Finite Group Databasebeta

← #13178014

Subgroups — #13178014

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1009C211yesmaximal, characteristic, complemented
2#11724563C211yesmaximal, characteristic, complemented
3#11723992C211yesmaximal, characteristic, complemented
4C3×#64159C211yesmaximal, characteristic, complemented
5#11723580C212yesmaximal, complemented
6#11723574C212yesmaximal, complemented
7#11723555C212yesmaximal, complemented
8D3×#3754C212yesmaximal, complemented
9C2×#88794C211yesmaximal, characteristic, complemented
10#11723364C211yesmaximal, characteristic, complemented
11#11723349C211yesmaximal, characteristic, complemented
12#10596307—31nomaximal
13#10595466—34nomaximal
14C33×#41C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C3×#3828C2211yescharacteristic, complemented
17C32×#239C2212yes—
18C32×D3×D4—22no—
19C32×D3×D4C2212yes—
20C32×#235—21no—
21C32×#235C2211yescharacteristic
22C2×C32×D12—21no—
23C2×C32×D12C2211yescharacteristic
24#89029—22no—
25#89021C2212yescomplemented
26C3⋊D3×Dic6—21no—
27#89014—21no—
28D3×#634C2212yescomplemented
29#88863C2212yescomplemented
30#88861C2212yescomplemented
31#88855C2211yescharacteristic, complemented
32C3×#3756C2211yescharacteristic, complemented
33C3×#3755C2211yescharacteristic, complemented
34C3×#3754C2211yescharacteristic, complemented
35C3×#3753C2211yescharacteristic, complemented
36C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
37#88794C2212yescomplemented
38#88794C2212yescomplemented
39#88792C2212yescomplemented
40#88789C2212yescomplemented
41#88786C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43C32×H(5,𝔽2)—31no—
44#64367—31no—
45C3×#1009D314yescomplemented
46#64298—34no—
47#64160—31no—
48#64160—34no—
49#64160—32no—
50#64160—64no—
51#64159—32no—
52#64159D311yescharacteristic, complemented
53#64157—31no—
54C2×#3754—32no—
55C2×#3754—31no—
56#64141—31no—
57#64141—32no—
58C3×#950—34no—
59#63947—38no—
60#63943—38no—
61#63932—38no—
62D3×#214—38no—
63C2×#3716—34no—
64#63823—34no—
65#63814—34no—
66C22×C32×D3—21no—
67C22×C32×D3—22no—
68C33×Q8C2311yescharacteristic
69C33×D4—21no—
70C33×D4C2311yescharacteristic
71C2×C33×C4—21no—
72C2×C33×C4C2311yescharacteristic
73C2×C3⋊(C3⋊Dic3)—21no—
74#6308—21no—
75#6308C2312yes—
76C3×#638—21no—
77C3×C4⋊(C3⋊D3)C2311yescharacteristic
78C3×C4×C3⋊D3—21no—
79C3×#634C2311yescharacteristic
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#6290—22no—
91Dic3×C3⋊D3—22no—
92D3×C3⋊Dic3—22no—
93C3⋊(C3⋊(C3⋊C8))C2312yes—
94C3×C3⋊(C3⋊C8)C2311yescharacteristic
95C3×C3⋊(C3⋊C8)C2311yescharacteristic
96#5836—94no—
97#5770—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—32no—
112#3828C2×D311yescharacteristic, complemented
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—32no—
116C2×#634—64no—
117C3×#239C2×D318yes—
118C3×D3×D4—28no—
119C3×D3×D4C2×D318yes—
120C3×#235—24no—
121C3×#235—34no—
122C3×#235C2×D314yes—
123C2×C3×D12—24no—
124C2×C3×D12C2×D314yes—
125#3805—68no—
126#3795—38no—
127#3795—64no—
128D3×Dic6—38no—
129D3×Dic6—64no—
130#3756—32no—
131#3756—38no—
132#3756—68no—
133#3756C2×D311yescharacteristic, complemented
134#3755C2×D311yescharacteristic, complemented
135#3754—32no—
136#3754—38no—
137#3754—32no—
138#3754—32no—
139#3754—32no—
140#3754—68no—
141#3754C2×D311yescharacteristic, complemented
142#3753—32no—
143#3753C2×D311yescharacteristic, complemented
144#3748—31no—
145#3748—34no—
146#3748—32no—
147#3748—64no—
148C2×C3⋊(C3⋊C8)—32no—
149C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
150C3×#216—34no—
151C3×#215—34no—
152C3×#214—34no—
153C3×#213—34no—
154C2×C3×C3⋊C8—34no—
155#3716—38no—
156#3716—38no—
157#3715—38no—
158#3713—38no—
159#3711—38no—
160D3×C3⋊C8—38no—
161C22×C33—21no—
162C22×C33C2×D411yescharacteristic
163C2×C3×C3⋊D3—21no—
164C2×C32×D3—22no—
165C2×C32×D3—82no—
166C2×C32×D3—81no—
167C2×C32×D3C2×D412yes—
168C33×C4—21no—
169C33×C4C2×D411yescharacteristic
170C33×C4C2411yescharacteristic
171C3⋊(C3⋊Dic3)—22no—
172C3×C3⋊Dic3—21no—
173C32×Dic3C2×D412yes—
174C3×H(5,𝔽2)—34no—
175#1010—94no—
176#1010—91no—
177#1009C3⋊D311yescharacteristic, complemented
178C3×#135—91no—
179#951—98no—
180#951—94no—
181#951—91no—
182#951—184no—
183#950—34no—
184#950—98no—
185#949—91no—
186#948—94no—
187C2×#214—94no—
188C2×#214—98no—
189#932—98no—
190#932—94no—
191#918—92no—
192#916—92no—
193D3×SD16—92no—
194#911—92no—
195C2×#204—91no—
196#901—91no—
197C23×C32—62no—
198C23×C32—61no—
199C22×C3×D3—24no—
200C22×C3×D3—28no—
201C32×Q8—24no—
202C32×Q8—31no—
203C32×Q8C22×D314yes—
204C32×Q8C22×D311yescharacteristic
205C32×D4—21no—
206C32×D4—24no—
207C32×D4—24no—
208C32×D4—32no—
209C32×D4—31no—
210C32×D4—32no—
211C32×D4—44no—
212C32×D4—62no—
213C32×D4—62no—
214C32×D4—61no—
215C32×D4C22×D314yes—
216C32×D4C22×D311yescharacteristic
217C2×C32×C4—21no—
218C2×C32×C4—24no—
219C2×C32×C4—24no—
220C2×C32×C4—32no—
221C2×C32×C4—44no—
222C2×C32×C4—62no—
223C2×C32×C4C22×D314yes—
224C2×C32×C4C22×D311yescharacteristic
225#638—21no—
226#638—62no—
227C2×C3⋊Dic3—62no—
228C2×C3⋊Dic3—61no—
229C2×C3⋊Dic3—64no—
230C2×C3⋊Dic3—124no—
231C4⋊(C3⋊D3)C22×D311yescharacteristic
232C4×C3⋊D3—21no—
233C4×C3⋊D3—62no—
234#634—32no—
235#634—38no—
236#634—68no—
237#634—62no—
238#634—61no—
239#634—64no—
240#634—124no—
241#634C22×D311yescharacteristic
242C3×#67—28no—
243C3×#67—64no—
244C3×#67#6718yes—
245C3×D12—28no—
246C3×D12—24no—
247C3×D12—34no—
248C3×D12#6714yes—
249C3×D12C22×D318yes—
250C3×C4×D3—28no—
251C3×C4×D3—64no—
252C3×C4×D3C22×D318yes—
253C3×Dic6—34no—
254C3×Dic6#6714yes—
255#627—68no—
256#625—68no—
257D3×Dic3—68no—
258D3×Dic3—68no—
259C3⋊(C3⋊C8)—32no—
260C3⋊(C3⋊C8)—38no—
261C3⋊(C3⋊C8)—68no—
262C3⋊(C3⋊C8)C22×D311yescharacteristic
263C3⋊(C3⋊C8)C22×D311yescharacteristic
264C3×C3⋊C8—34no—
265C3×C3⋊C8—34no—
266#577—271noSylow
267C2×C33—21no—
268C2×C33—41no—
269C2×C33C22×D411yescharacteristic
270C3×C3⋊D3—41no—
271C32×D3—22no—
272C32×D3—42no—
273C3×#41—24no—
274C3×#41—34no—
275C3×#41—38no—
276C3×#41—64no—
277C3×#41—91no—
278C3×#41D3214yes—
279C3×#41C2×C3⋊D311yescharacteristic
280C2×C3×D4—38no—
281C2×C3×D4—34no—
282C2×C3×D4—64no—
283C2×C3×D4—68no—
284#239C2×C3⋊D312yes—
285D3×Q8—92no—
286D3×Q8—184no—
287#237—92no—
288#237—184no—
289#237—188no—
290#237—182no—
291D3×D4—22no—
292D3×D4C2×C3⋊D312yes—
293#235—21no—
294#235—34no—
295#235—98no—
296#235—181no—
297#235C2×C3⋊D311yescharacteristic
298C2×D12—21no—
299C2×D12C2×C3⋊D311yescharacteristic
300C2×Dic6—98no—
301C2×Dic6—94no—
302C2×Dic6—91no—
303C2×Dic6—184no—
304C2×Dic6—181no—
305C3×Q16—91no—
306C3×SD16—91no—
307C3×SD16—91no—
308C3×D8—91no—
309C2×C3×C8—91no—
310#216—34no—
311#216—98no—
312#216—92no—
313#216—188no—
314#215—34no—
315#215—92no—
316#214—34no—
317#214—98no—
318#214—98no—
319#214—98no—
320#214—98no—
321#214—92no—
322#214—188no—
323#213—34no—
324#213—98no—
325#213—92no—
326#208—98no—
327#208—94no—
328#208—91no—
329#208—184no—
330C2×C3⋊C8—34no—
331C2×C3⋊C8—98no—
332#204—92no—
333#204—92no—
334ΓL(1,𝔽25)—92no—
335D3×C8—92no—
336C22×C32—21no—
337C22×C32—24no—
338C22×C32—24no—
339C22×C32—32no—
340C22×C32—44no—
341C22×C32—62no—
342C22×C32—241no—
343C22×C32—242no—
344C22×C32D3×D411yescharacteristic
345C22×C32C2×#6714yes—
346C2×C3⋊D3—21no—
347C2×C3×D3—28no—
348C2×C3×D3—64no—
349C2×C3×D3—88no—
350C2×C3×D3—84no—
351C2×C3×D3C2×#6718yes—
352C32×C4—21no—
353C32×C4—24no—
354C32×C4—24no—
355C32×C4—24no—
356C32×C4—32no—
357C32×C4—44no—
358C32×C4D3×D411yescharacteristic
359C32×C4C2×#6714yes—
360C32×C4C23×D314yes—
361C32×C4C23×D311yescharacteristic
362C3⋊Dic3—21no—
363C3⋊Dic3—62no—
364C3⋊Dic3—68no—
365C3⋊Dic3—128no—
366C3×Dic3—64no—
367C3×Dic3C2×#6718yes—
3682−1+4—271no—
369H(5,𝔽2)—31no—
370#137—272no—
371#137—271no—
372AGL(1,ℤ/8)—272no—
373AGL(1,ℤ/8)—271no—
374#135—91no—
375#135—272no—
376C2×SD16—271no—
377C2×SD16—272no—
378#131—271no—
379C33#57711yescharacteristic, complemented, Sylow
380C23×C3—68no—
381C23×C3—64no—
382C22×D3—21no—
383C22×D3—22no—
384C3×Q8—24no—
385C3×Q8—34no—
386C3×Q8—91no—
387C3×Q8C2×D3214yes—
388C3×Q8C22×C3⋊D311yescharacteristic
389C3×D4—24no—
390C3×D4—21no—
391C3×D4—24no—
392C3×D4—38no—
393C3×D4—38no—
394C3×D4—34no—
395C3×D4—44no—
396C3×D4—68no—
397C3×D4—68no—
398C3×D4—64no—
399C3×D4—91no—
400C3×D4—181no—
401C3×D4C2×D3214yes—
402C3×D4C22×C3⋊D311yescharacteristic
403C2×C3×C4—24no—
404C2×C3×C4—21no—
405C2×C3×C4—24no—
406C2×C3×C4—38no—
407C2×C3×C4—44no—
408C2×C3×C4—68no—
409C2×C3×C4—181no—
410C2×C3×C4C2×D3214yes—
411C2×C3×C4C22×C3⋊D311yescharacteristic
412#67—22no—
413#67—64no—
414#67—188no—
415#67—182no—
416#67#63812yes—
417C2×Dic3—188no—
418C2×Dic3—184no—
419C2×Dic3—181no—
420C2×Dic3—182no—
421C2×Dic3—364no—
422D12—22no—
423D12—21no—
424D12—34no—
425D12#63811yescharacteristic
426D12C22×C3⋊D312yes—
427C4×D3—22no—
428C4×D3—64no—
429C4×D3—188no—
430C4×D3—182no—
431C4×D3C22×C3⋊D312yes—
432Dic6—34no—
433Dic6—98no—
434Dic6—92no—
435Dic6—188no—
436Dic6—188no—
437Dic6—184no—
438Dic6—181no—
439Dic6—182no—
440Dic6—364no—
441Dic6#63811yescharacteristic
442C3×C8—91no—
443C3×C8—91no—
444C3⋊C8—34no—
445C3⋊C8—34no—
446C3⋊C8—98no—
447C3⋊C8—92no—
448C3⋊C8—188no—
449C2×C32—21no—
450C2×C32—24no—
451C2×C32—24no—
452C2×C32—44no—
453C2×C32—41no—
454C2×C32—44no—
455C2×C32—62no—
456C2×C32—84no—
457C2×C32—122no—
458C2×C32C2×D3×D411yescharacteristic
459C2×C32C22×#6714yes—
460C3⋊D3—41no—
461C3×D3—28no—
462C3×D3—48no—
463C3×D3—124no—
464#41—31no—
465#41—32no—
466#41—61no—
467#41—91no—
468#41—272no—
469#41—541no—
470#41—542no—
471#41D3×C3⋊D311yescharacteristic
472C2×Q8—272no—
473C2×Q8—271no—
474C2×Q8—541no—
475C2×D4—32no—
476C2×D4—31no—
477C2×D4—61no—
478C2×D4—62no—
479Q16—91no—
480Q16—272no—
481SD16—91no—
482SD16—91no—
483SD16—272no—
484SD16—272no—
485SD16—272no—
486SD16—272no—
487D8—91no—
488D8—272no—
489M16—272no—
490M16—271no—
491C2×C8—91no—
492C2×C8—272no—
493C22×C3—24no—
494C22×C3—21no—
495C22×C3—24no—
496C22×C3—38no—
497C22×C3—44no—
498C22×C3—68no—
499C22×C3—181no—
500C22×C3—244no—
501C22×C3—248no—
502C22×C3D3×#6714yes—
503C22×C3C2×#63811yescharacteristic
504C2×D3—22no—
505C2×D3—64no—
506C2×D3—82no—
507C2×D3—81no—
508C2×D3C2×#63812yes—
509C3×C4—24no—
510C3×C4—21no—
511C3×C4—24no—
512C3×C4—24no—
513C3×C4—38no—
514C3×C4—44no—
515C3×C4—181no—
516C3×C4D3×#6714yes—
517C3×C4C2×#63811yescharacteristic
518C3×C4C22×D3214yes—
519C3×C4C23×C3⋊D311yescharacteristic
520Dic3—64no—
521Dic3—188no—
522Dic3—182no—
523Dic3—368no—
524Dic3C2×#63812yes—
525C32—24no—
526C32#577011yescharacteristic, complemented
527C32#583614yescomplemented
528C23—62no—
529C23—61no—
530Q8—31no—
531Q8—91no—
532Q8—272no—
533Q8—542no—
534Q8—541no—
535Q8C2×D3×C3⋊D311yescharacteristic
536D4—21no—
537D4—32no—
538D4—32no—
539D4—31no—
540D4—62no—
541D4—62no—
542D4—61no—
543D4—91no—
544D4—181no—
545D4—542no—
546D4C2×D3×C3⋊D311yescharacteristic
547C2×C4—21no—
548C2×C4—32no—
549C2×C4—62no—
550C2×C4—181no—
551C2×C4—542no—
552C2×C4—542no—
553C2×C4—541no—
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C8—91no—
556C8—91no—
557C8—272no—
558C2×C3—24no—
559C2×C3—21no—
560C2×C3—24no—
561C2×C3—44no—
562C2×C3—44no—
563C2×C3—41no—
564C2×C3—68no—
565C2×C3—84no—
566C2×C3—128no—
567C2×C3—361no—
568C2×C3C2×D3×#6714yes—
569C2×C3C22×#63811yescharacteristic
570D3—22no—
571D3—42no—
572D3—124no—
573C22—21no—
574C22—32no—
575C22—62no—
576C22—181no—
577C22—241no—
578C22—242no—
579C22D3×#63811yescharacteristic
580C4—21no—
581C4—32no—
582C4—181no—
583C4—542no—
584C4D3×#63811yescharacteristic
585C4C22×D3×C3⋊D311yescharacteristic
586C3—24no—
587C3#1059546614yescomplemented
588C3#1059630711yescharacteristic, complemented
589C2—21no—
590C2—41no—
591C2—62no—
592C2—122no—
593C2—361no—
594C2C2×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. #11724563C2semidirect
3. #11723992C2semidirect
4. C3×#64159C2semidirect
5. #11723580C2semidirect
6. #11723574C2semidirect
7. #11723555C2semidirect
8. D3×#3754C2semidirect
9. C2×#88794C2semidirect
10. #11723364C2semidirect
11. #11723349C2semidirect
15. C2×#6308C22semidirect
16. C3×#3828C22semidirect
25. #89021C22semidirect
28. D3×#634C22semidirect
29. #88863C22semidirect
30. #88861C22semidirect
31. #88855C22semidirect
32. C3×#3756C22semidirect
33. C3×#3755C22semidirect
34. C3×#3754C22semidirect
35. C3×#3753C22semidirect
36. C2×C3×C3⋊(C3⋊C8)C22semidirect
37. #88794C22semidirect
38. #88794C22semidirect
39. #88792C22semidirect
40. #88789C22semidirect
41. #88786C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
45. C3×#1009D3semidirect
52. #64159D3semidirect
112. #3828C2×D3semidirect
133. #3756C2×D3semidirect
134. #3755C2×D3semidirect
141. #3754C2×D3semidirect
143. #3753C2×D3semidirect
149. C2×C3⋊(C3⋊C8)C2×D3semidirect
177. #1009C3⋊D3semidirect
379. C33#577semidirect
526. C32#5770semidirect
527. C32#5836semidirect
587. C3#10595466semidirect
588. C3#10596307semidirect

Maximal subgroups

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

subgroupindex
C32×#10092
#117245632
#117239922
C3×#641592
#117235802
#117235742
#117235552
D3×#37542
C2×#887942
#117233642
#117233492
#105963073
#105954663