Finite Group Databasebeta

← #120×C3⋊(C3⋊D3) #13179714

Subgroups — #120×C3⋊(C3⋊D3) #13179714

Order 1728.

9268 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 9268 conjugacy classes of subgroups — too many to draw. Only its 807 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
1C24×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×D4×C3⋊(C3⋊D3)C213yesmaximal, complemented
3C33×#120C211yesmaximal, characteristic, complemented
4#11723994C211yesmaximal, characteristic, complemented
5#11723978C213yesmaximal, complemented
6#11723923C213yesmaximal, complemented
7C22⋊C4×C3⋊(C3⋊D3)C213yesmaximal, complemented
8C3⋊D3×#120—313nomaximal
9C24×C33C2211yescharacteristic, complemented
10C23×C3⋊(C3⋊D3)—212no—
11C23×C3⋊(C3⋊D3)C2213yes—
12C23×C3⋊(C3⋊D3)C2213yescomplemented
13C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
14C2×C33×D4C2213yescomplemented
15C2×#6312C2213yescomplemented
16C2×#6312C2213yescomplemented
17D4×C3⋊(C3⋊D3)—212no—
18C2×C4⋊(C3⋊(C3⋊D3))C2213yescomplemented
19C2×C4×C3⋊(C3⋊D3)C2213yescomplemented
20C33×C22⋊C4C2213yescomplemented
21#88864C2213yescomplemented
22#88859C2216yescomplemented
23C24×C3⋊D3—313no—
24C2×D4×C3⋊D3—339no—
25C32×#120D3113yescomplemented
26#64162—313no—
27#64146—339no—
28#64091—339no—
29C22⋊C4×C3⋊D3—339no—
30C23×C33—26no—
31C23×C33C2313yes—
32C23×C33C2311yescharacteristic, complemented
33C22×C3⋊(C3⋊D3)—232no—
34C22×C3⋊(C3⋊D3)—26no—
35C22×C3⋊(C3⋊D3)—212no—
36C22×C3⋊(C3⋊D3)—212no—
37C22×C3⋊(C3⋊D3)—26no—
38C22×C3⋊(C3⋊D3)D4112yescomplemented
39C22×C3⋊(C3⋊D3)C2313yes—
40C22×C3⋊(C3⋊D3)C2311yescharacteristic
41C22×C3⋊(C3⋊D3)C2311yescharacteristic, complemented
42C33×D4—26no—
43C2×C33×C4C2313yescomplemented
44#6312—26no—
45#6312—26no—
46C2×C3⋊(C3⋊Dic3)C2313yescomplemented
47C4⋊(C3⋊(C3⋊D3))—26no—
48C4×C3⋊(C3⋊D3)—26no—
49D3×#120—913no—
50C24×C32C2×D3113yescomplemented
51C23×C3⋊D3—339no—
52C23×C3⋊D3—339no—
53C23×C3⋊D3—313no—
54C23×C3⋊D3—6156no—
55C2×C32×D4C2×D3139yescomplemented
56C2×#638—339no—
57C2×#638—339no—
58D4×C3⋊D3—6156no—
59C2×C4⋊(C3⋊D3)—339no—
60C2×C4×C3⋊D3—339no—
61C32×C22⋊C4C2×D3139yescomplemented
62#3757—339no—
63#3752—378no—
64C22×C33—28no—
65C22×C33—26no—
66C22×C33—23no—
67C22×C33C2×D416yescomplemented
68C22×C33C2411yescharacteristic
69C2×C3⋊(C3⋊D3)—224no—
70C2×C3⋊(C3⋊D3)—26no—
71C2×C3⋊(C3⋊D3)—224no—
72C2×C3⋊(C3⋊D3)—23no—
73C2×C3⋊(C3⋊D3)—44no—
74C2×C3⋊(C3⋊D3)C2×D416yes—
75C2×C3⋊(C3⋊D3)C2×D416yescomplemented
76C33×C4—23no—
77C3⋊(C3⋊Dic3)—23no—
78C24×D3—913no—
79C2×D3×D4—939no—
80C3×#120C3⋊D3113yescomplemented
81#953—913no—
82#937—939no—
83#882—939no—
84D3×C22⋊C4—939no—
85C23×C32—278no—
86C23×C32C22×D3139yes—
87C23×C32C22×D3113yescomplemented
88C22×C3⋊D3—313no—
89C22×C3⋊D3—3156no—
90C22×C3⋊D3—339no—
91C22×C3⋊D3—313no—
92C22×C3⋊D3—6156no—
93C22×C3⋊D3—6156no—
94C22×C3⋊D3—678no—
95C22×C3⋊D3—6416no—
96C22×C3⋊D3—678no—
97C32×D4—278no—
98C2×C32×C4C22×D3139yescomplemented
99#638—678no—
100#638—678no—
101C2×C3⋊Dic3—339no—
102C4⋊(C3⋊D3)—678no—
103C4×C3⋊D3—678no—
104C2×#120—271noSylow
105C2×C33—26no—
106C2×C33—41no—
107C2×C33C22×D413yes—
108C3⋊(C3⋊D3)—26no—
109C3⋊(C3⋊D3)—41no—
110C3⋊(C3⋊D3)#12014yesdirect product factor
111C24×C3C2×C3⋊D3113yescomplemented
112C23×D3—939no—
113C23×D3—939no—
114C23×D3—913no—
115C23×D3—18156no—
116C2×C3×D4C2×C3⋊D3139yescomplemented
117C2×#67—939no—
118C2×#67—939no—
119D3×D4—18156no—
120C2×D12—939no—
121C2×C4×D3—939no—
122C3×C22⋊C4C2×C3⋊D3139yescomplemented
123#217—939no—
124(C2×D3)⋊C4—978no—
125C22×C32—2104no—
126C22×C32—278no—
127C22×C32—239no—
128C22×C32D3×D4178yescomplemented
129C22×C32C23×D3113yes—
130C2×C3⋊D3—378no—
131C2×C3⋊D3—378no—
132C2×C3⋊D3—6312no—
133C2×C3⋊D3—6312no—
134C2×C3⋊D3—678no—
135C2×C3⋊D3—639no—
136C2×C3⋊D3—1252no—
137C32×C4—239no—
138C3⋊Dic3—639no—
139C25—271no—
140C22×D4—273no—
141#120—271no—
142#120—273no—
143#120—273no—
144#120C3⋊(C3⋊D3)11yescharacteristic, direct product factor
145C2×C22⋊C4—273no—
146C33C2×#12011yescharacteristic, complemented, Sylow
147C23×C3—278no—
148C23×C3C22×C3⋊D3139yes—
149C23×C3C22×C3⋊D3113yescomplemented
150C22×D3—913no—
151C22×D3—939no—
152C22×D3—9156no—
153C22×D3—913no—
154C22×D3—18156no—
155C22×D3—18156no—
156C22×D3—1878no—
157C22×D3—18416no—
158C22×D3—1878no—
159C3×D4—278no—
160C2×C3×C4C22×C3⋊D3139yescomplemented
161#67—1878no—
162#67—1878no—
163C2×Dic3—939no—
164D12—1878no—
165C4×D3—1878no—
166C2×C32—278no—
167C2×C32—413no—
168C2×C32C2×D3×D4139yes—
169C3⋊D3—352no—
170C3⋊D3—678no—
171C3⋊D3—1213no—
172C24—273no—
173C24—273no—
174C24—271no—
175C24—5412no—
176C24C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
177C2×D4—273no—
178C2×D4—273no—
179C2×D4—273no—
180C2×D4—5412no—
181C2×D4C2×C3⋊(C3⋊D3)13yescomplemented
182C22×C4—273no—
183C22⋊C4—273no—
184C22⋊C4—276no—
185C22⋊C4C2×C3⋊(C3⋊D3)13yescomplemented
186C22×C3—2104no—
187C22×C3—278no—
188C22×C3—239no—
189C22×C3D4×C3⋊D3178yescomplemented
190C22×C3C23×C3⋊D3113yes—
191C2×D3—978no—
192C2×D3—978no—
193C2×D3—18312no—
194C2×D3—18312no—
195C2×D3—1878no—
196C2×D3—1839no—
197C2×D3—3652no—
198C3×C4—239no—
199Dic3—1839no—
200C32D3×#120113yescomplemented
201C23—26no—
202C23—271no—
203C23—273no—
204C23—2712no—
205C23—271no—
206C23—5412no—
207C23—5412no—
208C23—546no—
209C23—5432no—
210C23—546no—
211C23C22×C3⋊(C3⋊D3)13yes—
212C23C22×C3⋊(C3⋊D3)11yescharacteristic, complemented
213D4—26no—
214D4—546no—
215D4—546no—
216D4—546no—
217C2×C4—273no—
218C2×C4—546no—
219C2×C4C22×C3⋊(C3⋊D3)13yescomplemented
220C2×C3—278no—
221C2×C3—413no—
222C2×C3C2×D4×C3⋊D3139yes—
223D3—952no—
224D3—1878no—
225D3—3613no—
226C22—28no—
227C22—26no—
228C22—23no—
229C22—276no—
230C22—276no—
231C22—5424no—
232C22—5424no—
233C22—546no—
234C22—543no—
235C22—1084no—
236C22D4×C3⋊(C3⋊D3)16yescomplemented
237C22C23×C3⋊(C3⋊D3)11yescharacteristic
238C4—23no—
239C4—543no—
240C3C3⋊D3×#120113yescomplemented
241C2—26no—
242C2—41no—
243C2—274no—
244C2—546no—
245C2—1081no—
246C2C2×D4×C3⋊(C3⋊D3)13yes—
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×#120C2
hypercenter#120C3⋊(C3⋊D3)
hypocenterC33C2×#120
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×C3⋊(C3⋊D3)C2semidirect
2. C2×D4×C3⋊(C3⋊D3)C2semidirect
3. C33×#120C2semidirect
4. #11723994C2semidirect
5. #11723978C2semidirect
6. #11723923C2semidirect
7. C22⋊C4×C3⋊(C3⋊D3)C2semidirect
9. C24×C33C22semidirect
12. C23×C3⋊(C3⋊D3)C22semidirect
13. C23×C3⋊(C3⋊D3)C22semidirect
14. C2×C33×D4C22semidirect
15. C2×#6312C22semidirect
16. C2×#6312C22semidirect
18. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×C4×C3⋊(C3⋊D3)C22semidirect
20. C33×C22⋊C4C22semidirect
21. #88864C22semidirect
22. #88859C22semidirect
25. C32×#120D3semidirect
32. C23×C33C23semidirect
38. C22×C3⋊(C3⋊D3)D4semidirect
41. C22×C3⋊(C3⋊D3)C23semidirect
43. C2×C33×C4C23semidirect
46. C2×C3⋊(C3⋊Dic3)C23semidirect
50. C24×C32C2×D3semidirect
55. C2×C32×D4C2×D3semidirect
61. C32×C22⋊C4C2×D3semidirect
67. C22×C33C2×D4semidirect
75. C2×C3⋊(C3⋊D3)C2×D4semidirect
80. C3×#120C3⋊D3semidirect
87. C23×C32C22×D3semidirect
98. C2×C32×C4C22×D3semidirect
110. C3⋊(C3⋊D3)#120direct (N × H)
111. C24×C3C2×C3⋊D3semidirect
116. C2×C3×D4C2×C3⋊D3semidirect
122. C3×C22⋊C4C2×C3⋊D3semidirect
128. C22×C32D3×D4semidirect
144. #120C3⋊(C3⋊D3)direct (N × H)
146. C33C2×#120semidirect
149. C23×C3C22×C3⋊D3semidirect
160. C2×C3×C4C22×C3⋊D3semidirect
176. C24C2×C3⋊(C3⋊D3)semidirect
181. C2×D4C2×C3⋊(C3⋊D3)semidirect
185. C22⋊C4C2×C3⋊(C3⋊D3)semidirect
189. C22×C3D4×C3⋊D3semidirect
200. C32D3×#120semidirect
212. C23C22×C3⋊(C3⋊D3)semidirect
219. C2×C4C22×C3⋊(C3⋊D3)semidirect
236. C22D4×C3⋊(C3⋊D3)semidirect
240. C3C3⋊D3×#120semidirect

Maximal subgroups

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

subgroupindex
C24×C3⋊(C3⋊D3)2
C2×D4×C3⋊(C3⋊D3)2
C33×#1202
#117239942
#117239782
#117239232
C22⋊C4×C3⋊(C3⋊D3)2
C3⋊D3×#1203