Finite Group Databasebeta

← #13178007

Subgroups — #13178007

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1009C211yesmaximal, characteristic, complemented
2#11724562C211yesmaximal, characteristic, complemented
3#11723991C211yesmaximal, characteristic, complemented
4C3×#64160C211yesmaximal, characteristic, complemented
5#11723582C212yesmaximal, complemented
6#11723574C212yesmaximal, complemented
7#11723557C212yesmaximal, complemented
8D3×#3754C212yesmaximal, complemented
9#11723378C211yesmaximal, characteristic, complemented
10C2×#88792C211yesmaximal, characteristic, complemented
11#11723350C211yesmaximal, characteristic, complemented
12#10596307—31nomaximal
13#10595464—34nomaximal
14C33×#41C2211yescharacteristic
15#89076C2211yescharacteristic, complemented
16C2×C3×#634C2211yescharacteristic, complemented
17C32×#239C2212yes—
18C32×D3×D4—22no—
19C32×D3×D4C2212yes—
20C32×#235—21no—
21C32×#235C2211yescharacteristic
22C2×C32×D12—21no—
23C2×C32×D12C2211yescharacteristic
24#89028—22no—
25#89022C2212yescomplemented
26#89019—21no—
27#89015—21no—
28D3×#634C2212yescomplemented
29#88863C2211yescharacteristic, complemented
30#88862C2211yescharacteristic, complemented
31#88861C2211yescharacteristic, complemented
32#88860C2211yescharacteristic, complemented
33C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
34C3×#3756C2212yescomplemented
35C3×#3754C2212yescomplemented
36C3×#3748C2211yescharacteristic, complemented
37#88794C2212yescomplemented
38#88792C2212yescomplemented
39#88792C2212yescomplemented
40#88790C2212yescomplemented
41#88786C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43C32×H(5,𝔽2)—31no—
44#64367—31no—
45C3×#1009D314yescomplemented
46#64297—34no—
47#64160—32no—
48#64160D311yescharacteristic, complemented
49#64159—31no—
50#64159—34no—
51#64159—32no—
52#64159—64no—
53#64157—31no—
54C2×#3754—31no—
55C2×#3754—32no—
56#64141—32no—
57#64141—31no—
58C3×#951—34no—
59#63949—38no—
60#63943—38no—
61#63934—38no—
62D3×#214—38no—
63#63830—34no—
64C2×#3715—34no—
65#63815—34no—
66C22×C32×D3—21no—
67C22×C32×D3—22no—
68C33×Q8C2311yescharacteristic
69C33×D4—21no—
70C33×D4C2311yescharacteristic
71C2×C33×C4—21no—
72C2×C33×C4C2311yescharacteristic
73#6312—21no—
74C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
75C4×C3⋊(C3⋊D3)—21no—
76#6308C2311yescharacteristic
77C2×C3×C3⋊Dic3—21no—
78C3×#634—21no—
79C3×#634C2312yes—
80C32×#67—22no—
81C32×#67D412yes—
82C32×D12—22no—
83C32×D12—21no—
84C32×D12D411yescharacteristic
85C32×D12C2312yes—
86C32×C4×D3—22no—
87C32×C4×D3C2312yes—
88C32×Dic6D411yescharacteristic
89#6293—22no—
90#6291—22no—
91#6289—22no—
92D3×C3⋊Dic3—22no—
93C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
94C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
95C3×C3⋊(C3⋊C8)C2312yes—
96#5836—94no—
97#5778—91no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41—32no—
101C32×#41—61no—
102C32×#41C2×D314yes—
103C32×#41C2×D311yescharacteristic
104C2×C32×D4—32no—
105C2×C32×D4—31no—
106C2×C32×D4—61no—
107C2×C32×D4—62no—
108Q8×C3⋊D3—61no—
109#3830—61no—
110#3830—62no—
111#3828—31no—
112#3828—34no—
113#3828—32no—
114#3828—64no—
115C2×#634—32no—
116C2×#634C2×D311yescharacteristic, complemented
117C3×#239C2×D318yes—
118C3×D3×D4—28no—
119C3×D3×D4C2×D318yes—
120C3×#235—24no—
121C3×#235C2×D314yes—
122C2×C3×D12—24no—
123C2×C3×D12C2×D314yes—
124C2×C3×Dic6—34no—
125#3804—68no—
126#3799—38no—
127#3797—64no—
128#3796—64no—
129D3×Dic6—38no—
130#3756—31no—
131#3756—34no—
132#3756—64no—
133#3756C2×D312yescomplemented
134#3755—31no—
135#3755—34no—
136#3755—64no—
137#3754—31no—
138#3754—34no—
139#3754—32no—
140#3754—32no—
141#3754—32no—
142#3754—64no—
143#3754C2×D312yescomplemented
144#3753—31no—
145#3753—34no—
146#3753—32no—
147#3753—64no—
148#3748—32no—
149#3748C2×D311yescharacteristic, complemented
150C2×C3⋊(C3⋊C8)—31no—
151C2×C3⋊(C3⋊C8)—34no—
152C2×C3⋊(C3⋊C8)—32no—
153C2×C3⋊(C3⋊C8)—64no—
154C3×#216—38no—
155C3×#214—38no—
156C3×#208—34no—
157#3716—38no—
158#3715—38no—
159#3715—38no—
160#3714—38no—
161#3711—38no—
162D3×C3⋊C8—38no—
163C22×C33—21no—
164C22×C33C2×D411yescharacteristic
165C2×C3⋊(C3⋊D3)—21no—
166C2×C32×D3—22no—
167C2×C32×D3—82no—
168C2×C32×D3—81no—
169C2×C32×D3C2×D412yes—
170C33×C4—21no—
171C33×C4C2×D411yescharacteristic
172C33×C4C2411yescharacteristic
173C3⋊(C3⋊Dic3)—21no—
174C3×C3⋊Dic3—22no—
175C32×Dic3C2×D412yes—
176C3×H(5,𝔽2)—34no—
177#1010—94no—
178#1009C3⋊D311yescharacteristic, complemented
179#1007—91no—
180C3×#137—91no—
181#951—34no—
182#951—98no—
183#950—98no—
184#950—94no—
185#950—91no—
186#950—184no—
187#948—94no—
188C2×#215—91no—
189C2×#214—98no—
190C2×#214—94no—
191#932—94no—
192#932—98no—
193#919—92no—
194#918—92no—
195#914—92no—
196D3×SD16—92no—
197#908—91no—
198#907—91no—
199C23×C32—62no—
200C23×C32—61no—
201C22×C3×D3—24no—
202C22×C3×D3—28no—
203C32×Q8—24no—
204C32×Q8—31no—
205C32×Q8C22×D314yes—
206C32×Q8C22×D311yescharacteristic
207C32×D4—21no—
208C32×D4—24no—
209C32×D4—24no—
210C32×D4—32no—
211C32×D4—31no—
212C32×D4—32no—
213C32×D4—44no—
214C32×D4—62no—
215C32×D4—62no—
216C32×D4—61no—
217C32×D4C22×D314yes—
218C32×D4C22×D311yescharacteristic
219C2×C32×C4—21no—
220C2×C32×C4—24no—
221C2×C32×C4—24no—
222C2×C32×C4—32no—
223C2×C32×C4—44no—
224C2×C32×C4—62no—
225C2×C32×C4C22×D314yes—
226C2×C32×C4C22×D311yescharacteristic
227#638—62no—
228#638—61no—
229#638—64no—
230#638—124no—
231C2×C3⋊Dic3—21no—
232C2×C3⋊Dic3—62no—
233C4⋊(C3⋊D3)—31no—
234C4⋊(C3⋊D3)—34no—
235C4⋊(C3⋊D3)—64no—
236C4×C3⋊D3—62no—
237C4×C3⋊D3—61no—
238C4×C3⋊D3—64no—
239C4×C3⋊D3—124no—
240#634—21no—
241#634—31no—
242#634—34no—
243#634—64no—
244#634—62no—
245#634C22×D312yes—
246C3×#67—28no—
247C3×#67#6718yes—
248C2×C3×Dic3—64no—
249C3×D12—28no—
250C3×D12—24no—
251C3×D12#6714yes—
252C3×D12C22×D318yes—
253C3×C4×D3—28no—
254C3×C4×D3C22×D318yes—
255C3×Dic6—38no—
256C3×Dic6—64no—
257C3×Dic6#6714yes—
258#627—68no—
259#626—68no—
260#624—68no—
261D3×Dic3—68no—
262C3⋊(C3⋊C8)—31no—
263C3⋊(C3⋊C8)—31no—
264C3⋊(C3⋊C8)—34no—
265C3⋊(C3⋊C8)—34no—
266C3⋊(C3⋊C8)—64no—
267C3⋊(C3⋊C8)—64no—
268C3⋊(C3⋊C8)C22×D312yes—
269C3×C3⋊C8—38no—
270#577—271noSylow
271C2×C33—21no—
272C2×C33—41no—
273C2×C33C22×D411yescharacteristic
274C3⋊(C3⋊D3)—41no—
275C32×D3—22no—
276C32×D3—42no—
277C3×#41—24no—
278C3×#41—34no—
279C3×#41—38no—
280C3×#41—64no—
281C3×#41D3214yes—
282C3×#41C2×C3⋊D311yescharacteristic
283C2×C3×Q8—91no—
284C2×C3×D4—38no—
285C2×C3×D4—34no—
286C2×C3×D4—64no—
287C2×C3×D4—68no—
288#239—92no—
289#239—181no—
290#239C2×C3⋊D312yes—
291D3×Q8—92no—
292D3×Q8—184no—
293D3×Q8—181no—
294#237—184no—
295#237—188no—
296D3×D4—22no—
297D3×D4C2×C3⋊D312yes—
298#235—21no—
299#235—98no—
300#235—94no—
301#235—91no—
302#235—184no—
303#235—182no—
304#235C2×C3⋊D311yescharacteristic
305C2×D12—21no—
306C2×D12C2×C3⋊D311yescharacteristic
307C2×Dic6—34no—
308C2×Dic6—98no—
309C3×Q16—92no—
310C3×SD16—92no—
311C3×M16—91no—
312#216—38no—
313#216—94no—
314#216—91no—
315#216—184no—
316#215—94no—
317#215—91no—
318#215—92no—
319#215—92no—
320#215—184no—
321#214—38no—
322#214—98no—
323#214—98no—
324#214—98no—
325#214—94no—
326#214—91no—
327#214—184no—
328#213—98no—
329#213—94no—
330#213—91no—
331#213—184no—
332#208—34no—
333#208—98no—
334C2×C3⋊C8—98no—
335C2×C3⋊C8—94no—
336C2×C3⋊C8—91no—
337C2×C3⋊C8—184no—
338D24—92no—
339#204—92no—
340ΓL(1,𝔽25)—92no—
341D3×C8—92no—
342C22×C32—21no—
343C22×C32—24no—
344C22×C32—24no—
345C22×C32—32no—
346C22×C32—44no—
347C22×C32—62no—
348C22×C32—241no—
349C22×C32—242no—
350C22×C32D3×D411yescharacteristic
351C22×C32C2×#6714yes—
352C2×C3⋊D3—61no—
353C2×C3⋊D3—64no—
354C2×C3⋊D3—124no—
355C2×C3×D3—28no—
356C2×C3×D3—88no—
357C2×C3×D3—84no—
358C2×C3×D3C2×#6718yes—
359C32×C4—21no—
360C32×C4—24no—
361C32×C4—24no—
362C32×C4—24no—
363C32×C4—32no—
364C32×C4—44no—
365C32×C4D3×D411yescharacteristic
366C32×C4C2×#6714yes—
367C32×C4C23×D314yes—
368C32×C4C23×D311yescharacteristic
369C3⋊Dic3—22no—
370C3⋊Dic3—61no—
371C3⋊Dic3—64no—
372C3⋊Dic3—124no—
373C3×Dic3—68no—
374C3×Dic3C2×#6718yes—
3752−1+4—271no—
376H(5,𝔽2)—31no—
377#137—91no—
378#137—272no—
379AGL(1,ℤ/8)—271no—
380AGL(1,ℤ/8)—272no—
381#135—272no—
382#135—271no—
383C2×SD16—272no—
384C2×SD16—271no—
385#131—271no—
386C33#57711yescharacteristic, complemented, Sylow
387C23×C3—68no—
388C23×C3—64no—
389C22×D3—21no—
390C22×D3—22no—
391C3×Q8—24no—
392C3×Q8—34no—
393C3×Q8—92no—
394C3×Q8—181no—
395C3×Q8C2×D3214yes—
396C3×Q8C22×C3⋊D311yescharacteristic
397C3×D4—24no—
398C3×D4—21no—
399C3×D4—24no—
400C3×D4—38no—
401C3×D4—38no—
402C3×D4—34no—
403C3×D4—44no—
404C3×D4—68no—
405C3×D4—68no—
406C3×D4—64no—
407C3×D4C2×D3214yes—
408C3×D4C22×C3⋊D311yescharacteristic
409C2×C3×C4—24no—
410C2×C3×C4—21no—
411C2×C3×C4—24no—
412C2×C3×C4—38no—
413C2×C3×C4—44no—
414C2×C3×C4—68no—
415C2×C3×C4—181no—
416C2×C3×C4C2×D3214yes—
417C2×C3×C4C22×C3⋊D311yescharacteristic
418#67—22no—
419#67—188no—
420#67—184no—
421#67—181no—
422#67—364no—
423#67#63812yes—
424C2×Dic3—64no—
425C2×Dic3—188no—
426D12—22no—
427D12—21no—
428D12—94no—
429D12—91no—
430D12—184no—
431D12—182no—
432D12#63811yescharacteristic
433D12C22×C3⋊D312yes—
434C4×D3—22no—
435C4×D3—188no—
436C4×D3—182no—
437C4×D3—184no—
438C4×D3—181no—
439C4×D3—182no—
440C4×D3—364no—
441C4×D3C22×C3⋊D312yes—
442Dic6—38no—
443Dic6—64no—
444Dic6—94no—
445Dic6—91no—
446Dic6—184no—
447Dic6—188no—
448Dic6—182no—
449Dic6#63811yescharacteristic
450C3×C8—92no—
451C3⋊C8—38no—
452C3⋊C8—94no—
453C3⋊C8—94no—
454C3⋊C8—91no—
455C3⋊C8—91no—
456C3⋊C8—184no—
457C3⋊C8—184no—
458C2×C32—21no—
459C2×C32—24no—
460C2×C32—24no—
461C2×C32—44no—
462C2×C32—41no—
463C2×C32—44no—
464C2×C32—62no—
465C2×C32—84no—
466C2×C32—122no—
467C2×C32C2×D3×D411yescharacteristic
468C2×C32C22×#6714yes—
469C3⋊D3—121no—
470C3⋊D3—124no—
471C3⋊D3—244no—
472C3×D3—28no—
473C3×D3—48no—
474#41—31no—
475#41—32no—
476#41—61no—
477#41—272no—
478#41—271no—
479#41—542no—
480#41—541no—
481#41D3×C3⋊D311yescharacteristic
482C2×Q8—91no—
483C2×Q8—272no—
484C2×Q8—541no—
485C2×D4—32no—
486C2×D4—31no—
487C2×D4—61no—
488C2×D4—62no—
489Q16—92no—
490Q16—271no—
491SD16—92no—
492SD16—272no—
493SD16—272no—
494SD16—272no—
495SD16—271no—
496SD16—271no—
497D8—272no—
498D8—271no—
499M16—91no—
500M16—272no—
501C2×C8—272no—
502C2×C8—271no—
503C22×C3—24no—
504C22×C3—21no—
505C22×C3—24no—
506C22×C3—38no—
507C22×C3—44no—
508C22×C3—68no—
509C22×C3—244no—
510C22×C3—248no—
511C22×C3D3×#6714yes—
512C22×C3C2×#63811yescharacteristic
513C2×D3—22no—
514C2×D3—82no—
515C2×D3—81no—
516C2×D3—184no—
517C2×D3—181no—
518C2×D3—364no—
519C2×D3C2×#63812yes—
520C3×C4—24no—
521C3×C4—21no—
522C3×C4—24no—
523C3×C4—24no—
524C3×C4—38no—
525C3×C4—44no—
526C3×C4—182no—
527C3×C4D3×#6714yes—
528C3×C4C2×#63811yescharacteristic
529C3×C4C22×D3214yes—
530C3×C4C23×C3⋊D311yescharacteristic
531Dic3—68no—
532Dic3—184no—
533Dic3—181no—
534Dic3—364no—
535Dic3C2×#63812yes—
536C32—24no—
537C32#577811yescharacteristic, complemented
538C32#583614yescomplemented
539C23—62no—
540C23—61no—
541Q8—31no—
542Q8—92no—
543Q8—181no—
544Q8—271no—
545Q8—542no—
546Q8C2×D3×C3⋊D311yescharacteristic
547D4—21no—
548D4—32no—
549D4—32no—
550D4—31no—
551D4—62no—
552D4—62no—
553D4—61no—
554D4—271no—
555D4—542no—
556D4—541no—
557D4C2×D3×C3⋊D311yescharacteristic
558C2×C4—21no—
559C2×C4—32no—
560C2×C4—62no—
561C2×C4—181no—
562C2×C4—542no—
563C2×C4—542no—
564C2×C4—541no—
565C2×C4C2×D3×C3⋊D311yescharacteristic
566C8—92no—
567C8—271no—
568C8—271no—
569C2×C3—24no—
570C2×C3—21no—
571C2×C3—24no—
572C2×C3—44no—
573C2×C3—44no—
574C2×C3—41no—
575C2×C3—68no—
576C2×C3—84no—
577C2×C3—128no—
578C2×C3C2×D3×#6714yes—
579C2×C3C22×#63811yescharacteristic
580D3—22no—
581D3—42no—
582D3—364no—
583D3—361no—
584D3—724no—
585C22—21no—
586C22—32no—
587C22—62no—
588C22—241no—
589C22—242no—
590C22—541no—
591C22D3×#63811yescharacteristic
592C4—21no—
593C4—32no—
594C4—182no—
595C4—541no—
596C4D3×#63811yescharacteristic
597C4C22×D3×C3⋊D311yescharacteristic
598C3—24no—
599C3#1059546414yescomplemented
600C3#1059630711yescharacteristic, complemented
601C2—21no—
602C2—41no—
603C2—62no—
604C2—122no—
605C2—1081no—
606C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC2C2×D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#577
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C32×#1009C2semidirect
2. #11724562C2semidirect
3. #11723991C2semidirect
4. C3×#64160C2semidirect
5. #11723582C2semidirect
6. #11723574C2semidirect
7. #11723557C2semidirect
8. D3×#3754C2semidirect
9. #11723378C2semidirect
10. C2×#88792C2semidirect
11. #11723350C2semidirect
15. #89076C22semidirect
16. C2×C3×#634C22semidirect
25. #89022C22semidirect
28. D3×#634C22semidirect
29. #88863C22semidirect
30. #88862C22semidirect
31. #88861C22semidirect
32. #88860C22semidirect
33. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
34. C3×#3756C22semidirect
35. C3×#3754C22semidirect
36. C3×#3748C22semidirect
37. #88794C22semidirect
38. #88792C22semidirect
39. #88792C22semidirect
40. #88790C22semidirect
41. #88786C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
45. C3×#1009D3semidirect
48. #64160D3semidirect
116. C2×#634C2×D3semidirect
133. #3756C2×D3semidirect
143. #3754C2×D3semidirect
149. #3748C2×D3semidirect
178. #1009C3⋊D3semidirect
386. C33#577semidirect
537. C32#5778semidirect
538. C32#5836semidirect
599. C3#10595464semidirect
600. C3#10596307semidirect

Maximal subgroups

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

subgroupindex
C32×#10092
#117245622
#117239912
C3×#641602
#117235822
#117235742
#117235572
D3×#37542
#117233782
C2×#887922
#117233502
#105963073
#105954643