1	2	yes	—
79	C₄⋊C₄	—	6	2	no	—
80	C₄⋊C₄	—	6	2	no	—
81	C₂²×C₃	—	2	2	no	—
82	C₂²×C₃	#121	1	1	yes	characteristic
83	C₂²×C₃	#126	1	2	yes	—
84	C₂×D₃	—	4	1	no	—
85	C₂×D₃	—	4	2	no	—
86	C₃×C₄	#123	1	2	yes	—
87	C₃×C₄	#123	1	1	yes	characteristic
88	C₃×C₄	#124	1	1	yes	characteristic
89	Dic₃	—	4	1	no	—
90	Dic₃	—	8	2	no	—
91	C₂³	—	6	1	no	—
92	C₂³	#239	1	1	yes	characteristic
93	Q₈	—	6	2	no	—
94	D₄	—	6	2	no	—
95	D₄	—	12	2	no	—
96	C₂×C₄	—	6	1	no	—
97	C₂×C₄	—	12	1	no	—
98	C₂×C₄	—	12	1	no	—
99	C₂×C₄	—	12	2	no	—
100	C₂×C₄	—	24	2	no	—
101	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
102	C₂×C₄	#235	1	2	yes	—
103	C₂×C₄	D₃×D₄	1	1	yes	characteristic
104	C₂×C₄	#237	1	2	yes	—
105	C₈	—	2	2	no	—
106	C₈	—	4	2	no	—
107	C₈	—	12	1	no	—
108	C₂×C₃	—	2	2	no	—
109	C₂×C₃	#397	1	1	yes	characteristic
110	C₂×C₃	#471	1	2	yes	—
111	D₃	—	8	1	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	12	2	no	—
114	C₂²	—	12	1	no	—
115	C₂²	#895	1	1	yes	characteristic
116	C₂²	#898	1	2	yes	—
117	C₄	—	12	1	no	—
118	C₄	—	24	2	no	—
119	C₄	#875	1	1	yes	characteristic
120	C₄	#883	1	2	yes	—
121	C₄	#886	1	1	yes	characteristic
122	C₃	#2079	1	1	yes	characteristic, complemented, Sylow
123	C₂	—	2	2	no	—
124	C₂	—	24	1	no	—
125	C₂	#4674	1	1	yes	characteristic
126	C₂	#4898	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#5133

C₂

yes

maximal

#5110

C₂

yes

maximal, characteristic

C₂×#817

C₂

yes

maximal, characteristic, complemented

C₃×#341

C₂

yes

maximal, characteristic, complemented

#4554

C₂

yes

maximal, complemented

#2079

—

maximal, Sylow

C₂×#235

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×#208

C₂²

yes

characteristic

#821

—

(C₂×D₃)⋊C₈

—

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×#67

—

C₂²×Dic₃

—

#235

—

#235

—

C₂×D₁₂

D₄

yes

characteristic

C₂×C₄×D₃

—

C₂×Dic₆

D₄

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

—

C₂²×C₈

C₂×D₃

yes

characteristic

C₂×C₄⋊C₄

—

#106

—

#103

—

#102

—

C₂²⋊C₈

—

C₂³×C₃

#41

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

#67

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²×C₃

—

C₂²×C₃

#121

yes

characteristic

C₂²×C₃

#126

yes

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

#123

yes

—

C₃×C₄

yes

C₃×C₄

yes

Dic₃

—

Dic₃

—

C₂³

—

C₂³

#239

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×D₁₂

yes

characteristic

102

C₂×C₄

#235

yes

—

103

C₂×C₄

D₃×D₄

yes

characteristic

104

C₂×C₄

#237

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂×C₃

—

109

C₂×C₃

#397

yes

characteristic

110

C₂×C₃

#471

yes

—

111

D₃

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

#895

yes

characteristic

116

C₂²

#898

yes

—

117

C₄

—

118

C₄

—

119

C₄

#875

yes

characteristic

120

C₄

#883

yes

—

121

C₄

#886

yes

characteristic

122

C₃

#2079

yes

characteristic, complemented, Sylow

123

C₂

—

124

C₂

—

125

C₂

#4674

yes

characteristic

126

C₂

#4898

yes

—

	subgroup	G/H
center	C₂²	#895
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#341	C₂
socle	C₂²×C₃	#121
prime radical	C₂×#235	C₂²
hypercenter	#341	D₃
hypocenter	C₃	#2079
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#895

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂²×C₃

C₂²

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
3. C₂×#817	C₂	semidirect
4. C₃×#341	C₂	semidirect
5. #4554	C₂	semidirect
20. #341	D₃	semidirect
122. C₃	#2079	semidirect

H ≅ G/N

type

3. C₂×#817

C₂

semidirect

4. C₃×#341

C₂

semidirect

5. #4554

C₂

semidirect

20. #341

D₃

semidirect

122. C₃

#2079

semidirect

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

subgroup	index
#5133	2
#5110	2
C₂×#817	2
C₃×#341	2
#4554	2
#2079	3

subgroup

index

Subgroups — #68922

Conjugacy classes of subgroups

Complements

Maximal subgroups