Group #68597 — subgroups

Subgroups — #68597

Order 384.

272 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 125 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×#815	C₂	1	2	yes	maximal
2	C₄²×Dic₃	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	#1745	—	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	C₄	1	2	yes	—
10	C₂×C₄×Dic₃	C₄	1	2	yes	—
11	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
14	#815	C₄	1	8	yes	complemented
15	C₂×#97	—	3	2	no	—
16	C₄³	—	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	C₂×C₄	1	4	yes	—
28	C₃×C₄²	—	2	2	no	—
29	C₄×Dic₃	—	2	4	no	—
30	C₄×Dic₃	—	2	1	no	—
31	C₄×Dic₃	—	2	1	no	—
32	C₄×Dic₃	—	2	4	no	—
33	C₄×Dic₃	C₂×C₄	1	4	yes	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃⋊C₈	—	2	2	no	—
36	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₈	—	3	2	no	—
38	C₂²×C₈	C₂×D₃	1	2	yes	—
39	C₂×C₄²	—	3	1	no	—
40	C₂×C₄²	—	3	2	no	—
41	C₂×C₄²	—	3	2	no	—
42	C₂×C₄²	—	3	1	no	—
43	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
44	#97	—	3	8	no	—
45	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	—	2	2	no	—
47	C₂×C₃×C₄	—	2	2	no	—
48	C₂×C₃×C₄	M₁₆	1	4	yes	—
49	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
50	C₂×C₃×C₄	#41	1	2	yes	—
51	C₂×C₃×C₄	#41	1	1	yes	characteristic
52	C₂×C₃×C₄	#41	1	1	yes	characteristic
53	C₂×Dic₃	—	2	2	no	—
54	C₂×Dic₃	—	2	2	no	—
55	C₂×Dic₃	—	2	2	no	—
56	C₂×Dic₃	—	2	2	no	—
57	C₂×Dic₃	C₄²	1	2	yes	—
58	C₂×Dic₃	C₄²	1	2	yes	—
59	C₂×Dic₃	M₁₆	1	4	yes	—
60	C₃×C₈	—	2	4	no	—
61	C₃⋊C₈	—	2	4	no	—
62	C₂²×C₄	—	3	1	no	—
63	C₂²×C₄	—	3	1	no	—
64	C₂²×C₄	—	3	2	no	—
65	C₂²×C₄	C₄×D₃	1	2	yes	—
66	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	3	4	no	—
70	C₂×C₈	—	6	2	no	—
71	C₂×C₈	—	6	2	no	—
72	C₂×C₈	C₄×D₃	1	4	yes	—
73	C₄²	—	2	2	no	—
74	C₄²	—	3	4	no	—
75	C₄²	—	6	4	no	—
76	C₄²	—	6	4	no	—
77	C₄²	—	6	1	no	—
78	C₄²	—	6	1	no	—
79	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
80	C₂²×C₃	#117	1	1	yes	characteristic
81	C₂²×C₃	#117	1	1	yes	characteristic
82	C₂²×C₃	C₂×M₁₆	1	2	yes	—
83	C₂²×C₃	C₂×M₁₆	1	2	yes	—
84	C₃×C₄	—	2	4	no	—
85	C₃×C₄	#117	1	2	yes	—
86	C₃×C₄	#117	1	2	yes	—
87	Dic₃	—	2	2	no	—
88	Dic₃	—	2	2	no	—
89	Dic₃	—	2	4	no	—
90	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	3	2	no	—
94	C₂×C₄	—	3	2	no	—
95	C₂×C₄	—	3	4	no	—
96	C₂×C₄	—	6	2	no	—
97	C₂×C₄	—	6	2	no	—
98	C₂×C₄	—	6	2	no	—
99	C₂×C₄	—	6	2	no	—
100	C₂×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
101	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
102	C₂×C₄	#237	1	1	yes	characteristic
103	C₂×C₄	#237	1	2	yes	—
104	C₂×C₄	#239	1	1	yes	characteristic
105	C₈	—	2	4	no	—
106	C₈	—	6	4	no	—
107	C₂×C₃	#376	1	1	yes	characteristic
108	C₂×C₃	C₄×M₁₆	1	1	yes	characteristic
109	C₂×C₃	C₄×M₁₆	1	1	yes	characteristic
110	C₂×C₃	#432	1	2	yes	—
111	C₂×C₃	#432	1	2	yes	—
112	C₂²	C₄²×D₃	1	1	yes	characteristic
113	C₂²	#877	1	1	yes	characteristic
114	C₂²	#892	1	1	yes	characteristic
115	C₂²	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
116	C₂²	D₃×M₁₆	1	2	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	6	2	no	—
119	C₄	—	6	2	no	—
120	C₄	—	6	4	no	—
121	C₄	#877	1	2	yes	—
122	C₄	#892	1	2	yes	—
123	C₃	#1745	1	1	yes	characteristic, complemented, Sylow
124	C₂	#4648	1	1	yes	characteristic
125	C₂	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
126	C₂	#4707	1	1	yes	characteristic
127	C₂	#4719	1	2	yes	—
128	C₂	#4835	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#815

C₂

yes

maximal

C₄²×Dic₃

C₂

yes

maximal, characteristic

C₃×#336

C₂

yes

maximal, characteristic

#4550

C₂

yes

maximal

#4524

C₂

yes

maximal, characteristic

#1745

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₄

yes

—

C₂×C₄×Dic₃

C₄

yes

—

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

—

#815

C₄

yes

complemented

C₂×#97

—

C₄³

—

#336

—

#336

—

#336

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

C₂×C₄

yes

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

C₂×C₄

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

#97

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₄²

yes

—

C₂×Dic₃

C₄²

yes

—

C₂×Dic₃

M₁₆

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

C₄×D₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂²×C₃

C₂×C₄²

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

C₂×M₁₆

yes

—

C₂²×C₃

C₂×M₁₆

yes

—

C₃×C₄

—

C₃×C₄

#117

yes

—

C₃×C₄

#117

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

—

100

C₂×C₄

ΓL(1,𝔽₂₅)

yes

—

101

C₂×C₄

C₂×C₄×D₃

yes

—

102

C₂×C₄

#237

yes

characteristic

103

C₂×C₄

#237

yes

—

104

C₂×C₄

#239

yes

characteristic

105

C₈

—

106

C₈

—

107

C₂×C₃

#376

yes

characteristic

108

C₂×C₃

C₄×M₁₆

yes

characteristic

109

C₂×C₃

C₄×M₁₆

yes

characteristic

110

C₂×C₃

#432

yes

—

111

C₂×C₃

#432

yes

—

112

C₂²

C₄²×D₃

yes

characteristic

113

C₂²

#877

yes

characteristic

114

C₂²

#892

yes

characteristic

115

C₂²

C₂×ΓL(1,𝔽₂₅)

yes

—

116

C₂²

D₃×M₁₆

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#877

yes

—

122

C₄

#892

yes

—

123

C₃

#1745

yes

characteristic, complemented, Sylow

124

C₂

#4648

yes

characteristic

125

C₂

C₄×ΓL(1,𝔽₂₅)

yes

characteristic

126

C₂

#4707

yes

characteristic

127

C₂

#4719

yes

—

128

C₂

#4835

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₃×#336	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#336	D₃
hypocenter	C₃	#1745
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃

commutator

C₂²×C₃

C₂×C₄²

Fitting

C₃×#336

C₂

socle, prime radical

C₂³×C₃

C₂²×C₄

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
14. #815	C₄	semidirect
123. C₃	#1745	semidirect

H ≅ G/N

type

14. #815

C₄

semidirect

123. C₃

#1745

semidirect

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

subgroup	index
C₂×#815	2
C₄²×Dic₃	2
C₃×#336	2
#4550	2
#4524	2
#1745	3

subgroup

index

C₄²×Dic₃

Subgroups — #68597

Conjugacy classes of subgroups

Complements

Maximal subgroups