Finite Group Databasebeta

← C2×D3×#67 #64328

Subgroups — C2×D3×#67 #64328

Order 288.

499 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D3C211yesmaximal, characteristic, complemented
2C22×D32C211yesmaximal, characteristic, complemented
3C2×#638C211yesmaximal, characteristic, complemented
4C2×C3×#67C211yesmaximal, characteristic, complemented
5D3×#67C218yesmaximal, direct product factor
6C2×#626C211yesmaximal, characteristic, complemented
7C2×#625C211yesmaximal, characteristic, complemented
8C2×D3×Dic3C211yesmaximal, characteristic, complemented
9C22×#67—31nomaximal
10C2×D3×D4—31nomaximal
11C23×C32C2211yescharacteristic, complemented
12C22×C3⋊D3C2211yescharacteristic, complemented
13C22×C3×D3—24no—
14C22×C3×D3C2211yescharacteristic
15C22×C3×D3C2214yescomplemented
16C22×C3×D3C2211yescharacteristic, complemented
17C22×C3×D3C2211yescharacteristic, complemented
18C2×D32—24no—
19C2×D32C2214yescomplemented
20#638C2214yescomplemented
21C2×C3⋊Dic3C2211yescharacteristic, complemented
22C3×#67C2214yescomplemented
23C2×C3×Dic3C2211yescharacteristic, complemented
24#626C2214yescomplemented
25#625C2214yescomplemented
26D3×Dic3C2214yescomplemented
27C24×C3—31no—
28C23×D3—31no—
29C23×D3—31no—
30C23×D3D311yescharacteristic, complemented
31C2×C3×D4—31no—
32C2×#67—31no—
33C2×#67—31no—
34C2×#67—31no—
35C2×#67—31no—
36C2×#67—38no—
37C2×#67—31no—
38C2×#67—61no—
39C2×#67D311yescharacteristic, direct product factor
40C22×Dic3—31no—
41D3×D4—38no—
42C2×D12—31no—
43C2×C4×D3—31no—
44C22×C32—22no—
45C22×C32C2311yescharacteristic
46C22×C32C2312yescomplemented
47C2×C3⋊D3—22no—
48C2×C3⋊D3C2312yescomplemented
49C2×C3×D3—22no—
50C2×C3×D3—22no—
51C2×C3×D3—28no—
52C2×C3×D3D414yescomplemented
53C2×C3×D3C2312yes—
54C2×C3×D3C2312yescomplemented
55C2×C3×D3C2312yescomplemented
56D32—28no—
57C3⋊Dic3C2312yescomplemented
58C3×Dic3C2312yescomplemented
59C22×D4—91noSylow
60C23×C3—21no—
61C23×C3—31no—
62C23×C3—31no—
63C23×C3—31no—
64C23×C3—34no—
65C23×C3—64no—
66C23×C3C2×D311yescharacteristic, complemented
67C23×C3C2×D311yescharacteristic, complemented
68C22×D3—24no—
69C22×D3—31no—
70C22×D3—31no—
71C22×D3—34no—
72C22×D3—34no—
73C22×D3—61no—
74C22×D3—64no—
75C22×D3—64no—
76C22×D3C2×D311yescharacteristic
77C22×D3C2×D314yescomplemented
78C22×D3C2×D311yescharacteristic, complemented
79C22×D3C2×D311yescharacteristic, complemented
80C3×D4—34no—
81C2×C3×C4—31no—
82#67—34no—
83#67—34no—
84#67—34no—
85#67—34no—
86#67—34no—
87#67—64no—
88#67C2×D314yesdirect product factor
89C2×Dic3—31no—
90C2×Dic3—31no—
91C2×Dic3—34no—
92C2×Dic3—61no—
93C2×Dic3C2×D311yescharacteristic, complemented
94D12—34no—
95C4×D3—34no—
96C2×C32—22no—
97C2×C32C2×D412yescomplemented
98C2×C32C2411yescharacteristic
99C3⋊D3—22no—
100C3×D3—22no—
101C3×D3—22no—
102C3×D3C2×D414yescomplemented
103C24—31no—
104C24—91no—
105C2×D4—31no—
106C2×D4—91no—
107C2×D4—98no—
108C2×D4—91no—
109C2×D4—91no—
110C22×C4—91no—
111C22×C3—22no—
112C22×C3—22no—
113C22×C3—21no—
114C22×C3—22no—
115C22×C3—34no—
116C22×C3—32no—
117C22×C3—32no—
118C22×C3—32no—
119C22×C3—42no—
120C22×C3—68no—
121C22×C3—62no—
122C22×C3—62no—
123C22×C3C22×D311yescharacteristic
124C22×C3C22×D311yescharacteristic
125C22×C3C22×D312yescomplemented
126C22×C3C22×D312yescomplemented
127C2×D3—22no—
128C2×D3—22no—
129C2×D3—28no—
130C2×D3—32no—
131C2×D3—32no—
132C2×D3—62no—
133C2×D3—68no—
134C2×D3—62no—
135C2×D3—62no—
136C2×D3—68no—
137C2×D3—122no—
138C2×D3#6714yesdirect product factor
139C2×D3C22×D312yes—
140C2×D3C22×D312yescomplemented
141C2×D3C22×D312yescomplemented
142C3×C4—32no—
143Dic3—32no—
144Dic3—32no—
145Dic3—62no—
146Dic3C22×D312yescomplemented
147C32C22×D411yescharacteristic, complemented, Sylow
148C23—31no—
149C23—31no—
150C23—31no—
151C23—34no—
152C23—64no—
153C23—94no—
154C23—91no—
155C23—184no—
156C23D3211yescharacteristic, complemented
157D4—34no—
158D4—94no—
159D4—94no—
160D4—94no—
161C2×C4—31no—
162C2×C4—94no—
163C2×C4—91no—
164C2×C3—22no—
165C2×C3—22no—
166C2×C3—21no—
167C2×C3—22no—
168C2×C3—34no—
169C2×C3—42no—
170C2×C3—62no—
171C2×C3—62no—
172C2×C3D3×D412yescomplemented
173C2×C3C2×#6712yescomplemented
174C2×C3C23×D311yescharacteristic
175C2×C3C23×D311yescharacteristic
176D3—22no—
177D3—22no—
178D3—62no—
179D3—62no—
180D3—122no—
181D3C2×#6714yesdirect product factor
182C22—22no—
183C22—32no—
184C22—34no—
185C22—32no—
186C22—32no—
187C22—68no—
188C22—62no—
189C22—62no—
190C22—92no—
191C22—188no—
192C22—182no—
193C22C2×D3211yescharacteristic
194C22C2×D3212yescomplemented
195C4—32no—
196C4—92no—
197C3—21no—
198C3C2×D3×D411yescharacteristic, complemented
199C3C22×#6711yescharacteristic, complemented
200C2—22no—
201C2—34no—
202C2—62no—
203C2—62no—
204C2—182no—
205C2D3×#6712yesdirect product factor
206C2C22×D3211yescharacteristic
subgroupG/H
centerC22C2×D32
commutator, cosocleC2×C32C24
FrattiniC2C22×D32
FittingC23×C32C22
socleC22×C32C23
hypercenterC23D32
hypocenterC32C22×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×C3×D3C2semidirect
2. C22×D32C2semidirect
3. C2×#638C2semidirect
4. C2×C3×#67C2semidirect
5. D3×#67C2direct (N × H)
6. C2×#626C2semidirect
7. C2×#625C2semidirect
8. C2×D3×Dic3C2semidirect
11. C23×C32C22semidirect
12. C22×C3⋊D3C22semidirect
15. C22×C3×D3C22semidirect
16. C22×C3×D3C22semidirect
17. C22×C3×D3C22semidirect
19. C2×D32C22semidirect
20. #638C22semidirect
21. C2×C3⋊Dic3C22semidirect
22. C3×#67C22semidirect
23. C2×C3×Dic3C22semidirect
24. #626C22semidirect
25. #625C22semidirect
26. D3×Dic3C22semidirect
30. C23×D3D3semidirect
39. C2×#67D3direct (N × H)
46. C22×C32C23semidirect
48. C2×C3⋊D3C23semidirect
52. C2×C3×D3D4semidirect
54. C2×C3×D3C23semidirect
55. C2×C3×D3C23semidirect
57. C3⋊Dic3C23semidirect
58. C3×Dic3C23semidirect
66. C23×C3C2×D3semidirect
67. C23×C3C2×D3semidirect
77. C22×D3C2×D3semidirect
78. C22×D3C2×D3semidirect
79. C22×D3C2×D3semidirect
88. #67C2×D3direct (N × H)
93. C2×Dic3C2×D3semidirect
97. C2×C32C2×D4semidirect
102. C3×D3C2×D4semidirect
125. C22×C3C22×D3semidirect
126. C22×C3C22×D3semidirect
138. C2×D3#67direct (N × H)
140. C2×D3C22×D3semidirect
141. C2×D3C22×D3semidirect
146. Dic3C22×D3semidirect
147. C32C22×D4semidirect
156. C23D32semidirect
172. C2×C3D3×D4semidirect
173. C2×C3C2×#67semidirect
181. D3C2×#67direct (N × H)
194. C22C2×D32semidirect
198. C3C2×D3×D4semidirect
199. C3C22×#67semidirect
205. C2D3×#67direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D32
C22×D322
C2×#6382
C2×C3×#672
D3×#672
C2×#6262
C2×#6252
C2×D3×Dic32
C22×#673
C2×D3×D43