Finite Group Databasebeta

← #13151914

Subgroups — #13151914

Order 1728.

834 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
1C3×#63715C211yesmaximal, characteristic, complemented
2#11722465C211yesmaximal, characteristic, complemented
3#11721866C211yesmaximal, characteristic, complemented
4C9×#950C211yesmaximal, characteristic, complemented
5#11721219C212yesmaximal, complemented
6#11721215C212yesmaximal, complemented
7D9×#214C212yesmaximal, complemented
8#11721176C212yesmaximal, complemented
9#11720882C211yesmaximal, characteristic, complemented
10C2×#88415C211yesmaximal, characteristic, complemented
11#11720858C211yesmaximal, characteristic, complemented
12#10595466—31nomaximal
13#10589872—31nomaximal
14C3×C9×#41C2211yescharacteristic
15C2×#6226C2211yescharacteristic, complemented
16C3×#3701C2212yes—
17C3×D4×D9—22no—
18C3×D4×D9C2212yes—
19C9×#235C2211yescharacteristic, complemented
20C3×#3697—21no—
21C3×#3697C2211yescharacteristic
22C2×C3×D36—21no—
23C2×C3×D36C2211yescharacteristic
24#88663—22no—
25#88641—21no—
26#88638C2212yescomplemented
27D3×Dic18—21no—
28D9×Dic6C2212yescomplemented
29#88547C2212yescomplemented
30#88543C2212yescomplemented
31#88532C2211yescharacteristic, complemented
32C9×#216C2211yescharacteristic, complemented
33C9×#215C2211yescharacteristic, complemented
34C9×#214C2211yescharacteristic, complemented
35C9×#213C2211yescharacteristic, complemented
36C2×C9×C3⋊C8C2211yescharacteristic, complemented
37#88425C2212yescomplemented
38#88421C2212yescomplemented
39#88415C2212yescomplemented
40#88415C2212yescomplemented
41#88413C2212yescomplemented
42D9×C3⋊C8C2212yescomplemented
43C3×#1009—31no—
44#64298—31no—
45#64160—31no—
46C3×#950D311yescharacteristic
47#63947—32no—
48#63943—32no—
49#63932—32no—
50D3×#214—32no—
51C2×#3716—31no—
52#63823—31no—
53#63814—31no—
54#63716—31no—
55#63715D311yescharacteristic, complemented
56C9×#135—31no—
57#63512—31no—
58#63511—31no—
59#63511—61no—
60#63480—32no—
61#63478—32no—
62SD16×D9—32no—
63#63473—32no—
64C2×#3664—31no—
65#63464—31no—
66C22×C3×D9—21no—
67C22×C3×D9—22no—
68C3×Q8×C9C2311yescharacteristic
69C3×D4×C9—21no—
70C3×D4×C9C2311yescharacteristic
71C2×C3×C4×C9—21no—
72C2×C3×C4×C9C2311yescharacteristic
73C2×C3⋊Dic9—21no—
74#6226—21no—
75#6226C2312yes—
76C9×#67—21no—
77C3×#611—22no—
78C3×#611D412yes—
79C9×D12C2311yescharacteristic
80C4×D3×C9—21no—
81C3×D36—22no—
82C3×D36—21no—
83C3×D36D411yescharacteristic
84C3×D36C2312yes—
85C3×C4×D9—22no—
86C3×C4×D9C2312yes—
87C9×Dic6C2311yescharacteristic
88C3×Dic18D411yescharacteristic
89#6194—22no—
90D3×Dic9—22no—
91Dic3×D9—22no—
92#6189—22no—
93C3⋊(C9⋊C8)C2312yes—
94C9×C3⋊C8C2311yescharacteristic
95C9×C3⋊C8C2311yescharacteristic
96#5836—91no—
97#5770—91no—
98C32×#41C2×D311yescharacteristic
99C2×#634—31no—
100C3×#239—32no—
101C3×D3×D4—32no—
102C3×D3×D4—62no—
103C3×#235—31no—
104C3×#235—61no—
105C3×#235C2×D311yescharacteristic
106C2×C3×D12—31no—
107C2×C3×D12—61no—
108#3805—62no—
109#3795—32no—
110#3795—61no—
111D3×Dic6—32no—
112D3×Dic6—61no—
113#3756—32no—
114#3754—32no—
115#3748—31no—
116C3×#216C2×D311yescharacteristic
117C3×#215C2×D311yescharacteristic
118C3×#214C2×D311yescharacteristic
119C3×#213C2×D311yescharacteristic
120C2×C3×C3⋊C8C2×D311yescharacteristic
121#3716—32no—
122#3716—32no—
123#3715—32no—
124#3713—32no—
125#3711—32no—
126D3×C3⋊C8—32no—
127C9×#41—21no—
128C9×#41—31no—
129C9×#41C2×D311yescharacteristic
130#3701C2×D312yes—
131Q8×D9—32no—
132#3699—32no—
133#3699—62no—
134D4×D9—22no—
135D4×D9C2×D312yes—
136#3697—21no—
137#3697—61no—
138#3697C2×D311yescharacteristic
139C2×D36—21no—
140C2×D36C2×D311yescharacteristic
141C2×Dic18—31no—
142C2×Dic18—61no—
143C2×Dic18—61no—
144C9×Q16—31no—
145C9×SD16—31no—
146C9×SD16—31no—
147C9×D8—31no—
148C2×C8×C9—31no—
149#3675—32no—
150#3674—32no—
151#3674—62no—
152#3673—32no—
153#3672—32no—
154#3672—62no—
155#3667—31no—
156#3667—61no—
157#3664—32no—
158#3664—32no—
159#3663—32no—
160C8×D9—32no—
161C22×C3×C9—21no—
162C22×C3×C9C2×D411yescharacteristic
163C2×D3×C9—21no—
164C2×C3×D9—22no—
165C2×C3×D9—82no—
166C2×C3×D9—81no—
167C2×C3×D9C2×D412yes—
168C3×C4×C9—21no—
169C3×C4×C9C2×D411yescharacteristic
170C3×C4×C9C2411yescharacteristic
171C3⋊Dic9—22no—
172C9×Dic3—21no—
173C3×Dic9C2×D412yes—
174C3×H(5,𝔽2)—91no—
175#1010—91no—
176#1010—91no—
177#1009—31no—
178C3×#135—31no—
179#951—91no—
180#951—91no—
181#951—92no—
182#951—181no—
183#950—92no—
184#950D911yescharacteristic, complemented
185#949—91no—
186#948—91no—
187C2×#214—92no—
188C2×#214—91no—
189#932—91no—
190#932—92no—
191#918—92no—
192#916—92no—
193D3×SD16—92no—
194#911—92no—
195C2×#204—91no—
196#901—91no—
197C22×C3×D3—62no—
198C22×C3×D3—61no—
199C32×Q8C22×D311yescharacteristic
200C32×D4—21no—
201C32×D4C22×D311yescharacteristic
202C2×C32×C4—21no—
203C2×C32×C4C22×D311yescharacteristic
204C2×C3⋊Dic3—61no—
205#634—32no—
206#634—61no—
207C3×#67—21no—
208C3×#67—32no—
209C3×#67—62no—
210C3×D12—31no—
211C3×D12—32no—
212C3×D12—62no—
213C3×D12—61no—
214C3×D12C22×D311yescharacteristic
215C3×C4×D3—21no—
216C3×C4×D3—32no—
217C3×C4×D3—62no—
218C3×Dic6—31no—
219C3×Dic6C22×D311yescharacteristic
220#627—62no—
221#625—62no—
222D3×Dic3—62no—
223D3×Dic3—62no—
224C22×D9—21no—
225C22×D9—22no—
226C3⋊(C3⋊C8)—32no—
227C3×C3⋊C8C22×D311yescharacteristic
228C3×C3⋊C8C22×D311yescharacteristic
229Q8×C9—21no—
230Q8×C9—31no—
231Q8×C9C22×D311yescharacteristic
232D4×C9—21no—
233D4×C9—21no—
234D4×C9—31no—
235D4×C9—41no—
236D4×C9—61no—
237D4×C9C22×D311yescharacteristic
238C2×C4×C9—21no—
239C2×C4×C9—21no—
240C2×C4×C9—41no—
241C2×C4×C9—61no—
242C2×C4×C9C22×D311yescharacteristic
243#611—22no—
244#611—62no—
245#611#6712yes—
246C2×Dic9—61no—
247C2×Dic9—62no—
248C2×Dic9—121no—
249D36—22no—
250D36—21no—
251D36#6711yescharacteristic
252D36C22×D312yes—
253C4×D9—22no—
254C4×D9—62no—
255C4×D9C22×D312yes—
256Dic18—32no—
257Dic18—62no—
258Dic18—61no—
259Dic18—62no—
260Dic18—121no—
261Dic18#6711yescharacteristic
262C8×C9—31no—
263C8×C9—31no—
264C9⋊C8—32no—
265C9⋊C8—62no—
266#577—271noSylow
267C2×C3×C9—21no—
268C2×C3×C9—41no—
269C2×C3×C9C22×D411yescharacteristic
270D3×C9—41no—
271C3×D9—22no—
272C3×D9—42no—
273C3×#41—21no—
274C3×#41—31no—
275C3×#41—91no—
276C3×#41—92no—
277C3×#41—181no—
278C3×#41C2×D911yescharacteristic
279C3×#41D3211yescharacteristic
280C2×C3×D4—92no—
281C2×C3×D4—91no—
282C2×C3×D4—181no—
283C2×C3×D4—182no—
284#239—32no—
285D3×Q8—92no—
286D3×Q8—181no—
287#237—92no—
288#237—181no—
289#237—182no—
290#237—182no—
291D3×D4—32no—
292D3×D4—62no—
293#235—31no—
294#235—61no—
295#235—92no—
296#235—181no—
297#235C2×D911yescharacteristic, complemented
298C2×D12—31no—
299C2×D12—61no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C2×Dic6—92no—
303C2×Dic6—181no—
304C2×Dic6—181no—
305C3×Q16—31no—
306C3×SD16—31no—
307C3×SD16—31no—
308C3×D8—31no—
309C2×C3×C8—31no—
310#216—92no—
311#216—92no—
312#216—182no—
313#216C2×D911yescharacteristic, complemented
314#215—92no—
315#215C2×D911yescharacteristic, complemented
316#214—92no—
317#214—92no—
318#214—92no—
319#214—92no—
320#214—92no—
321#214—182no—
322#214C2×D911yescharacteristic, complemented
323#213—92no—
324#213—92no—
325#213C2×D911yescharacteristic, complemented
326#208—91no—
327#208—91no—
328#208—92no—
329#208—181no—
330C2×C3⋊C8—92no—
331C2×C3⋊C8C2×D911yescharacteristic, complemented
332#204—92no—
333#204—92no—
334ΓL(1,𝔽25)—92no—
335D3×C8—92no—
336C22×C32—21no—
337C22×C32D3×D411yescharacteristic
338C2×C3×D3—21no—
339C2×C3×D3—32no—
340C2×C3×D3—62no—
341C2×C3×D3—242no—
342C2×C3×D3—241no—
343C32×C4—21no—
344C32×C4D3×D411yescharacteristic
345C32×C4C23×D311yescharacteristic
346C3⋊Dic3—62no—
347C3×Dic3—21no—
348C3×Dic3—32no—
349C22×C9—21no—
350C22×C9—21no—
351C22×C9—41no—
352C22×C9—61no—
353C22×C9C2×#6711yescharacteristic
354C2×D9—22no—
355C2×D9—82no—
356C2×D9—81no—
357C2×D9C2×#6712yes—
358C4×C9—21no—
359C4×C9—21no—
360C4×C9—21no—
361C4×C9—41no—
362C4×C9—61no—
363C4×C9C2×#6711yescharacteristic
364C4×C9C23×D311yescharacteristic
365Dic9—62no—
366Dic9—122no—
367Dic9C2×#6712yes—
3682−1+4—271no—
369H(5,𝔽2)—91no—
370#137—272no—
371#137—271no—
372AGL(1,ℤ/8)—272no—
373AGL(1,ℤ/8)—271no—
374#135—31no—
375#135—272no—
376C2×SD16—271no—
377C2×SD16—272no—
378#131—271no—
379C3×C9#57711yescharacteristic, complemented, Sylow
380C23×C3—182no—
381C23×C3—181no—
382C22×D3—62no—
383C22×D3—61no—
384C3×Q8—21no—
385C3×Q8—31no—
386C3×Q8—91no—
387C3×Q8C22×D911yescharacteristic
388C3×Q8C2×D3211yescharacteristic
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—31no—
393C3×D4—41no—
394C3×D4—61no—
395C3×D4—92no—
396C3×D4—91no—
397C3×D4—92no—
398C3×D4—182no—
399C3×D4—182no—
400C3×D4—181no—
401C3×D4C22×D911yescharacteristic
402C3×D4C2×D3211yescharacteristic
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—41no—
407C2×C3×C4—61no—
408C2×C3×C4—92no—
409C2×C3×C4—182no—
410C2×C3×C4C22×D911yescharacteristic
411C2×C3×C4C2×D3211yescharacteristic
412#67—21no—
413#67—32no—
414#67—62no—
415#67—182no—
416#67—182no—
417C2×Dic3—182no—
418C2×Dic3—182no—
419C2×Dic3—181no—
420C2×Dic3—181no—
421C2×Dic3—361no—
422D12—32no—
423D12—31no—
424D12—62no—
425D12—61no—
426D12C22×D911yescharacteristic
427C4×D3—21no—
428C4×D3—32no—
429C4×D3—62no—
430C4×D3—182no—
431C4×D3—182no—
432Dic6—31no—
433Dic6—92no—
434Dic6—92no—
435Dic6—182no—
436Dic6—182no—
437Dic6—182no—
438Dic6—181no—
439Dic6—181no—
440Dic6—361no—
441Dic6C22×D911yescharacteristic
442C3×C8—31no—
443C3×C8—31no—
444C3⋊C8—92no—
445C3⋊C8—92no—
446C3⋊C8—182no—
447C3⋊C8C22×D911yescharacteristic
448C3⋊C8C22×D911yescharacteristic
449C2×C32—21no—
450C2×C32—41no—
451C2×C32C2×D3×D411yescharacteristic
452C3×D3—41no—
453C3×D3—62no—
454C3×D3—122no—
455C2×C9—21no—
456C2×C9—21no—
457C2×C9—41no—
458C2×C9—41no—
459C2×C9—81no—
460C2×C9—121no—
461C2×C9C22×#6711yescharacteristic
462D9—22no—
463D9—42no—
464#41—31no—
465#41—91no—
466#41—92no—
467#41—181no—
468#41—272no—
469#41—541no—
470#41—542no—
471#41D3×D911yescharacteristic
472C2×Q8—272no—
473C2×Q8—271no—
474C2×Q8—541no—
475C2×D4—92no—
476C2×D4—91no—
477C2×D4—181no—
478C2×D4—182no—
479Q16—31no—
480Q16—272no—
481SD16—31no—
482SD16—31no—
483SD16—272no—
484SD16—272no—
485SD16—272no—
486SD16—272no—
487D8—31no—
488D8—272no—
489M16—272no—
490M16—271no—
491C2×C8—31no—
492C2×C8—272no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—41no—
497C22×C3—61no—
498C22×C3—92no—
499C22×C3—182no—
500C22×C3—721no—
501C22×C3—722no—
502C22×C3D4×D911yescharacteristic
503C22×C3D3×#6711yescharacteristic
504C2×D3—21no—
505C2×D3—32no—
506C2×D3—62no—
507C2×D3—242no—
508C2×D3—241no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—41no—
514C3×C4—61no—
515C3×C4—92no—
516C3×C4D4×D911yescharacteristic
517C3×C4C23×D911yescharacteristic
518C3×C4D3×#6711yescharacteristic
519C3×C4C22×D3211yescharacteristic
520Dic3—21no—
521Dic3—32no—
522Dic3—182no—
523Dic3—182no—
524Dic3—362no—
525C32#577011yescharacteristic
526C9—21no—
527C9#583611yescharacteristic, complemented
528C23—182no—
529C23—181no—
530Q8—31no—
531Q8—91no—
532Q8—272no—
533Q8—542no—
534Q8—541no—
535Q8C2×D3×D911yescharacteristic
536D4—21no—
537D4—31no—
538D4—61no—
539D4—92no—
540D4—92no—
541D4—91no—
542D4—182no—
543D4—182no—
544D4—181no—
545D4—542no—
546D4C2×D3×D911yescharacteristic
547C2×C4—21no—
548C2×C4—61no—
549C2×C4—92no—
550C2×C4—182no—
551C2×C4—542no—
552C2×C4—542no—
553C2×C4—541no—
554C2×C4C2×D3×D911yescharacteristic
555C8—31no—
556C8—31no—
557C8—272no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—81no—
565C2×C3—121no—
566C2×C3—182no—
567C2×C3—362no—
568C2×C3C2×D4×D911yescharacteristic
569C2×C3C2×D3×#6711yescharacteristic
570D3—41no—
571D3—62no—
572D3—122no—
573C22—21no—
574C22—61no—
575C22—92no—
576C22—182no—
577C22—721no—
578C22—722no—
579C22D9×#6711yescharacteristic
580C4—21no—
581C4—61no—
582C4—92no—
583C4—542no—
584C4D9×#6711yescharacteristic
585C4C22×D3×D911yescharacteristic
586C3—21no—
587C3#1058987211yescharacteristic, complemented
588C3#1059546611yescharacteristic
589C2—21no—
590C2—41no—
591C2—121no—
592C2—182no—
593C2—362no—
594C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC2C2×D9×#67
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×#41C22
socleC2×C32C2×D3×D4
hypercenter#41D3×D9
hypocenterC3×C9#577
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#63715C2semidirect
2. #11722465C2semidirect
3. #11721866C2semidirect
4. C9×#950C2semidirect
5. #11721219C2semidirect
6. #11721215C2semidirect
7. D9×#214C2semidirect
8. #11721176C2semidirect
9. #11720882C2semidirect
10. C2×#88415C2semidirect
11. #11720858C2semidirect
15. C2×#6226C22semidirect
19. C9×#235C22semidirect
26. #88638C22semidirect
28. D9×Dic6C22semidirect
29. #88547C22semidirect
30. #88543C22semidirect
31. #88532C22semidirect
32. C9×#216C22semidirect
33. C9×#215C22semidirect
34. C9×#214C22semidirect
35. C9×#213C22semidirect
36. C2×C9×C3⋊C8C22semidirect
37. #88425C22semidirect
38. #88421C22semidirect
39. #88415C22semidirect
40. #88415C22semidirect
41. #88413C22semidirect
42. D9×C3⋊C8C22semidirect
55. #63715D3semidirect
184. #950D9semidirect
297. #235C2×D9semidirect
313. #216C2×D9semidirect
315. #215C2×D9semidirect
322. #214C2×D9semidirect
325. #213C2×D9semidirect
331. C2×C3⋊C8C2×D9semidirect
379. C3×C9#577semidirect
527. C9#5836semidirect
587. C3#10589872semidirect

Maximal subgroups

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

subgroupindex
C3×#637152
#117224652
#117218662
C9×#9502
#117212192
#117212152
D9×#2142
#117211762
#117208822
C2×#884152
#117208582
#105954663
#105898723