Finite Group Databasebeta

← #13179732

Subgroups — #13179732

Order 1728.

4844 conjugacy classes of subgroups; only the 675 normal subgroups are shown.

normal subgroups only

This group has 4844 conjugacy classes of subgroups — too many to draw. Only its 675 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
1C2×D4×C3⋊(C3⋊D3)C212yesmaximal, complemented
2C33×#121C211yesmaximal, characteristic, complemented
3#11723980C212yesmaximal, complemented
4#11723978C212yesmaximal, complemented
5#11723971C211yesmaximal, characteristic, complemented
6C2×#88859C211yesmaximal, characteristic, complemented
7#11723935C212yesmaximal, complemented
8#11723924C212yesmaximal, complemented
9#11723923C212yesmaximal, complemented
10#10596076—313nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C23×C3⋊(C3⋊D3)C2212yescomplemented
13C2×C33×D4C2212yescomplemented
14C2×C33×D4C2211yescharacteristic, complemented
15C22×C33×C4C2211yescharacteristic, complemented
16C2×#6312C2212yes—
17C2×#6312C2212yescomplemented
18C2×#6312C2212yescomplemented
19C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
20D4×C3⋊(C3⋊D3)—28no—
21C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
22C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
23C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
24C33×C4⋊C4C2211yescharacteristic, complemented
25C33×C22⋊C4C2212yescomplemented
26#88864C2212yescomplemented
27#88859C2212yescomplemented
28#88859C2212yescomplemented
29#88859C2212yescomplemented
30#88859C2212yescomplemented
31C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
32#88857C2212yescomplemented
33C2×D4×C3⋊D3—326no—
34C32×#121D3113yescomplemented
35#64148—326no—
36#64146—326no—
37#64139—313no—
38C2×#3752—313no—
39#64103—326no—
40#64092—326no—
41#64091—326no—
42C23×C33C2312yes—
43C23×C33C2311yescharacteristic
44C22×C3⋊(C3⋊D3)—24no—
45C22×C3⋊(C3⋊D3)—41no—
46C22×C3⋊(C3⋊D3)—41no—
47C22×C3⋊(C3⋊D3)—42no—
48C22×C3⋊(C3⋊D3)—42no—
49C22×C3⋊(C3⋊D3)D414yescomplemented
50C22×C3⋊(C3⋊D3)C2312yes—
51C22×C3⋊(C3⋊D3)C2312yes—
52C33×D4—24no—
53C33×D4—41no—
54C2×C33×C4—41no—
55C2×C33×C4C2312yes—
56C2×C33×C4C2311yescharacteristic
57C2×C33×C4C2311yescharacteristic
58#6312—24no—
59#6312—24no—
60#6312—42no—
61C2×C3⋊(C3⋊Dic3)—41no—
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C2×C3⋊(C3⋊Dic3)C2312yes—
64C4⋊(C3⋊(C3⋊D3))—24no—
65C4⋊(C3⋊(C3⋊D3))—41no—
66C4×C3⋊(C3⋊D3)—24no—
67#5605—913no—
68C23×C3⋊D3—313no—
69C23×C3⋊D3—326no—
70C2×C32×D4C2×D3126yescomplemented
71C2×C32×D4C2×D3113yescomplemented
72C22×C32×C4C2×D3113yescomplemented
73C2×#638—326no—
74C2×#638—326no—
75C2×#638—326no—
76C22×C3⋊Dic3—313no—
77D4×C3⋊D3—6104no—
78C2×C4⋊(C3⋊D3)—326no—
79C2×C4⋊(C3⋊D3)—313no—
80C2×C4×C3⋊D3—326no—
81C32×C4⋊C4C2×D3113yescomplemented
82C32×C22⋊C4C2×D3126yescomplemented
83#3757—326no—
84#3752—326no—
85#3752—326no—
86#3752—326no—
87#3752—326no—
88C4⋊(C3⋊Dic3)—313no—
89#3750—326no—
90C22×C33—21no—
91C22×C33—21no—
92C22×C33—22no—
93C22×C33—22no—
94C22×C33—22no—
95C22×C33C2×D412yes—
96C22×C33C2411yescharacteristic
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—22no—
100C2×C3⋊(C3⋊D3)—22no—
101C2×C3⋊(C3⋊D3)—22no—
102C2×C3⋊(C3⋊D3)—44no—
103C2×C3⋊(C3⋊D3)—48no—
104C2×C3⋊(C3⋊D3)C2×D414yes—
105C33×C4—22no—
106C33×C4—21no—
107C33×C4—21no—
108C3⋊(C3⋊Dic3)—22no—
109C3⋊(C3⋊Dic3)—22no—
110C2×D3×D4—926no—
111C3×#121C3⋊D3113yescomplemented
112#939—926no—
113#937—926no—
114#930—913no—
115C2×(C2×D3)⋊C4—913no—
116#894—926no—
117#883—926no—
118#882—926no—
119C23×C32C22×D3126yes—
120C23×C32C22×D3113yes—
121C22×C3⋊D3—326no—
122C22×C3⋊D3—352no—
123C22×C3⋊D3—326no—
124C22×C3⋊D3—652no—
125C22×C3⋊D3—1213no—
126C22×C3⋊D3—1226no—
127C22×C3⋊D3—1213no—
128C22×C3⋊D3—1226no—
129C32×D4—252no—
130C32×D4—413no—
131C2×C32×C4—413no—
132C2×C32×C4C22×D3126yes—
133C2×C32×C4C22×D3113yes—
134C2×C32×C4C22×D3113yes—
135#638—652no—
136#638—652no—
137#638—1226no—
138C2×C3⋊Dic3—326no—
139C2×C3⋊Dic3—326no—
140C2×C3⋊Dic3—1213no—
141C4⋊(C3⋊D3)—652no—
142C4⋊(C3⋊D3)—1213no—
143C4×C3⋊D3—652no—
144#534—271noSylow
145C2×C33—22no—
146C2×C33—42no—
147C2×C33C22×D411yescharacteristic
148C2×C33H(5,𝔽2)11yescharacteristic
149C2×C33H(5,𝔽2)11yescharacteristic
150C3⋊(C3⋊D3)—24no—
151C3⋊(C3⋊D3)—42no—
152C23×D3—913no—
153C23×D3—926no—
154C2×C3×D4C2×C3⋊D3126yescomplemented
155C2×C3×D4C2×C3⋊D3113yescomplemented
156C22×C3×C4C2×C3⋊D3113yescomplemented
157C2×#67—926no—
158C2×#67—926no—
159C2×#67—926no—
160C22×Dic3—913no—
161D3×D4—18104no—
162C2×D12—926no—
163C2×D12—913no—
164C2×C4×D3—926no—
165C3×C4⋊C4C2×C3⋊D3113yescomplemented
166C3×C22⋊C4C2×C3⋊D3126yescomplemented
167#217—926no—
168(C2×D3)⋊C4—926no—
169(C2×D3)⋊C4—926no—
170(C2×D3)⋊C4—926no—
171(C2×D3)⋊C4—926no—
172C4⋊Dic3—913no—
173Dic3⋊C4—926no—
174C22×C32—213no—
175C22×C32—213no—
176C22×C32—226no—
177C22×C32—226no—
178C22×C32—226no—
179C22×C32D3×D4126yes—
180C22×C32C23×D3113yes—
181C2×C3⋊D3—352no—
182C2×C3⋊D3—626no—
183C2×C3⋊D3—626no—
184C2×C3⋊D3—626no—
185C2×C3⋊D3—626no—
186C2×C3⋊D3—626no—
187C2×C3⋊D3—1252no—
188C2×C3⋊D3—12104no—
189C32×C4—226no—
190C32×C4—213no—
191C32×C4—213no—
192C3⋊Dic3—626no—
193C3⋊Dic3—626no—
194C22×D4—272no—
195#123—272no—
196#123—272no—
197#121—272no—
198#121—271no—
199#121C3⋊(C3⋊D3)11yescharacteristic, complemented
200#120—272no—
201#120—272no—
202C2×C22⋊C4—271no—
203C33#53411yescharacteristic, complemented, Sylow
204C23×C3C22×C3⋊D3126yes—
205C23×C3C22×C3⋊D3113yes—
206C22×D3—926no—
207C22×D3—952no—
208C22×D3—926no—
209C22×D3—1852no—
210C22×D3—3613no—
211C22×D3—3626no—
212C22×D3—3613no—
213C22×D3—3626no—
214C3×D4—252no—
215C3×D4—413no—
216C2×C3×C4—413no—
217C2×C3×C4C22×C3⋊D3126yes—
218C2×C3×C4C22×C3⋊D3113yes—
219C2×C3×C4C22×C3⋊D3113yes—
220#67—1852no—
221#67—1852no—
222#67—3626no—
223C2×Dic3—926no—
224C2×Dic3—926no—
225C2×Dic3—3613no—
226D12—1852no—
227D12—3613no—
228C4×D3—1852no—
229C2×C32—226no—
230C2×C32—426no—
231C2×C32C2×D3×D4113yes—
232C2×C32#1004113yes—
233C2×C32#1009113yes—
234C3⋊D3—652no—
235C3⋊D3—1226no—
236C24—271no—
237C24—272no—
238C2×D4—272no—
239C2×D4—272no—
240C2×D4—272no—
241C2×D4—272no—
242C2×D4—271no—
243C2×D4—548no—
244C2×D4C2×C3⋊(C3⋊D3)12yescomplemented
245C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
246C22×C4—271no—
247C22×C4—272no—
248C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
249C4⋊C4—272no—
250C4⋊C4—271no—
251C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
252C22⋊C4—272no—
253C22⋊C4—272no—
254C22⋊C4—272no—
255C22⋊C4—272no—
256C22⋊C4—272no—
257C22⋊C4C2×C3⋊(C3⋊D3)12yescomplemented
258C22×C3—213no—
259C22×C3—213no—
260C22×C3—226no—
261C22×C3—226no—
262C22×C3—226no—
263C22×C3D4×C3⋊D3126yes—
264C22×C3C23×C3⋊D3113yes—
265C2×D3—952no—
266C2×D3—1826no—
267C2×D3—1826no—
268C2×D3—1826no—
269C2×D3—1826no—
270C2×D3—1826no—
271C2×D3—3652no—
272C2×D3—36104no—
273C3×C4—226no—
274C3×C4—213no—
275C3×C4—213no—
276Dic3—1826no—
277Dic3—1826no—
278C32#5605113yescomplemented
279C23—272no—
280C23—274no—
281C23—272no—
282C23—544no—
283C23—1081no—
284C23—1081no—
285C23—1082no—
286C23—1082no—
287C23C22×C3⋊(C3⋊D3)12yes—
288C23C22×C3⋊(C3⋊D3)11yescharacteristic
289D4—24no—
290D4—41no—
291D4—544no—
292D4—544no—
293D4—544no—
294D4—1081no—
295D4—1082no—
296C2×C4—41no—
297C2×C4—272no—
298C2×C4—272no—
299C2×C4—544no—
300C2×C4—1081no—
301C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
302C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
303C2×C4C22×C3⋊(C3⋊D3)12yes—
304C2×C3—226no—
305C2×C3—426no—
306C2×C3C2×D4×C3⋊D3113yes—
307C2×C3#64361113yes—
308C2×C3#64366113yes—
309D3—1852no—
310D3—3626no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—274no—
317C22—542no—
318C22—542no—
319C22—542no—
320C22—542no—
321C22—542no—
322C22—1084no—
323C22—1088no—
324C22D4×C3⋊(C3⋊D3)12yes—
325C22C23×C3⋊(C3⋊D3)11yescharacteristic
326C4—22no—
327C4—21no—
328C4—21no—
329C4—542no—
330C4—542no—
331C3#10596076113yescomplemented
332C2—22no—
333C2—42no—
334C2—544no—
335C2—1082no—
336C2C2×D4×C3⋊(C3⋊D3)11yescharacteristic
337C2#1172467911yescharacteristic
338C2#1172468411yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×#121C2
hypercenter#121C3⋊(C3⋊D3)
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×C3⋊(C3⋊D3)C2semidirect
2. C33×#121C2semidirect
3. #11723980C2semidirect
4. #11723978C2semidirect
5. #11723971C2semidirect
6. C2×#88859C2semidirect
7. #11723935C2semidirect
8. #11723924C2semidirect
9. #11723923C2semidirect
12. C23×C3⋊(C3⋊D3)C22semidirect
13. C2×C33×D4C22semidirect
14. C2×C33×D4C22semidirect
15. C22×C33×C4C22semidirect
17. C2×#6312C22semidirect
18. C2×#6312C22semidirect
19. C22×C3⋊(C3⋊Dic3)C22semidirect
22. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
23. C2×C4×C3⋊(C3⋊D3)C22semidirect
24. C33×C4⋊C4C22semidirect
25. C33×C22⋊C4C22semidirect
26. #88864C22semidirect
27. #88859C22semidirect
28. #88859C22semidirect
29. #88859C22semidirect
30. #88859C22semidirect
31. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
32. #88857C22semidirect
34. C32×#121D3semidirect
49. C22×C3⋊(C3⋊D3)D4semidirect
70. C2×C32×D4C2×D3semidirect
71. C2×C32×D4C2×D3semidirect
72. C22×C32×C4C2×D3semidirect
81. C32×C4⋊C4C2×D3semidirect
82. C32×C22⋊C4C2×D3semidirect
111. C3×#121C3⋊D3semidirect
154. C2×C3×D4C2×C3⋊D3semidirect
155. C2×C3×D4C2×C3⋊D3semidirect
156. C22×C3×C4C2×C3⋊D3semidirect
165. C3×C4⋊C4C2×C3⋊D3semidirect
166. C3×C22⋊C4C2×C3⋊D3semidirect
199. #121C3⋊(C3⋊D3)semidirect
203. C33#534semidirect
244. C2×D4C2×C3⋊(C3⋊D3)semidirect
245. C2×D4C2×C3⋊(C3⋊D3)semidirect
248. C22×C4C2×C3⋊(C3⋊D3)semidirect
251. C4⋊C4C2×C3⋊(C3⋊D3)semidirect
257. C22⋊C4C2×C3⋊(C3⋊D3)semidirect
278. C32#5605semidirect
331. C3#10596076semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C33×#1212
#117239802
#117239782
#117239712
C2×#888592
#117239352
#117239242
#117239232
#105960763