Finite Group Databasebeta

← #13151107

Subgroups — #13151107

Order 1728.

880 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×#63710C211yesmaximal, characteristic, complemented
2#11722478C211yesmaximal, characteristic, complemented
3C2×#88544C211yesmaximal, characteristic, complemented
4C9×#932C211yesmaximal, characteristic, complemented
5#11721185C212yesmaximal, complemented
6#11721184C212yesmaximal, complemented
7#11721173C212yesmaximal, complemented
8D9×#213C212yesmaximal, complemented
9#11720881C211yesmaximal, characteristic, complemented
10#11720873C211yesmaximal, characteristic, complemented
11#11720859C211yesmaximal, characteristic, complemented
12#10595097—31nomaximal
13#10589876—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
16C2×C3×#611—22no—
17C3×#3699—21no—
18C3×#3699C2212yescomplemented
19C3×D4×D9—21no—
20C3×D4×D9C2212yescomplemented
21C9×#235C2211yescharacteristic, complemented
22C3×#3697C2211yescharacteristic
23C3×#3697C2211yescharacteristic, complemented
24#88668—22no—
25D9×D12C2212yescomplemented
26D3×D36—21no—
27#88642—21no—
28#88635C2212yescomplemented
29#88544C2212yescomplemented
30#88544C2212yescomplemented
31C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
32C9×#214C2212yescomplemented
33C9×#213C2212yescomplemented
34C9×#208C2211yescharacteristic, complemented
35#88427C2211yescharacteristic, complemented
36#88425C2211yescharacteristic, complemented
37#88423C2211yescharacteristic, complemented
38#88421C2211yescharacteristic, complemented
39#88417C2212yescomplemented
40#88416C2212yescomplemented
41#88413C2212yescomplemented
42D9×C3⋊C8C2212yescomplemented
43C3×#1004—31no—
44#64308—31no—
45C2×#3753—31no—
46C3×#932D311yescharacteristic
47#63935—32no—
48#63934—32no—
49#63925—32no—
50D3×#213—32no—
51#63825—31no—
52#63822—31no—
53#63815—31no—
54#63715—31no—
55#63710D311yescharacteristic, complemented
56C9×AGL(1,ℤ/8)—31no—
57#63513—31no—
58C2×#3673—31no—
59C2×#3673—61no—
60#63478—32no—
61#63476—32no—
62#63473—32no—
63D8×D9—32no—
64#63470—31no—
65#63469—31no—
66C23×C3×C9—21no—
67C22×C3⋊D9—21no—
68C22×C3×D9—22no—
69C2×D3×D9—22no—
70C3×D4×C9—21no—
71C3×D4×C9C2312yes—
72C2×C3×C4×C9C2311yescharacteristic
73C4⋊(C3⋊D9)—21no—
74C4⋊(C3⋊D9)C2312yes—
75C9×#67—21no—
76C3×#611—21no—
77C3×#611—22no—
78C3×#611—22no—
79C3×#611D412yes—
80C2×C3×Dic9—22no—
81C9×D12C2311yescharacteristic
82C4×D3×C9—21no—
83C3×D36D411yescharacteristic
84C3×D36C2311yescharacteristic
85C3×C4×D9—21no—
86C3×C4×D9C2312yes—
87C9×Dic6C2311yescharacteristic
88C3×Dic18D411yescharacteristic
89C3×Dic18C2311yescharacteristic
90#6195—22no—
91#6192—22no—
92(C3⋊D9)⋊C4—22no—
93C3⋊(C9⋊C8)C2311yescharacteristic
94C3⋊(C9⋊C8)C2311yescharacteristic
95C9×C3⋊C8C2312yes—
96#5835—91no—
97#5774—91no—
98C2×C32×D4C2×D311yescharacteristic
99C2×C4⋊(C3⋊D3)—31no—
100C2×C3×#67—62no—
101C3×#237—32no—
102C3×#237—61no—
103C3×D3×D4—32no—
104C3×D3×D4—61no—
105C3×#235—31no—
106C3×#235—31no—
107C3×#235C2×D311yescharacteristic
108#3811—62no—
109D3×D12—32no—
110D3×D12—61no—
111#3799—32no—
112#3799—61no—
113#3753—32no—
114#3753—32no—
115C2×C3⋊(C3⋊C8)—31no—
116C3×#214C2×D312yes—
117C3×#213C2×D312yes—
118C3×#208C2×D311yescharacteristic
119#3718—31no—
120#3717—32no—
121#3715—31no—
122#3715—31no—
123#3714—32no—
124#3713—31no—
125#3711—32no—
126D3×C3⋊C8—32no—
127C9×#41—31no—
128C2×D4×C9—21no—
129C2×D4×C9C2×D311yescharacteristic, complemented
130C2×#611—22no—
131#3701—32no—
132#3699—21no—
133#3699C2×D312yescomplemented
134D4×D9—21no—
135D4×D9—32no—
136D4×D9—62no—
137D4×D9C2×D312yescomplemented
138#3697—61no—
139#3697C2×D311yescharacteristic
140#3697C2×D311yescharacteristic, complemented
141C2×D36—31no—
142C2×D36—61no—
143C2×D36—61no—
144C9×SD16—32no—
145C9×D8—32no—
146C9×M16—31no—
147#3675—31no—
148#3674—31no—
149#3673—32no—
150#3673—32no—
151#3673—31no—
152#3673—62no—
153#3673—62no—
154#3672—31no—
155C2×C9⋊C8—31no—
156C2×C9⋊C8—61no—
157D72—32no—
158#3664—32no—
159#3663—32no—
160C8×D9—32no—
161C22×C3×C9—22no—
162C22×C3×C9—81no—
163C22×C3×C9C2×D411yescharacteristic
164C2×C3⋊D9—22no—
165C2×C3⋊D9—81no—
166C2×D3×C9—21no—
167C2×C3×D9—21no—
168C2×C3×D9—82no—
169C2×C3×D9C2×D412yes—
170D3×D9—82no—
171C3×C4×C9C2×D411yescharacteristic
172C3×C4×C9C2411yescharacteristic
173C9×Dic3—21no—
174C3×Dic9—21no—
175C3×Dic9C2×D412yes—
176C3×H(5,𝔽2)—91no—
177#1009—91no—
178#1009—91no—
179#1004—31no—
180C3×AGL(1,ℤ/8)—31no—
181#950—91no—
182#950—91no—
183#950—92no—
184#949—92no—
185#949—91no—
186#948—91no—
187#932—92no—
188#932D911yescharacteristic, complemented
189C2×#213—91no—
190C2×#213—91no—
191C2×#213—92no—
192C2×#213—181no—
193#916—92no—
194#914—92no—
195#911—92no—
196D3×D8—92no—
197#908—91no—
198#907—91no—
199C23×C32—21no—
200C22×C3⋊D3—61no—
201C22×C3×D3—62no—
202C2×D32—62no—
203C32×D4—21no—
204C32×D4C22×D312yes—
205C2×C32×C4C22×D311yescharacteristic
206C4⋊(C3⋊D3)—32no—
207C4⋊(C3⋊D3)—61no—
208C3×#67—21no—
209C3×#67—32no—
210C3×#67—62no—
211C3×#67—62no—
212C3×#67—61no—
213C2×C3×Dic3—62no—
214C3×D12—31no—
215C3×D12—31no—
216C3×D12C22×D311yescharacteristic
217C3×C4×D3—21no—
218C3×C4×D3—32no—
219C3×C4×D3—61no—
220C3×Dic6—31no—
221C3×Dic6—31no—
222C3×Dic6C22×D311yescharacteristic
223#626—62no—
224#626—62no—
225#624—62no—
226C23×C9—21no—
227C23×C9—41no—
228C22×D9—22no—
229C22×D9—61no—
230C22×D9—62no—
231C22×D9—121no—
232C3⋊(C3⋊C8)—31no—
233C3⋊(C3⋊C8)—31no—
234C3×C3⋊C8C22×D312yes—
235Q8×C9—31no—
236D4×C9—21no—
237D4×C9—22no—
238D4×C9—31no—
239D4×C9—41no—
240D4×C9—61no—
241D4×C9C22×D312yes—
242C2×C4×C9—21no—
243C2×C4×C9—61no—
244C2×C4×C9C22×D311yescharacteristic
245#611—21no—
246#611—22no—
247#611—22no—
248#611—62no—
249#611#6712yes—
250C2×Dic9—22no—
251D36—32no—
252D36—62no—
253D36—62no—
254D36—61no—
255D36—121no—
256D36#6711yescharacteristic
257D36C22×D311yescharacteristic
258C4×D9—21no—
259C4×D9—62no—
260C4×D9C22×D312yes—
261Dic18#6711yescharacteristic
262Dic18C22×D311yescharacteristic
263C8×C9—32no—
264C9⋊C8—31no—
265C9⋊C8—31no—
266C9⋊C8—61no—
267C9⋊C8—61no—
268#576—271noSylow
269C2×C3×C9—21no—
270C2×C3×C9—42no—
271C2×C3×C9C22×D411yescharacteristic
272C3⋊D9—42no—
273D3×C9—41no—
274C3×D9—22no—
275C3×D9—41no—
276C3×#41—31no—
277C3×#41—91no—
278C3×#41—92no—
279C3×#41—91no—
280C3×#41—181no—
281C2×C3×D4—21no—
282C2×C3×D4—92no—
283C2×C3×D4—181no—
284C2×C3×D4—182no—
285C2×C3×D4C2×D911yescharacteristic, complemented
286C2×C3×D4D3211yescharacteristic
287C2×#67—62no—
288#239—92no—
289#239—181no—
290#237—32no—
291#237—61no—
292D3×D4—32no—
293D3×D4—61no—
294D3×D4—92no—
295D3×D4—181no—
296D3×D4—182no—
297D3×D4—182no—
298#235—31no—
299#235—31no—
300#235—92no—
301#235—181no—
302#235C2×D911yescharacteristic, complemented
303C2×D12—91no—
304C2×D12—91no—
305C2×D12—92no—
306C2×D12—181no—
307C2×D12—181no—
308C3×SD16—32no—
309C3×D8—32no—
310C3×M16—31no—
311#216—91no—
312#216—91no—
313#215—91no—
314#215—91no—
315#215—92no—
316#214—91no—
317#214—91no—
318#214C2×D912yescomplemented
319#213—91no—
320#213—92no—
321#213—92no—
322#213—92no—
323#213—92no—
324#213—91no—
325#213—92no—
326#213—182no—
327#213—182no—
328#213C2×D912yescomplemented
329#208—92no—
330#208C2×D911yescharacteristic, complemented
331C2×C3⋊C8—91no—
332C2×C3⋊C8—91no—
333C2×C3⋊C8—92no—
334C2×C3⋊C8—181no—
335D24—92no—
336#204—92no—
337ΓL(1,𝔽25)—92no—
338D3×C8—92no—
339C22×C32—22no—
340C22×C32—81no—
341C22×C32D3×D411yescharacteristic
342C2×C3⋊D3—62no—
343C2×C3⋊D3—241no—
344C2×C3×D3—21no—
345C2×C3×D3—32no—
346C2×C3×D3—61no—
347C2×C3×D3—242no—
348D32—242no—
349C32×C4D3×D411yescharacteristic
350C32×C4C23×D311yescharacteristic
351C3×Dic3—21no—
352C3×Dic3—32no—
353C3×Dic3—61no—
354C22×C9—22no—
355C22×C9—21no—
356C22×C9—42no—
357C22×C9—61no—
358C22×C9—81no—
359C22×C9—82no—
360C22×C9C2×#6711yescharacteristic
361C2×D9—21no—
362C2×D9—62no—
363C2×D9—82no—
364C2×D9—122no—
365C2×D9—242no—
366C2×D9—241no—
367C2×D9—242no—
368C2×D9C2×#6712yes—
369C4×C9—21no—
370C4×C9—21no—
371C4×C9—61no—
372C4×C9C2×#6711yescharacteristic
373C4×C9C23×D311yescharacteristic
374Dic9—21no—
375Dic9C2×#6712yes—
376H(5,𝔽2)—91no—
377H(5,𝔽2)—271no—
378AGL(1,ℤ/8)—31no—
379AGL(1,ℤ/8)—271no—
380AGL(1,ℤ/8)—272no—
381AGL(1,ℤ/8)—272no—
382#135—272no—
383#135—271no—
384C2×D8—272no—
385C2×D8—271no—
386#131—271no—
387C3×C9#57611yescharacteristic, complemented, Sylow
388C23×C3—21no—
389C23×C3—21no—
390C23×C3—41no—
391C23×C3—182no—
392C22×D3—62no—
393C22×D3—182no—
394C22×D3—182no—
395C22×D3—181no—
396C22×D3—181no—
397C22×D3—361no—
398C3×Q8—31no—
399C3×Q8—91no—
400C3×Q8—91no—
401C3×D4—21no—
402C3×D4—21no—
403C3×D4—22no—
404C3×D4—31no—
405C3×D4—41no—
406C3×D4—61no—
407C3×D4—92no—
408C3×D4—91no—
409C3×D4—91no—
410C3×D4—182no—
411C3×D4—182no—
412C3×D4—181no—
413C3×D4C22×D912yes—
414C3×D4C2×D3212yes—
415C2×C3×C4—21no—
416C2×C3×C4—61no—
417C2×C3×C4—92no—
418C2×C3×C4—182no—
419C2×C3×C4—181no—
420C2×C3×C4C22×D911yescharacteristic
421C2×C3×C4C2×D3211yescharacteristic
422#67—21no—
423#67—32no—
424#67—62no—
425#67—62no—
426#67—61no—
427#67—182no—
428#67—182no—
429C2×Dic3—62no—
430D12—31no—
431D12—31no—
432D12—92no—
433D12—92no—
434D12—182no—
435D12—182no—
436D12—182no—
437D12—181no—
438D12—181no—
439D12—361no—
440D12C22×D911yescharacteristic
441C4×D3—21no—
442C4×D3—32no—
443C4×D3—61no—
444C4×D3—182no—
445C4×D3—182no—
446Dic6—31no—
447Dic6—31no—
448Dic6C22×D911yescharacteristic
449C3×C8—32no—
450C3⋊C8—91no—
451C3⋊C8—91no—
452C3⋊C8—91no—
453C3⋊C8—91no—
454C3⋊C8—181no—
455C3⋊C8—181no—
456C3⋊C8C22×D912yes—
457C2×C32—21no—
458C2×C32—42no—
459C2×C32C2×D3×D411yescharacteristic
460C3⋊D3—122no—
461C3×D3—41no—
462C3×D3—62no—
463C3×D3—121no—
464C2×C9—22no—
465C2×C9—21no—
466C2×C9—21no—
467C2×C9—42no—
468C2×C9—82no—
469C2×C9—121no—
470C2×C9C22×#6711yescharacteristic
471D9—22no—
472D9—41no—
473D9—122no—
474D9—242no—
475#41—31no—
476#41—91no—
477#41—92no—
478#41—91no—
479#41—181no—
480#41—272no—
481#41—541no—
482C2×D4—92no—
483C2×D4—181no—
484C2×D4—182no—
485C2×D4—272no—
486C2×D4—271no—
487C2×D4—541no—
488C2×D4—542no—
489C2×D4D3×D911yescharacteristic, complemented
490Q16—271no—
491SD16—32no—
492SD16—272no—
493SD16—271no—
494SD16—271no—
495D8—32no—
496D8—272no—
497D8—271no—
498D8—272no—
499D8—272no—
500M16—31no—
501M16—272no—
502C2×C8—272no—
503C2×C8—271no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—21no—
507C22×C3—42no—
508C22×C3—61no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3—82no—
512C22×C3—92no—
513C22×C3—181no—
514C22×C3—722no—
515C22×C3D4×D911yescharacteristic
516C22×C3D3×#6711yescharacteristic
517C2×D3—21no—
518C2×D3—32no—
519C2×D3—61no—
520C2×D3—182no—
521C2×D3—182no—
522C2×D3—242no—
523C2×D3—362no—
524C2×D3—722no—
525C2×D3—722no—
526C2×D3—721no—
527C2×D3—721no—
528C2×D3—722no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—61no—
532C3×C4—92no—
533C3×C4—181no—
534C3×C4D4×D911yescharacteristic
535C3×C4C23×D911yescharacteristic
536C3×C4D3×#6711yescharacteristic
537C3×C4C22×D3211yescharacteristic
538Dic3—21no—
539Dic3—32no—
540Dic3—61no—
541C32#577411yescharacteristic
542C9—21no—
543C9#583511yescharacteristic, complemented
544C23—21no—
545C23—182no—
546C23—542no—
547C23—541no—
548Q8—31no—
549Q8—91no—
550Q8—91no—
551D4—21no—
552D4—31no—
553D4—61no—
554D4—92no—
555D4—91no—
556D4—91no—
557D4—182no—
558D4—182no—
559D4—181no—
560D4—272no—
561D4—542no—
562D4—542no—
563D4—541no—
564D4C2×D3×D912yes—
565C2×C4—61no—
566C2×C4—92no—
567C2×C4—182no—
568C2×C4—181no—
569C2×C4—542no—
570C2×C4C2×D3×D911yescharacteristic
571C8—32no—
572C8—271no—
573C8—271no—
574C2×C3—22no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—42no—
579C2×C3—42no—
580C2×C3—82no—
581C2×C3—121no—
582C2×C3—182no—
583C2×C3—361no—
584C2×C3C2×D4×D911yescharacteristic
585C2×C3C2×D3×#6711yescharacteristic
586D3—41no—
587D3—62no—
588D3—121no—
589D3—362no—
590D3—362no—
591D3—722no—
592C22—22no—
593C22—61no—
594C22—81no—
595C22—92no—
596C22—181no—
597C22—542no—
598C22—722no—
599C22—2161no—
600C22—2162no—
601C22D9×#6711yescharacteristic
602C4—61no—
603C4—92no—
604C4—181no—
605C4D9×#6711yescharacteristic
606C4C22×D3×D911yescharacteristic
607C3—21no—
608C3#1058987611yescharacteristic, complemented
609C3#1059509711yescharacteristic
610C2—21no—
611C2—42no—
612C2—121no—
613C2—182no—
614C2—361no—
615C2—1082no—
616C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC2C2×D9×#67
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC2×C3×D4×C9C22
socleC2×C32C2×D3×D4
hypercenterC2×D4D3×D9
hypocenterC3×C9#576
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#63710C2semidirect
2. #11722478C2semidirect
3. C2×#88544C2semidirect
4. C9×#932C2semidirect
5. #11721185C2semidirect
6. #11721184C2semidirect
7. #11721173C2semidirect
8. D9×#213C2semidirect
9. #11720881C2semidirect
10. #11720873C2semidirect
11. #11720859C2semidirect
14. C2×C3×D4×C9C22semidirect
15. C2×C4⋊(C3⋊D9)C22semidirect
18. C3×#3699C22semidirect
20. C3×D4×D9C22semidirect
21. C9×#235C22semidirect
23. C3×#3697C22semidirect
25. D9×D12C22semidirect
28. #88635C22semidirect
29. #88544C22semidirect
30. #88544C22semidirect
31. C2×C3⋊(C9⋊C8)C22semidirect
32. C9×#214C22semidirect
33. C9×#213C22semidirect
34. C9×#208C22semidirect
35. #88427C22semidirect
36. #88425C22semidirect
37. #88423C22semidirect
38. #88421C22semidirect
39. #88417C22semidirect
40. #88416C22semidirect
41. #88413C22semidirect
42. D9×C3⋊C8C22semidirect
55. #63710D3semidirect
129. C2×D4×C9C2×D3semidirect
133. #3699C2×D3semidirect
137. D4×D9C2×D3semidirect
140. #3697C2×D3semidirect
188. #932D9semidirect
285. C2×C3×D4C2×D9semidirect
302. #235C2×D9semidirect
318. #214C2×D9semidirect
328. #213C2×D9semidirect
330. #208C2×D9semidirect
387. C3×C9#576semidirect
489. C2×D4D3×D9semidirect
543. C9#5835semidirect
608. C3#10589876semidirect

Maximal subgroups

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

subgroupindex
C3×#637102
#117224782
C2×#885442
C9×#9322
#117211852
#117211842
#117211732
D9×#2132
#117208812
#117208732
#117208592
#105950973
#105898763