Finite Group Databasebeta

← #13176782

Subgroups — #13176782

Order 1728.

1856 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724597C212yesmaximal, complemented
2#11723960C211yesmaximal, characteristic, complemented
3C3×#64128C211yesmaximal, characteristic, complemented
4C32×#919C211yesmaximal, characteristic, complemented
5#11723572C212yesmaximal, complemented
6#11723570C212yesmaximal, complemented
7#11723568C212yesmaximal, complemented
8#11723331C211yesmaximal, characteristic, complemented
9#11723322C211yesmaximal, characteristic, complemented
10C3⋊D3×D24C211yesmaximal, characteristic, complemented
11D3×#3745C211yesmaximal, characteristic, complemented
12#10596240—31nomaximal
13#10594771—34nomaximal
14#89080C2212yescomplemented
15C3×#3832C2212yescomplemented
16C32×#239C2212yescomplemented
17#89026—22no—
18#89026C2212yescomplemented
19C3⋊D3×D12—22no—
20C3⋊D3×D12C2212yescomplemented
21D3×C4⋊(C3⋊D3)—22no—
22D3×C4⋊(C3⋊D3)C2212yescomplemented
23#89020—22no—
24#89020C2211yescharacteristic
25C33×Q16C2211yescharacteristic, complemented
26#88862C2212yescomplemented
27#88852C2211yescharacteristic, complemented
28C8×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
29C3×#3755C2212yescomplemented
30C3×#3745C2211yescharacteristic, complemented
31C3×C8×C3⋊D3C2211yescharacteristic, complemented
32C32×#215C2212yescomplemented
33C32×D24C2211yescharacteristic, complemented
34C32×D3×C8C2211yescharacteristic, complemented
35#88791C2212yescomplemented
36#88790C2212yescomplemented
37#88789C2212yescomplemented
38#88788C2211yescharacteristic, complemented
39#88787C2211yescharacteristic, complemented
40#88786C2211yescharacteristic, complemented
41#64366—32no—
42#64320—38no—
43C32×#135—31no—
44#64158—32no—
45#64128—31no—
46#64128—34no—
47#64128—64no—
48#64128D311yescharacteristic, complemented
49#64123—32no—
50#64123—32no—
51D8×C3⋊D3—31no—
52D8×C3⋊D3—31no—
53C2×#3745—31no—
54#64110—31no—
55C3×#919—34no—
56C3×#919D314yescomplemented
57#63941—38no—
58#63939—38no—
59#63939—38no—
60#63803—34no—
61#63797—34no—
62D3×D24—34no—
63D3×D24—34no—
64C2×D3×C3⋊D3—22no—
65C2×D3×C3⋊D3—22no—
66C2×D3×C3⋊D3—22no—
67C33×Q8C2312yes—
68C4⋊(C3⋊(C3⋊D3))—22no—
69C4⋊(C3⋊(C3⋊D3))C2312yes—
70C4×C3⋊(C3⋊D3)—22no—
71C4×C3⋊(C3⋊D3)C2311yescharacteristic
72C3×C4⋊(C3⋊D3)—22no—
73C3×C4⋊(C3⋊D3)C2312yes—
74C3×C4×C3⋊D3—22no—
75C3×C4×C3⋊D3C2311yescharacteristic
76C32×D12—22no—
77C32×D12C2312yes—
78C32×C4×D3—22no—
79C32×C4×D3C2311yescharacteristic
80#6293D411yescharacteristic
81#6292—22no—
82#6292D411yescharacteristic
83#6291—22no—
84#6291D411yescharacteristic
85#6290—22no—
86#6290D411yescharacteristic
87C33×C8C2311yescharacteristic
88C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
89C3×C3⋊(C3⋊C8)C2311yescharacteristic
90C32×C3⋊C8C2311yescharacteristic
91#5769—94no—
92#5769—91no—
93C32×#41—32no—
94#3832—32no—
95#3832—38no—
96#3832—68no—
97#3832C2×D312yescomplemented
98D4×C3⋊D3—32no—
99D4×C3⋊D3—32no—
100D4×C3⋊D3—62no—
101D4×C3⋊D3—62no—
102#3828—31no—
103#3828—62no—
104C2×C4⋊(C3⋊D3)—32no—
105C2×C4⋊(C3⋊D3)—62no—
106C3×#239—38no—
107C3×#239C2×D318yescomplemented
108#3802—38no—
109#3802—68no—
110D3×D12—38no—
111D3×D12—38no—
112D3×D12—68no—
113D3×D12—68no—
114#3798—34no—
115#3798—68no—
116C32×Q16—24no—
117C32×Q16C2×D314yescomplemented
118C32×Q16C2×D311yescharacteristic, complemented
119C32×SD16—32no—
120C32×D8—31no—
121C2×C32×C8—31no—
122#3755—32no—
123#3755—38no—
124#3755—68no—
125#3755C2×D312yescomplemented
126#3753—32no—
127#3753—32no—
128#3748—31no—
129#3745—32no—
130#3745—31no—
131#3745—34no—
132#3745—64no—
133#3745C2×D311yescharacteristic, complemented
134#3743—31no—
135#3743—31no—
136C8×C3⋊D3—31no—
137C8×C3⋊D3—34no—
138C8×C3⋊D3—64no—
139C8×C3⋊D3C2×D311yescharacteristic, complemented
140C3×#215—38no—
141C3×#215C2×D318yescomplemented
142C3×D24—34no—
143C3×D24C2×D314yescomplemented
144C3×D3×C8—34no—
145C3×D3×C8C2×D314yescomplemented
146#3714—38no—
147#3714—38no—
148#3713—38no—
149#3712—34no—
150#3711—34no—
151#3711—34no—
152C2×C3⋊(C3⋊D3)—22no—
153C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3C2×D411yescharacteristic
156C2×C32×D3—22no—
157C2×C32×D3C2×D411yescharacteristic
158D3×C3⋊D3—82no—
159D3×C3⋊D3—82no—
160D3×C3⋊D3—82no—
161C33×C4—22no—
162C33×C4C2411yescharacteristic
163C3⋊(C3⋊Dic3)C2×D411yescharacteristic
164C3×C3⋊Dic3C2×D411yescharacteristic
165C32×Dic3C2×D411yescharacteristic
166#1009—98no—
167#1009—92no—
168C3×#135—34no—
169C3×#135—91no—
170#949—98no—
171#949—92no—
172#919—34no—
173#919—94no—
174#919—91no—
175#919—184no—
176#919C3⋊D311yescharacteristic, complemented
177#914—98no—
178#914—98no—
179#914—92no—
180#914—92no—
181D3×D8—94no—
182D3×D8—94no—
183D3×D8—91no—
184D3×D8—91no—
185C2×D24—94no—
186C2×D24—91no—
187#901—94no—
188#901—91no—
189C22×C3⋊D3—62no—
190C22×C3⋊D3—62no—
191C22×C3⋊D3—62no—
192C2×D32—68no—
193C2×D32—68no—
194C2×D32—68no—
195C32×Q8—28no—
196C32×Q8C22×D318yes—
197C32×Q8C22×D312yes—
198C32×D4—32no—
199C32×D4—62no—
200C2×C32×C4—31no—
201C2×C32×C4—62no—
202#638—31no—
203#638—31no—
204#638—62no—
205#638—62no—
206C4⋊(C3⋊D3)—22no—
207C4⋊(C3⋊D3)—31no—
208C4⋊(C3⋊D3)—32no—
209C4⋊(C3⋊D3)—38no—
210C4⋊(C3⋊D3)—68no—
211C4⋊(C3⋊D3)—62no—
212C4⋊(C3⋊D3)—62no—
213C4⋊(C3⋊D3)—68no—
214C4⋊(C3⋊D3)—128no—
215C4⋊(C3⋊D3)C22×D312yes—
216C4×C3⋊D3—22no—
217C4×C3⋊D3—31no—
218C4×C3⋊D3—34no—
219C4×C3⋊D3—64no—
220C4×C3⋊D3—62no—
221C4×C3⋊D3—68no—
222C4×C3⋊D3—128no—
223C4×C3⋊D3C22×D311yescharacteristic
224#634—31no—
225C3×D12—28no—
226C3×D12—38no—
227C3×D12—68no—
228C3×D12C22×D318yes—
229C3×C4×D3—28no—
230C3×C4×D3—34no—
231C3×C4×D3—68no—
232C3×C4×D3C22×D314yes—
233#627—34no—
234#626—34no—
235#626—34no—
236#626—68no—
237#626—68no—
238#625—34no—
239#625—68no—
240C32×C8—24no—
241C32×C8—31no—
242C32×C8C22×D314yes—
243C32×C8C22×D311yescharacteristic
244C3⋊(C3⋊C8)—31no—
245C3⋊(C3⋊C8)—34no—
246C3⋊(C3⋊C8)—64no—
247C3⋊(C3⋊C8)C22×D311yescharacteristic
248C3×C3⋊C8—34no—
249C3×C3⋊C8C22×D314yes—
250#576—271noSylow
251C2×C33C22×D411yescharacteristic
252C3⋊(C3⋊D3)—21no—
253C3⋊(C3⋊D3)—42no—
254C3×C3⋊D3—21no—
255C3×C3⋊D3—42no—
256C32×D3—21no—
257C32×D3—42no—
258C3×#41—38no—
259C3×#41—92no—
260#239—38no—
261#239—98no—
262#239—92no—
263#239—188no—
264#239C2×C3⋊D312yescomplemented
265D3×D4—98no—
266D3×D4—98no—
267D3×D4—92no—
268D3×D4—92no—
269D3×D4—182no—
270D3×D4—188no—
271D3×D4—182no—
272D3×D4—188no—
273#235—94no—
274#235—91no—
275#235—182no—
276#235—188no—
277C2×D12—98no—
278C2×D12—92no—
279C2×D12—182no—
280C2×D12—188no—
281C3×Q16—24no—
282C3×Q16D3214yescomplemented
283C3×Q16C2×C3⋊D311yescharacteristic, complemented
284C3×SD16—38no—
285C3×SD16—92no—
286C3×D8—34no—
287C3×D8—91no—
288C2×C3×C8—34no—
289C2×C3×C8—91no—
290#215—38no—
291#215—98no—
292#215—92no—
293#215—188no—
294#215C2×C3⋊D312yescomplemented
295#213—98no—
296#213—98no—
297#213—92no—
298#213—92no—
299#208—94no—
300#208—91no—
301D24—34no—
302D24—98no—
303D24—92no—
304D24—94no—
305D24—91no—
306D24—184no—
307D24C2×C3⋊D311yescharacteristic, complemented
308ΓL(1,𝔽25)—94no—
309ΓL(1,𝔽25)—94no—
310ΓL(1,𝔽25)—91no—
311ΓL(1,𝔽25)—91no—
312D3×C8—34no—
313D3×C8—94no—
314D3×C8—91no—
315D3×C8—184no—
316D3×C8C2×C3⋊D311yescharacteristic, complemented
317C22×C32—31no—
318C22×C32—62no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3—31no—
321C2×C3⋊D3—34no—
322C2×C3⋊D3—64no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—68no—
325C2×C3⋊D3—128no—
326C2×C3⋊D3—242no—
327C2×C3⋊D3—242no—
328C2×C3⋊D3—242no—
329C2×C3⋊D3D3×D411yescharacteristic
330C2×C3×D3—28no—
331C2×C3×D3—34no—
332C2×C3×D3—68no—
333C2×C3×D3D3×D414yes—
334D32—248no—
335D32—248no—
336D32—248no—
337C32×C4—22no—
338C32×C4—24no—
339C32×C4—28no—
340C32×C4—31no—
341C32×C4—48no—
342C32×C4C23×D314yes—
343C32×C4C23×D311yescharacteristic
344C3⋊Dic3—31no—
345C3⋊Dic3—34no—
346C3⋊Dic3—64no—
347C3⋊Dic3D3×D411yescharacteristic
348C3×Dic3—34no—
349C3×Dic3D3×D414yes—
350H(5,𝔽2)—272no—
351AGL(1,ℤ/8)—272no—
352AGL(1,ℤ/8)—272no—
353AGL(1,ℤ/8)—272no—
354#135—31no—
355#135—91no—
356#135—271no—
357C2×D8—271no—
358C2×D8—271no—
359C2×D8—271no—
360#131—271no—
361C33#57611yescharacteristic, complemented, Sylow
362C22×D3—182no—
363C22×D3—182no—
364C22×D3—188no—
365C22×D3—188no—
366C22×D3—188no—
367C22×D3—182no—
368C3×Q8—28no—
369C3×Q8C2×D3218yes—
370C3×Q8C22×C3⋊D312yes—
371C3×D4—38no—
372C3×D4—68no—
373C3×D4—92no—
374C3×D4—182no—
375C2×C3×C4—34no—
376C2×C3×C4—68no—
377C2×C3×C4—91no—
378C2×C3×C4—182no—
379#67—94no—
380#67—94no—
381#67—91no—
382#67—91no—
383#67—188no—
384#67—182no—
385#67—188no—
386#67—182no—
387D12—22no—
388D12—38no—
389D12—68no—
390D12—94no—
391D12—91no—
392D12—98no—
393D12—92no—
394D12—188no—
395D12—182no—
396D12—188no—
397D12—188no—
398D12—182no—
399D12—368no—
400D12C22×C3⋊D312yes—
401C4×D3—22no—
402C4×D3—34no—
403C4×D3—68no—
404C4×D3—94no—
405C4×D3—91no—
406C4×D3—184no—
407C4×D3—182no—
408C4×D3—188no—
409C4×D3—368no—
410C4×D3C22×C3⋊D311yescharacteristic
411Dic6—94no—
412Dic6—91no—
413C3×C8—24no—
414C3×C8—34no—
415C3×C8—91no—
416C3×C8C2×D3214yes—
417C3×C8C22×C3⋊D311yescharacteristic
418C3⋊C8—34no—
419C3⋊C8—94no—
420C3⋊C8—91no—
421C3⋊C8—184no—
422C3⋊C8C22×C3⋊D311yescharacteristic
423C2×C32—24no—
424C2×C32—61no—
425C2×C32—122no—
426C2×C32C2×D3×D414yes—
427C2×C32C2×D3×D411yescharacteristic
428C3⋊D3—21no—
429C3⋊D3—42no—
430C3⋊D3—61no—
431C3⋊D3—64no—
432C3⋊D3—124no—
433C3⋊D3—122no—
434C3⋊D3—128no—
435C3⋊D3—248no—
436C3×D3—24no—
437C3×D3—48no—
438C3×D3—64no—
439C3×D3—128no—
440#41—32no—
441#41—92no—
442#41—271no—
443#41—272no—
444#41—542no—
445C2×D4—272no—
446C2×D4—272no—
447C2×D4—272no—
448C2×D4—542no—
449C2×D4—542no—
450C2×D4—542no—
451Q16D3×C3⋊D311yescharacteristic, complemented
452SD16—32no—
453SD16—92no—
454SD16—272no—
455D8—31no—
456D8—91no—
457D8—272no—
458D8—272no—
459D8—272no—
460D8—271no—
461M16—271no—
462M16—271no—
463M16—271no—
464C2×C8—31no—
465C2×C8—91no—
466C2×C8—271no—
467C22×C3—34no—
468C22×C3—68no—
469C22×C3—91no—
470C22×C3—182no—
471C2×D3—22no—
472C2×D3—34no—
473C2×D3—68no—
474C2×D3—94no—
475C2×D3—91no—
476C2×D3—184no—
477C2×D3—188no—
478C2×D3—182no—
479C2×D3—368no—
480C2×D3—722no—
481C2×D3—722no—
482C2×D3—722no—
483C2×D3—728no—
484C2×D3—728no—
485C2×D3—728no—
486C2×D3D4×C3⋊D311yescharacteristic
487C3×C4—28no—
488C3×C4—22no—
489C3×C4—24no—
490C3×C4—34no—
491C3×C4—48no—
492C3×C4—91no—
493C3×C4C22×D3214yes—
494C3×C4C23×C3⋊D311yescharacteristic
495Dic3—34no—
496Dic3—94no—
497Dic3—91no—
498Dic3—184no—
499Dic3D4×C3⋊D311yescharacteristic
500C32—24no—
501C32#576914yescomplemented
502C32#576911yescharacteristic, complemented
503C23—542no—
504C23—542no—
505C23—542no—
506Q8—271no—
507Q8C2×D3×C3⋊D312yes—
508D4—32no—
509D4—62no—
510D4—92no—
511D4—182no—
512D4—271no—
513D4—271no—
514D4—271no—
515D4—272no—
516D4—542no—
517D4—542no—
518D4—542no—
519D4—542no—
520C2×C4—31no—
521C2×C4—62no—
522C2×C4—91no—
523C2×C4—182no—
524C2×C4—271no—
525C2×C4—542no—
526C8—31no—
527C8—91no—
528C8—271no—
529C8C2×D3×C3⋊D311yescharacteristic
530C2×C3—24no—
531C2×C3—64no—
532C2×C3—128no—
533C2×C3—181no—
534C2×C3—362no—
535C2×C3D32×D414yes—
536C2×C3C2×D4×C3⋊D311yescharacteristic
537D3—21no—
538D3—42no—
539D3—64no—
540D3—128no—
541D3—181no—
542D3—184no—
543D3—364no—
544D3—362no—
545D3—368no—
546D3—728no—
547C22—31no—
548C22—62no—
549C22—91no—
550C22—182no—
551C22—271no—
552C22—542no—
553C22—2162no—
554C22—2162no—
555C22—2162no—
556C4—22no—
557C4—31no—
558C4—91no—
559C4—271no—
560C4C22×D3×C3⋊D311yescharacteristic
561C3—24no—
562C3#1059477114yescomplemented
563C3#1059624011yescharacteristic, complemented
564C2—61no—
565C2—122no—
566C2—181no—
567C2—362no—
568C2—541no—
569C2—1082no—
570C2D3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2D3×D4×C3⋊D3
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C22×D4
hypercenterQ16D3×C3⋊D3
hypocenterC33#576
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. #11724597C2semidirect
2. #11723960C2semidirect
3. C3×#64128C2semidirect
4. C32×#919C2semidirect
5. #11723572C2semidirect
6. #11723570C2semidirect
7. #11723568C2semidirect
8. #11723331C2semidirect
9. #11723322C2semidirect
10. C3⋊D3×D24C2semidirect
11. D3×#3745C2semidirect
14. #89080C22semidirect
15. C3×#3832C22semidirect
16. C32×#239C22semidirect
18. #89026C22semidirect
20. C3⋊D3×D12C22semidirect
22. D3×C4⋊(C3⋊D3)C22semidirect
25. C33×Q16C22semidirect
26. #88862C22semidirect
27. #88852C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
29. C3×#3755C22semidirect
30. C3×#3745C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C32×#215C22semidirect
33. C32×D24C22semidirect
34. C32×D3×C8C22semidirect
35. #88791C22semidirect
36. #88790C22semidirect
37. #88789C22semidirect
38. #88788C22semidirect
39. #88787C22semidirect
40. #88786C22semidirect
48. #64128D3semidirect
56. C3×#919D3semidirect
97. #3832C2×D3semidirect
107. C3×#239C2×D3semidirect
117. C32×Q16C2×D3semidirect
118. C32×Q16C2×D3semidirect
125. #3755C2×D3semidirect
133. #3745C2×D3semidirect
139. C8×C3⋊D3C2×D3semidirect
141. C3×#215C2×D3semidirect
143. C3×D24C2×D3semidirect
145. C3×D3×C8C2×D3semidirect
176. #919C3⋊D3semidirect
264. #239C2×C3⋊D3semidirect
282. C3×Q16D32semidirect
283. C3×Q16C2×C3⋊D3semidirect
294. #215C2×C3⋊D3semidirect
307. D24C2×C3⋊D3semidirect
316. D3×C8C2×C3⋊D3semidirect
361. C33#576semidirect
451. Q16D3×C3⋊D3semidirect
501. C32#5769semidirect
502. C32#5769semidirect
562. C3#10594771semidirect
563. C3#10596240semidirect

Maximal subgroups

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

subgroupindex
#117245972
#117239602
C3×#641282
C32×#9192
#117235722
#117235702
#117235682
#117233312
#117233222
C3⋊D3×D242
D3×#37452
#105962403
#105947713