Finite Group Databasebeta

← #13151898

Subgroups — #13151898

Order 1728.

872 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63715C211yesmaximal, characteristic, complemented
2#11722478C211yesmaximal, characteristic, complemented
3#11721867C211yesmaximal, characteristic, complemented
4C9×#950C211yesmaximal, characteristic, complemented
5#11721223C212yesmaximal, complemented
6#11721211C212yesmaximal, complemented
7#11721184C212yesmaximal, complemented
8D9×#213C212yesmaximal, complemented
9#11720882C211yesmaximal, characteristic, complemented
10C2×#88417C211yesmaximal, characteristic, complemented
11#11720858C211yesmaximal, characteristic, complemented
12#10595456—31nomaximal
13#10589871—31nomaximal
14C3×C9×#41C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
16C3×#3701C2212yescomplemented
17C3×D4×D9—22no—
18C3×D4×D9C2212yescomplemented
19C9×#235C2211yescharacteristic, complemented
20C3×#3697—21no—
21C3×#3697C2211yescharacteristic
22C2×C3×D36—21no—
23C2×C3×D36C2211yescharacteristic, complemented
24#88668—22no—
25D9×D12C2212yescomplemented
26D3×D36—21no—
27#88642—21no—
28#88635C2212yescomplemented
29#88548C2212yescomplemented
30#88544C2212yescomplemented
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#88417C2212yescomplemented
40#88417C2212yescomplemented
41#88413C2212yescomplemented
42D9×C3⋊C8C2212yescomplemented
43C3×#1009—31no—
44#64308—31no—
45#64158—31no—
46C3×#950D311yescharacteristic
47#63949—32no—
48#63939—32no—
49#63934—32no—
50D3×#213—32no—
51C2×#3714—31no—
52#63823—31no—
53#63814—31no—
54#63715—31no—
55#63715D311yescharacteristic, complemented
56C9×#135—31no—
57#63512—31no—
58#63512—31no—
59#63512—61no—
60#63481—32no—
61#63476—32no—
62#63476—32no—
63D8×D9—32no—
64C2×D72—31no—
65#63464—31no—
66C22×C3⋊D9—21no—
67C22×C3×D9—21no—
68C22×C3×D9—22no—
69C2×D3×D9—22no—
70C3×Q8×C9C2311yescharacteristic
71C3×D4×C9—21no—
72C3×D4×C9C2311yescharacteristic
73C2×C3×C4×C9—21no—
74C2×C3×C4×C9C2311yescharacteristic
75C4⋊(C3⋊D9)—21no—
76C4⋊(C3⋊D9)C2312yes—
77C9×#67—21no—
78C3×#611—22no—
79C3×#611D412yes—
80C9×D12C2311yescharacteristic
81C4×D3×C9—21no—
82C3×D36—22no—
83C3×D36—21no—
84C3×D36D411yescharacteristic
85C3×D36C2312yes—
86C3×C4×D9—22no—
87C3×C4×D9C2312yes—
88C9×Dic6C2311yescharacteristic
89C3×Dic18D411yescharacteristic
90#6195—22no—
91#6192—22no—
92(C3⋊D9)⋊C4—22no—
93C3⋊(C9⋊C8)C2312yes—
94C9×C3⋊C8C2311yescharacteristic
95C9×C3⋊C8C2311yescharacteristic
96#5835—91no—
97#5769—91no—
98C32×#41C2×D311yescharacteristic
99C2×C4⋊(C3⋊D3)—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#3811—62no—
109D3×D12—32no—
110D3×D12—61no—
111#3799—32no—
112#3799—61no—
113#3755—32no—
114#3753—32no—
115#3748—31no—
116C3×#216C2×D311yescharacteristic
117C3×#215C2×D311yescharacteristic
118C3×#214C2×D311yescharacteristic
119C3×#213C2×D311yescharacteristic
120C2×C3×C3⋊C8C2×D311yescharacteristic
121#3715—32no—
122#3714—32no—
123#3714—32no—
124#3713—32no—
125#3711—32no—
126D3×C3⋊C8—32no—
127C9×#41—21no—
128C9×#41—31no—
129C9×#41C2×D311yescharacteristic, complemented
130#3701—32no—
131#3701C2×D312yescomplemented
132D4×D9—22no—
133D4×D9—32no—
134D4×D9—62no—
135D4×D9C2×D312yescomplemented
136#3697—21no—
137#3697—61no—
138#3697C2×D311yescharacteristic
139C2×D36—21no—
140C2×D36—31no—
141C2×D36—61no—
142C2×D36—61no—
143C2×D36C2×D311yescharacteristic, complemented
144C9×Q16—31no—
145C9×SD16—31no—
146C9×SD16—31no—
147C9×D8—31no—
148C2×C8×C9—31no—
149#3675—32no—
150#3675—32no—
151#3675—62no—
152#3673—32no—
153#3673—32no—
154#3673—62no—
155#3667—31no—
156#3667—61no—
157D72—32no—
158D72—32no—
159#3663—32no—
160C8×D9—32no—
161C22×C3×C9—21no—
162C22×C3×C9C2×D411yescharacteristic
163C2×C3⋊D9—22no—
164C2×C3⋊D9—81no—
165C2×D3×C9—21no—
166C2×C3×D9—22no—
167C2×C3×D9—82no—
168C2×C3×D9—81no—
169C2×C3×D9C2×D412yes—
170D3×D9—82no—
171C3×C4×C9—21no—
172C3×C4×C9C2×D411yescharacteristic
173C3×C4×C9C2411yescharacteristic
174C9×Dic3—21no—
175C3×Dic9C2×D412yes—
176C3×H(5,𝔽2)—91no—
177#1009—31no—
178#1009—91no—
179#1009—91no—
180C3×#135—31no—
181#950—92no—
182#950D911yescharacteristic, complemented
183#949—91no—
184#949—91no—
185#949—91no—
186#949—92no—
187#949—181no—
188#948—91no—
189#932—91no—
190#932—92no—
191C2×#213—92no—
192C2×#213—91no—
193#919—92no—
194#914—92no—
195#914—92no—
196D3×D8—92no—
197C2×D24—91no—
198#901—91no—
199C22×C3⋊D3—61no—
200C22×C3×D3—62no—
201C22×C3×D3—61no—
202C2×D32—62no—
203C32×Q8C22×D311yescharacteristic
204C32×D4—21no—
205C32×D4C22×D311yescharacteristic
206C2×C32×C4—21no—
207C2×C32×C4C22×D311yescharacteristic
208C4⋊(C3⋊D3)—32no—
209C4⋊(C3⋊D3)—61no—
210C3×#67—21no—
211C3×#67—32no—
212C3×#67—62no—
213C3×D12—31no—
214C3×D12—32no—
215C3×D12—62no—
216C3×D12—61no—
217C3×D12C22×D311yescharacteristic
218C3×C4×D3—21no—
219C3×C4×D3—32no—
220C3×C4×D3—62no—
221C3×Dic6—31no—
222C3×Dic6C22×D311yescharacteristic
223#626—62no—
224#626—62no—
225#624—62no—
226C22×D9—21no—
227C22×D9—22no—
228C22×D9—61no—
229C22×D9—62no—
230C22×D9—121no—
231C3⋊(C3⋊C8)—32no—
232C3×C3⋊C8C22×D311yescharacteristic
233C3×C3⋊C8C22×D311yescharacteristic
234Q8×C9—21no—
235Q8×C9—31no—
236Q8×C9C22×D311yescharacteristic
237D4×C9—21no—
238D4×C9—21no—
239D4×C9—31no—
240D4×C9—41no—
241D4×C9—61no—
242D4×C9C22×D311yescharacteristic
243C2×C4×C9—21no—
244C2×C4×C9—21no—
245C2×C4×C9—41no—
246C2×C4×C9—61no—
247C2×C4×C9C22×D311yescharacteristic
248#611—22no—
249#611—62no—
250#611#6712yes—
251D36—22no—
252D36—21no—
253D36—32no—
254D36—62no—
255D36—62no—
256D36—61no—
257D36—121no—
258D36#6711yescharacteristic
259D36C22×D312yes—
260C4×D9—22no—
261C4×D9—62no—
262C4×D9C22×D312yes—
263Dic18#6711yescharacteristic
264C8×C9—31no—
265C8×C9—31no—
266C9⋊C8—32no—
267C9⋊C8—62no—
268#576—271noSylow
269C2×C3×C9—21no—
270C2×C3×C9—41no—
271C2×C3×C9C22×D411yescharacteristic
272C3⋊D9—42no—
273D3×C9—41no—
274C3×D9—22no—
275C3×D9—42no—
276C3×#41—21no—
277C3×#41—31no—
278C3×#41—91no—
279C3×#41—92no—
280C3×#41—181no—
281C3×#41C2×D911yescharacteristic, complemented
282C3×#41D3211yescharacteristic
283C2×C3×D4—92no—
284C2×C3×D4—91no—
285C2×C3×D4—181no—
286C2×C3×D4—182no—
287#239—32no—
288#239—92no—
289#239—181no—
290D3×D4—32no—
291D3×D4—62no—
292D3×D4—92no—
293D3×D4—181no—
294D3×D4—182no—
295D3×D4—182no—
296#235—31no—
297#235—61no—
298#235—92no—
299#235—181no—
300#235C2×D911yescharacteristic, complemented
301C2×D12—31no—
302C2×D12—61no—
303C2×D12—91no—
304C2×D12—91no—
305C2×D12—92no—
306C2×D12—181no—
307C2×D12—181no—
308C3×Q16—31no—
309C3×SD16—31no—
310C3×SD16—31no—
311C3×D8—31no—
312C2×C3×C8—31no—
313#216C2×D911yescharacteristic, complemented
314#215—92no—
315#215—92no—
316#215—92no—
317#215—182no—
318#215C2×D911yescharacteristic, complemented
319#214—92no—
320#214C2×D911yescharacteristic, complemented
321#213—92no—
322#213—92no—
323#213—92no—
324#213—92no—
325#213—92no—
326#213—92no—
327#213—182no—
328#213C2×D911yescharacteristic, complemented
329#208—91no—
330#208—91no—
331#208—92no—
332#208—181no—
333C2×C3⋊C8—92no—
334C2×C3⋊C8C2×D911yescharacteristic, complemented
335D24—92no—
336D24—92no—
337ΓL(1,𝔽25)—92no—
338D3×C8—92no—
339C22×C32—21no—
340C22×C32D3×D411yescharacteristic
341C2×C3⋊D3—62no—
342C2×C3⋊D3—241no—
343C2×C3×D3—21no—
344C2×C3×D3—32no—
345C2×C3×D3—62no—
346C2×C3×D3—242no—
347C2×C3×D3—241no—
348D32—242no—
349C32×C4—21no—
350C32×C4D3×D411yescharacteristic
351C32×C4C23×D311yescharacteristic
352C3×Dic3—21no—
353C3×Dic3—32no—
354C22×C9—21no—
355C22×C9—21no—
356C22×C9—41no—
357C22×C9—61no—
358C22×C9C2×#6711yescharacteristic
359C2×D9—22no—
360C2×D9—62no—
361C2×D9—82no—
362C2×D9—81no—
363C2×D9—122no—
364C2×D9—242no—
365C2×D9—241no—
366C2×D9—242no—
367C2×D9C2×#6712yes—
368C4×C9—21no—
369C4×C9—21no—
370C4×C9—21no—
371C4×C9—41no—
372C4×C9—61no—
373C4×C9C2×#6711yescharacteristic
374C4×C9C23×D311yescharacteristic
375Dic9C2×#6712yes—
376H(5,𝔽2)—91no—
377H(5,𝔽2)—271no—
378AGL(1,ℤ/8)—272no—
379AGL(1,ℤ/8)—272no—
380AGL(1,ℤ/8)—271no—
381AGL(1,ℤ/8)—271no—
382#135—31no—
383#135—272no—
384C2×D8—271no—
385C2×D8—272no—
386#131—271no—
387C3×C9#57611yescharacteristic, complemented, Sylow
388C23×C3—182no—
389C23×C3—181no—
390C22×D3—62no—
391C22×D3—61no—
392C22×D3—182no—
393C22×D3—182no—
394C22×D3—181no—
395C22×D3—181no—
396C22×D3—361no—
397C3×Q8—21no—
398C3×Q8—31no—
399C3×Q8—91no—
400C3×Q8C22×D911yescharacteristic
401C3×Q8C2×D3211yescharacteristic
402C3×D4—21no—
403C3×D4—21no—
404C3×D4—21no—
405C3×D4—31no—
406C3×D4—41no—
407C3×D4—61no—
408C3×D4—92no—
409C3×D4—91no—
410C3×D4—92no—
411C3×D4—182no—
412C3×D4—182no—
413C3×D4—181no—
414C3×D4C22×D911yescharacteristic
415C3×D4C2×D3211yescharacteristic
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—41no—
420C2×C3×C4—61no—
421C2×C3×C4—92no—
422C2×C3×C4—182no—
423C2×C3×C4C22×D911yescharacteristic
424C2×C3×C4C2×D3211yescharacteristic
425#67—21no—
426#67—32no—
427#67—62no—
428#67—182no—
429#67—182no—
430D12—32no—
431D12—31no—
432D12—62no—
433D12—61no—
434D12—92no—
435D12—92no—
436D12—182no—
437D12—182no—
438D12—182no—
439D12—181no—
440D12—181no—
441D12—361no—
442D12C22×D911yescharacteristic
443C4×D3—21no—
444C4×D3—32no—
445C4×D3—62no—
446C4×D3—182no—
447C4×D3—182no—
448Dic6—31no—
449Dic6C22×D911yescharacteristic
450C3×C8—31no—
451C3×C8—31no—
452C3⋊C8—92no—
453C3⋊C8—92no—
454C3⋊C8—182no—
455C3⋊C8C22×D911yescharacteristic
456C3⋊C8C22×D911yescharacteristic
457C2×C32—21no—
458C2×C32—41no—
459C2×C32C2×D3×D411yescharacteristic
460C3⋊D3—122no—
461C3×D3—41no—
462C3×D3—62no—
463C3×D3—122no—
464C2×C9—21no—
465C2×C9—21no—
466C2×C9—41no—
467C2×C9—41no—
468C2×C9—81no—
469C2×C9—121no—
470C2×C9C22×#6711yescharacteristic
471D9—22no—
472D9—42no—
473D9—122no—
474D9—242no—
475#41—31no—
476#41—91no—
477#41—92no—
478#41—181no—
479#41—272no—
480#41—541no—
481#41D3×D911yescharacteristic, complemented
482C2×D4—92no—
483C2×D4—91no—
484C2×D4—181no—
485C2×D4—182no—
486C2×D4—272no—
487C2×D4—271no—
488C2×D4—541no—
489C2×D4—542no—
490Q16—31no—
491SD16—31no—
492SD16—31no—
493SD16—272no—
494SD16—272no—
495D8—31no—
496D8—272no—
497D8—272no—
498D8—272no—
499D8—272no—
500M16—272no—
501M16—271no—
502C2×C8—31no—
503C2×C8—272no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—41no—
508C22×C3—61no—
509C22×C3—92no—
510C22×C3—182no—
511C22×C3—721no—
512C22×C3—722no—
513C22×C3D4×D911yescharacteristic
514C22×C3D3×#6711yescharacteristic
515C2×D3—21no—
516C2×D3—32no—
517C2×D3—62no—
518C2×D3—182no—
519C2×D3—182no—
520C2×D3—242no—
521C2×D3—241no—
522C2×D3—362no—
523C2×D3—722no—
524C2×D3—722no—
525C2×D3—721no—
526C2×D3—721no—
527C2×D3—722no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—41no—
533C3×C4—61no—
534C3×C4—92no—
535C3×C4D4×D911yescharacteristic
536C3×C4C23×D911yescharacteristic
537C3×C4D3×#6711yescharacteristic
538C3×C4C22×D3211yescharacteristic
539Dic3—21no—
540Dic3—32no—
541C32#576911yescharacteristic
542C9—21no—
543C9#583511yescharacteristic, complemented
544C23—182no—
545C23—181no—
546C23—542no—
547C23—541no—
548Q8—31no—
549Q8—91no—
550Q8C2×D3×D911yescharacteristic
551D4—21no—
552D4—31no—
553D4—61no—
554D4—92no—
555D4—92no—
556D4—91no—
557D4—182no—
558D4—182no—
559D4—181no—
560D4—272no—
561D4—542no—
562D4—542no—
563D4—541no—
564D4C2×D3×D911yescharacteristic
565C2×C4—21no—
566C2×C4—61no—
567C2×C4—92no—
568C2×C4—182no—
569C2×C4—542no—
570C2×C4C2×D3×D911yescharacteristic
571C8—31no—
572C8—31no—
573C8—272no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—41no—
578C2×C3—41no—
579C2×C3—41no—
580C2×C3—81no—
581C2×C3—121no—
582C2×C3—182no—
583C2×C3—362no—
584C2×C3C2×D4×D911yescharacteristic
585C2×C3C2×D3×#6711yescharacteristic
586D3—41no—
587D3—62no—
588D3—122no—
589D3—362no—
590D3—362no—
591D3—722no—
592C22—21no—
593C22—61no—
594C22—92no—
595C22—182no—
596C22—542no—
597C22—721no—
598C22—722no—
599C22—2161no—
600C22—2162no—
601C22D9×#6711yescharacteristic
602C4—21no—
603C4—61no—
604C4—92no—
605C4D9×#6711yescharacteristic
606C4C22×D3×D911yescharacteristic
607C3—21no—
608C3#1058987111yescharacteristic, complemented
609C3#1059545611yescharacteristic
610C2—21no—
611C2—41no—
612C2—121no—
613C2—182no—
614C2—362no—
615C2—1082no—
616C2C2×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#576
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. #11722478C2semidirect
3. #11721867C2semidirect
4. C9×#950C2semidirect
5. #11721223C2semidirect
6. #11721211C2semidirect
7. #11721184C2semidirect
8. D9×#213C2semidirect
9. #11720882C2semidirect
10. C2×#88417C2semidirect
11. #11720858C2semidirect
14. C3×C9×#41C22semidirect
15. C2×C4⋊(C3⋊D9)C22semidirect
16. C3×#3701C22semidirect
18. C3×D4×D9C22semidirect
19. C9×#235C22semidirect
23. C2×C3×D36C22semidirect
25. D9×D12C22semidirect
28. #88635C22semidirect
29. #88548C22semidirect
30. #88544C22semidirect
31. #88532C22semidirect
32. C9×#216C22semidirect
33. C9×#215C22semidirect
34. C9×#214C22semidirect
35. C9×#213C22semidirect
36. C2×C9×C3⋊C8C22semidirect
37. #88425C22semidirect
38. #88421C22semidirect
39. #88417C22semidirect
40. #88417C22semidirect
41. #88413C22semidirect
42. D9×C3⋊C8C22semidirect
55. #63715D3semidirect
129. C9×#41C2×D3semidirect
131. #3701C2×D3semidirect
135. D4×D9C2×D3semidirect
143. C2×D36C2×D3semidirect
182. #950D9semidirect
281. C3×#41C2×D9semidirect
300. #235C2×D9semidirect
313. #216C2×D9semidirect
318. #215C2×D9semidirect
320. #214C2×D9semidirect
328. #213C2×D9semidirect
334. C2×C3⋊C8C2×D9semidirect
387. C3×C9#576semidirect
481. #41D3×D9semidirect
543. C9#5835semidirect
608. C3#10589871semidirect

Maximal subgroups

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

subgroupindex
C3×#637152
#117224782
#117218672
C9×#9502
#117212232
#117212112
#117211842
D9×#2132
#117208822
C2×#884172
#117208582
#105954563
#105898713