Finite Group Databasebeta

← C3×D3×#67 #89011

Subgroups — C3×D3×#67 #89011

Order 432.

290 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C22×C32×D3C211yesmaximal, characteristic, complemented
2C2×C3×D32C211yesmaximal, characteristic, complemented
3C3×#638C211yesmaximal, characteristic, complemented
4C32×#67C211yesmaximal, characteristic, complemented
5C3×#626C211yesmaximal, characteristic, complemented
6C3×#625C211yesmaximal, characteristic, complemented
7C3×D3×Dic3C211yesmaximal, characteristic, complemented
8C2×C3×#67—31nomaximal
9C3×D3×D4—31nomaximal
10D3×#67C311yesmaximal, characteristic, direct product factor
11C22×C33C2211yescharacteristic, complemented
12C2×C3×C3⋊D3C2211yescharacteristic, complemented
13C2×C32×D3—22no—
14C2×C32×D3C2211yescharacteristic
15C2×C32×D3C2211yescharacteristic, complemented
16C2×C32×D3C2211yescharacteristic, complemented
17C3×D32—22no—
18C3×C3⋊Dic3C2211yescharacteristic, complemented
19C32×Dic3C2211yescharacteristic, complemented
20C23×C32—31no—
21C22×C3×D3—21no—
22C22×C3×D3—31no—
23C22×C3×D3—31no—
24C22×C3×D3D311yescharacteristic, complemented
25C22×C3×D3C2×C311yescharacteristic, complemented
26C2×D32C2×C311yescharacteristic, complemented
27C32×D4—31no—
28#638C2×C311yescharacteristic, complemented
29C3×#67—21no—
30C3×#67—31no—
31C3×#67—31no—
32C3×#67—31no—
33C3×#67—31no—
34C3×#67—31no—
35C3×#67—61no—
36C3×#67D311yescharacteristic, direct product factor
37C3×#67C2×C311yescharacteristic, complemented
38C2×C3×Dic3—31no—
39C3×D12—31no—
40C3×C4×D3—31no—
41#626C2×C311yescharacteristic, complemented
42#625C2×C311yescharacteristic, complemented
43D3×Dic3C2×C311yescharacteristic, complemented
44C2×C33—21no—
45C2×C33C2311yescharacteristic
46C3×C3⋊D3—21no—
47C32×D3—21no—
48C32×D3—21no—
49C32×D3D412yescomplemented
50C2×C3×D4—91no—
51C2×#67—31no—
52D3×D4—31no—
53C22×C32—21no—
54C22×C32—21no—
55C22×C32—21no—
56C22×C32—31no—
57C22×C32—31no—
58C22×C32—31no—
59C22×C32—41no—
60C22×C32—62no—
61C22×C32C2×D311yescharacteristic, complemented
62C22×C32C2×D311yescharacteristic, complemented
63C22×C32C22×C311yescharacteristic, complemented
64C2×C3⋊D3C22×C311yescharacteristic, complemented
65C2×C3×D3—24no—
66C2×C3×D3—22no—
67C2×C3×D3—22no—
68C2×C3×D3—21no—
69C2×C3×D3—21no—
70C2×C3×D3—21no—
71C2×C3×D3—31no—
72C2×C3×D3—31no—
73C2×C3×D3—61no—
74C2×C3×D3—62no—
75C2×C3×D3—62no—
76C2×C3×D3C2×D311yescharacteristic
77C2×C3×D3C2×D311yescharacteristic, complemented
78C2×C3×D3C2×D311yescharacteristic, complemented
79C2×C3×D3C22×C311yescharacteristic
80C2×C3×D3C22×C311yescharacteristic, complemented
81C2×C3×D3C22×C311yescharacteristic, complemented
82D32—22no—
83C32×C4—31no—
84C3⋊Dic3C22×C311yescharacteristic, complemented
85C3×Dic3—21no—
86C3×Dic3—31no—
87C3×Dic3—31no—
88C3×Dic3—61no—
89C3×Dic3C2×D311yescharacteristic, complemented
90C3×Dic3C22×C311yescharacteristic, complemented
91C33C2×D411yescharacteristic, complemented, Sylow
92C23×C3—31no—
93C23×C3—31no—
94C23×C3—61no—
95C23×C3—91no—
96C22×D3—31no—
97C22×D3—31no—
98C22×D3C3×D311yescharacteristic, complemented
99C3×D4—31no—
100C3×D4—31no—
101C3×D4—61no—
102C3×D4—91no—
103C3×D4—91no—
104C3×D4—91no—
105C2×C3×C4—91no—
106#67—31no—
107#67—31no—
108#67—31no—
109#67—31no—
110#67—31no—
111#67—61no—
112#67C3×D311yescharacteristic, direct product factor
113C2×Dic3—31no—
114D12—31no—
115C4×D3—31no—
116C2×C32—21no—
117C2×C32—22no—
118C2×C32—21no—
119C2×C32—21no—
120C2×C32—21no—
121C2×C32—21no—
122C2×C32—21no—
123C2×C32—32no—
124C2×C32—41no—
125C2×C32—41no—
126C2×C32—41no—
127C2×C32—42no—
128C2×C32—61no—
129C2×C32—61no—
130C2×C32C22×D311yescharacteristic
131C2×C32C22×D311yescharacteristic
132C2×C32C23×C311yescharacteristic
133C3⋊D3—21no—
134C3×D3—22no—
135C3×D3—21no—
136C3×D3—21no—
137C3×D3—21no—
138C3×D3—21no—
139C3×D3—22no—
140C3×D3—41no—
141C3×D3—61no—
142C3×D3—61no—
143C3×D3—121no—
144C3×D3#6712yesdirect product factor
145C3×D3C3×D412yescomplemented
146C2×D4—91noSylow
147C22×C3—21no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—31no—
151C22×C3—31no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—31no—
155C22×C3—31no—
156C22×C3—41no—
157C22×C3—61no—
158C22×C3—61no—
159C22×C3—61no—
160C22×C3—62no—
161C22×C3—62no—
162C22×C3—64no—
163C22×C3—91no—
164C22×C3—182no—
165C22×C3D3211yescharacteristic, complemented
166C22×C3C2×C3×D311yescharacteristic, complemented
167C22×C3C2×C3×D311yescharacteristic, complemented
168C2×D3—22no—
169C2×D3—31no—
170C2×D3—31no—
171C2×D3—61no—
172C2×D3—62no—
173C2×D3—62no—
174C2×D3C2×C3×D311yescharacteristic
175C2×D3C2×C3×D311yescharacteristic, complemented
176C2×D3C2×C3×D311yescharacteristic, complemented
177C3×C4—31no—
178C3×C4—31no—
179C3×C4—61no—
180C3×C4—91no—
181Dic3—31no—
182Dic3—31no—
183Dic3—61no—
184Dic3C2×C3×D311yescharacteristic, complemented
185C32—21no—
186C32—21no—
187C32—21no—
188C32—41no—
189C32D3×D411yescharacteristic, complemented
190C32C2×#6711yescharacteristic, complemented
191C32C2×C3×D411yescharacteristic, complemented
192C23—31no—
193C23—91no—
194D4—31no—
195D4—91no—
196D4—91no—
197D4—91no—
198C2×C4—91no—
199C2×C3—22no—
200C2×C3—21no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3—21no—
206C2×C3—32no—
207C2×C3—32no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—42no—
211C2×C3—41no—
212C2×C3—62no—
213C2×C3—62no—
214C2×C3—61no—
215C2×C3—61no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3—121no—
219C2×C3—181no—
220C2×C3C2×D3211yescharacteristic
221C2×C3C22×C3×D311yescharacteristic
222C2×C3C22×C3×D311yescharacteristic
223D3—21no—
224D3—21no—
225D3—61no—
226D3—61no—
227D3—121no—
228D3C3×#6712yesdirect product factor
229C22—31no—
230C22—31no—
231C22—31no—
232C22—62no—
233C22—91no—
234C22—182no—
235C22C3×D3211yescharacteristic, complemented
236C4—31no—
237C4—91no—
238C3—21no—
239C3—21no—
240C3—21no—
241C3—41no—
242C3D3×#6711yescharacteristic, direct product factor
243C3C3×D3×D411yescharacteristic, complemented
244C3C2×C3×#6711yescharacteristic, complemented
245C2—21no—
246C2—32no—
247C2—61no—
248C2—61no—
249C2—181no—
250C2C2×C3×D3211yescharacteristic
subgroupG/H
centerC2×C3C2×D32
commutator, cosocleC2×C32C23×C3
FrattiniC2C2×C3×D32
FittingC22×C33C22
socleC2×C33C23
hypercenterC22×C3D32
hypocenterC32C2×C3×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C32×D3C2semidirect
2. C2×C3×D32C2semidirect
3. C3×#638C2semidirect
4. C32×#67C2semidirect
5. C3×#626C2semidirect
6. C3×#625C2semidirect
7. C3×D3×Dic3C2semidirect
10. D3×#67C3direct (N × H)
11. C22×C33C22semidirect
12. C2×C3×C3⋊D3C22semidirect
15. C2×C32×D3C22semidirect
16. C2×C32×D3C22semidirect
18. C3×C3⋊Dic3C22semidirect
19. C32×Dic3C22semidirect
24. C22×C3×D3D3semidirect
25. C22×C3×D3C2×C3semidirect
26. C2×D32C2×C3semidirect
28. #638C2×C3semidirect
36. C3×#67D3direct (N × H)
37. C3×#67C2×C3semidirect
41. #626C2×C3semidirect
42. #625C2×C3semidirect
43. D3×Dic3C2×C3semidirect
49. C32×D3D4semidirect
61. C22×C32C2×D3semidirect
62. C22×C32C2×D3semidirect
63. C22×C32C22×C3semidirect
64. C2×C3⋊D3C22×C3semidirect
77. C2×C3×D3C2×D3semidirect
78. C2×C3×D3C2×D3semidirect
80. C2×C3×D3C22×C3semidirect
81. C2×C3×D3C22×C3semidirect
84. C3⋊Dic3C22×C3semidirect
89. C3×Dic3C2×D3semidirect
90. C3×Dic3C22×C3semidirect
91. C33C2×D4semidirect
98. C22×D3C3×D3semidirect
112. #67C3×D3direct (N × H)
144. C3×D3#67direct (N × H)
145. C3×D3C3×D4semidirect
165. C22×C3D32semidirect
166. C22×C3C2×C3×D3semidirect
167. C22×C3C2×C3×D3semidirect
175. C2×D3C2×C3×D3semidirect
176. C2×D3C2×C3×D3semidirect
184. Dic3C2×C3×D3semidirect
189. C32D3×D4semidirect
190. C32C2×#67semidirect
191. C32C2×C3×D4semidirect
228. D3C3×#67direct (N × H)
235. C22C3×D32semidirect
242. C3D3×#67direct (N × H)
243. C3C3×D3×D4semidirect
244. C3C2×C3×#67semidirect

Maximal subgroups

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

subgroupindex
C22×C32×D32
C2×C3×D322
C3×#6382
C32×#672
C3×#6262
C3×#6252
C3×D3×Dic32
C2×C3×#673
C3×D3×D43
D3×#673