Finite Group Databasebeta

← #13151899

Subgroups — #13151899

Order 1728.

842 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63716C211yesmaximal, characteristic, complemented
2#11722478C211yesmaximal, characteristic, complemented
3#11721867C211yesmaximal, characteristic, complemented
4C9×#950C211yesmaximal, characteristic, complemented
5#11721222C212yesmaximal, complemented
6D9×#215C212yesmaximal, complemented
7#11721185C212yesmaximal, complemented
8#11721173C212yesmaximal, complemented
9#11720887C211yesmaximal, characteristic, complemented
10C2×#88416C211yesmaximal, characteristic, complemented
11#11720858C211yesmaximal, characteristic, complemented
12#10595457—31nomaximal
13#10589872—31nomaximal
14C3×C9×#41C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C3×Q8×D9C2212yescomplemented
17C3×#3699—22no—
18C3×#3699C2212yescomplemented
19C9×#235C2211yescharacteristic
20C3×#3697—21no—
21C3×#3697C2211yescharacteristic
22C2×C3×Dic18—21no—
23C2×C3×Dic18C2211yescharacteristic, complemented
24#88668—22no—
25D9×D12C2212yes—
26D3×D36—21no—
27#88642—21no—
28#88635C2212yes—
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#88427C2212yescomplemented
38#88423C2212yescomplemented
39#88416C2212yescomplemented
40#88416C2212yescomplemented
41#88413C2212yescomplemented
42D9×C3⋊C8C2212yescomplemented
43C3×#1010—31no—
44#64308—31no—
45#64158—31no—
46C3×#950D311yescharacteristic
47#63948—32no—
48D3×#215—32no—
49#63935—32no—
50#63925—32no—
51C2×#3717—31no—
52#63827—31no—
53#63814—31no—
54#63716D311yescharacteristic, complemented
55#63715—31no—
56C9×#135—31no—
57#63512—31no—
58#63512—61no—
59#63511—31no—
60#63480—32no—
61#63478—32no—
62SD16×D9—32no—
63#63473—32no—
64C2×#3664—31no—
65#63464—31no—
66C22×C3⋊D9—21no—
67C2×D3×D9—22no—
68C3×Q8×C9C2311yescharacteristic
69C3×D4×C9—21no—
70C3×D4×C9C2311yescharacteristic
71C2×C3×C4×C9—21no—
72C2×C3×C4×C9C2311yescharacteristic
73C4⋊(C3⋊D9)—21no—
74C4⋊(C3⋊D9)C2312yes—
75C9×#67—21no—
76C3×#611—22no—
77C3×#611D412yes—
78C2×C3×Dic9—21no—
79C2×C3×Dic9—22no—
80C9×D12C2311yescharacteristic
81C4×D3×C9—21no—
82C3×D36D411yescharacteristic
83C3×C4×D9—22no—
84C3×C4×D9C2312yes—
85C9×Dic6C2311yescharacteristic
86C3×Dic18—21no—
87C3×Dic18—22no—
88C3×Dic18D411yescharacteristic
89C3×Dic18C2312yes—
90#6195—22no—
91#6192—22no—
92(C3⋊D9)⋊C4—22no—
93C3⋊(C9⋊C8)C2312yes—
94C9×C3⋊C8C2311yescharacteristic
95C9×C3⋊C8C2311yescharacteristic
96#5837—91no—
97#5770—91no—
98C32×#41C2×D311yescharacteristic
99C2×C4⋊(C3⋊D3)—31no—
100C3×D3×Q8—32no—
101C3×#237—32no—
102C3×#237—62no—
103C3×#235—31no—
104C3×#235—61no—
105C3×#235C2×D311yescharacteristic
106C2×C3×Dic6—31no—
107C2×C3×Dic6—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#3718—32no—
122#3717—32no—
123#3717—32no—
124#3715—32no—
125#3711—32no—
126D3×C3⋊C8—32no—
127C9×#41—21no—
128C9×#41—31no—
129C9×#41C2×D311yescharacteristic, complemented
130#3701—32no—
131Q8×D9C2×D312yescomplemented
132#3699—22no—
133#3699C2×D312yescomplemented
134D4×D9—32no—
135D4×D9—62no—
136#3697—21no—
137#3697—61no—
138#3697C2×D311yescharacteristic
139C2×D36—31no—
140C2×D36—61no—
141C2×D36—61no—
142C2×Dic18—21no—
143C2×Dic18C2×D311yescharacteristic, complemented
144C9×Q16—31no—
145C9×SD16—31no—
146C9×SD16—31no—
147C9×D8—31no—
148C2×C8×C9—31no—
149#3675—32no—
150#3675—62no—
151#3674—32no—
152#3673—32no—
153#3673—62no—
154#3672—32no—
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×C3⋊D9—22no—
164C2×C3⋊D9—81no—
165C2×D3×C9—21no—
166C2×C3×D9C2×D412yes—
167D3×D9—82no—
168C3×C4×C9—21no—
169C3×C4×C9C2×D411yescharacteristic
170C3×C4×C9C2411yescharacteristic
171C9×Dic3—21no—
172C3×Dic9—22no—
173C3×Dic9C2×D412yes—
174C3×2−1+4—91no—
175#1010—31no—
176#1009—91no—
177#1009—91no—
178C3×#135—31no—
179#951—91no—
180#950—92no—
181#950D911yescharacteristic, complemented
182#949—91no—
183#949—91no—
184#949—92no—
185#949—181no—
186#948—91no—
187#942—91no—
188#942—92no—
189C2×#215—92no—
190C2×#215—91no—
191#918—92no—
192#916—92no—
193D3×SD16—92no—
194#911—92no—
195C2×#204—91no—
196#901—91no—
197C22×C3⋊D3—61no—
198C2×D32—62no—
199C32×Q8C22×D311yescharacteristic
200C32×D4—21no—
201C32×D4C22×D311yescharacteristic
202C2×C32×C4—21no—
203C2×C32×C4C22×D311yescharacteristic
204C4⋊(C3⋊D3)—32no—
205C4⋊(C3⋊D3)—61no—
206C3×#67—21no—
207C3×#67—32no—
208C3×#67—62no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—61no—
211C3×D12—31no—
212C3×D12C22×D311yescharacteristic
213C3×C4×D3—21no—
214C3×C4×D3—32no—
215C3×C4×D3—62no—
216C3×Dic6—31no—
217C3×Dic6—32no—
218C3×Dic6—62no—
219C3×Dic6—61no—
220C3×Dic6C22×D311yescharacteristic
221#626—62no—
222#626—62no—
223#624—62no—
224C22×D9—61no—
225C22×D9—62no—
226C22×D9—121no—
227C3⋊(C3⋊C8)—32no—
228C3×C3⋊C8C22×D311yescharacteristic
229C3×C3⋊C8C22×D311yescharacteristic
230Q8×C9—21no—
231Q8×C9—31no—
232Q8×C9C22×D311yescharacteristic
233D4×C9—21no—
234D4×C9—21no—
235D4×C9—31no—
236D4×C9—41no—
237D4×C9—61no—
238D4×C9C22×D311yescharacteristic
239C2×C4×C9—21no—
240C2×C4×C9—21no—
241C2×C4×C9—41no—
242C2×C4×C9—61no—
243C2×C4×C9C22×D311yescharacteristic
244#611—22no—
245#611—62no—
246#611#6712yes—
247C2×Dic9—21no—
248C2×Dic9—22no—
249D36—32no—
250D36—62no—
251D36—62no—
252D36—61no—
253D36—121no—
254D36#6711yescharacteristic
255C4×D9—22no—
256C4×D9—62no—
257C4×D9C22×D312yes—
258Dic18—21no—
259Dic18—22no—
260Dic18#6711yescharacteristic
261Dic18C22×D312yes—
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
270C3⋊D9—42no—
271D3×C9—41no—
272C3×D9—22no—
273C3×#41—21no—
274C3×#41—31no—
275C3×#41—91no—
276C3×#41—92no—
277C3×#41—181no—
278C3×#41—182no—
279C3×#41C2×D911yescharacteristic, complemented
280C3×#41D3211yescharacteristic
281C2×C3×Q8—92no—
282C2×C3×Q8—91no—
283C2×C3×Q8—181no—
284#239—92no—
285#239—181no—
286D3×Q8—32no—
287#237—32no—
288#237—62no—
289D3×D4—92no—
290D3×D4—181no—
291D3×D4—182no—
292D3×D4—182no—
293#235—31no—
294#235—61no—
295#235—92no—
296#235—181no—
297#235C2×D911yescharacteristic
298C2×D12—91no—
299C2×D12—91no—
300C2×D12—92no—
301C2×D12—181no—
302C2×D12—181no—
303C2×Dic6—31no—
304C2×Dic6—61no—
305C3×Q16—31no—
306C3×SD16—31no—
307C3×SD16—31no—
308C3×D8—31no—
309C2×C3×C8—31no—
310#216—92no—
311#216—92no—
312#216C2×D911yescharacteristic, complemented
313#215—92no—
314#215—92no—
315#215—92no—
316#215—92no—
317#215—92no—
318#215—182no—
319#215C2×D911yescharacteristic, complemented
320#214—92no—
321#214C2×D911yescharacteristic, complemented
322#213—92no—
323#213—92no—
324#213—182no—
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—62no—
339C2×C3⋊D3—241no—
340C2×C3×D3—21no—
341C2×C3×D3—32no—
342D32—242no—
343C32×C4—21no—
344C32×C4D3×D411yescharacteristic
345C32×C4C23×D311yescharacteristic
346C3×Dic3—21no—
347C3×Dic3—32no—
348C3×Dic3—62no—
349C22×C9—21no—
350C22×C9—21no—
351C22×C9—41no—
352C22×C9—61no—
353C22×C9C2×#6711yescharacteristic
354C2×D9—62no—
355C2×D9—122no—
356C2×D9—242no—
357C2×D9—241no—
358C2×D9—242no—
359C2×D9C2×#6712yes—
360C4×C9—21no—
361C4×C9—21no—
362C4×C9—21no—
363C4×C9—41no—
364C4×C9—61no—
365C4×C9C2×#6711yescharacteristic
366C4×C9C23×D311yescharacteristic
367Dic9—22no—
368Dic9C2×#6712yes—
3692−1+4—91no—
370H(5,𝔽2)—271no—
371#137—272no—
372#137—271no—
373AGL(1,ℤ/8)—272no—
374AGL(1,ℤ/8)—271no—
375#135—31no—
376#135—272no—
377C2×SD16—271no—
378C2×SD16—272no—
379#131—271no—
380C3×C9#57711yescharacteristic, complemented, Sylow
381C22×D3—182no—
382C22×D3—182no—
383C22×D3—181no—
384C22×D3—181no—
385C22×D3—361no—
386C3×Q8—21no—
387C3×Q8—31no—
388C3×Q8—91no—
389C3×Q8—92no—
390C3×Q8—182no—
391C3×Q8—181no—
392C3×Q8C22×D911yescharacteristic
393C3×Q8C2×D3211yescharacteristic
394C3×D4—21no—
395C3×D4—21no—
396C3×D4—21no—
397C3×D4—31no—
398C3×D4—41no—
399C3×D4—61no—
400C3×D4—92no—
401C3×D4—91no—
402C3×D4—182no—
403C3×D4C22×D911yescharacteristic
404C3×D4C2×D3211yescharacteristic
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—41no—
409C2×C3×C4—61no—
410C2×C3×C4—92no—
411C2×C3×C4—182no—
412C2×C3×C4—182no—
413C2×C3×C4—181no—
414C2×C3×C4C22×D911yescharacteristic
415C2×C3×C4C2×D3211yescharacteristic
416#67—21no—
417#67—32no—
418#67—62no—
419#67—182no—
420#67—182no—
421C2×Dic3—62no—
422C2×Dic3—61no—
423D12—31no—
424D12—92no—
425D12—92no—
426D12—182no—
427D12—182no—
428D12—182no—
429D12—181no—
430D12—181no—
431D12—361no—
432D12C22×D911yescharacteristic
433C4×D3—21no—
434C4×D3—32no—
435C4×D3—62no—
436C4×D3—182no—
437C4×D3—182no—
438Dic6—31no—
439Dic6—32no—
440Dic6—62no—
441Dic6—61no—
442Dic6C22×D911yescharacteristic
443C3×C8—31no—
444C3×C8—31no—
445C3⋊C8—92no—
446C3⋊C8—92no—
447C3⋊C8—182no—
448C3⋊C8C22×D911yescharacteristic
449C3⋊C8C22×D911yescharacteristic
450C2×C32—21no—
451C2×C32—41no—
452C2×C32C2×D3×D411yescharacteristic
453C3⋊D3—122no—
454C3×D3—41no—
455C3×D3—62no—
456C2×C9—21no—
457C2×C9—21no—
458C2×C9—41no—
459C2×C9—41no—
460C2×C9—81no—
461C2×C9—121no—
462C2×C9C22×#6711yescharacteristic
463D9—22no—
464D9—122no—
465D9—242no—
466#41—31no—
467#41—91no—
468#41—92no—
469#41—181no—
470#41—182no—
471#41—272no—
472#41—541no—
473#41D3×D911yescharacteristic, complemented
474C2×Q8—92no—
475C2×Q8—91no—
476C2×Q8—181no—
477C2×D4—272no—
478C2×D4—271no—
479C2×D4—541no—
480C2×D4—542no—
481Q16—31no—
482Q16—272no—
483SD16—31no—
484SD16—31no—
485SD16—272no—
486SD16—272no—
487SD16—272no—
488SD16—272no—
489D8—31no—
490D8—272no—
491M16—272no—
492M16—271no—
493C2×C8—31no—
494C2×C8—272no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—41no—
499C22×C3—61no—
500C22×C3—92no—
501C22×C3D4×D911yescharacteristic
502C22×C3D3×#6711yescharacteristic
503C2×D3—21no—
504C2×D3—32no—
505C2×D3—182no—
506C2×D3—182no—
507C2×D3—362no—
508C2×D3—722no—
509C2×D3—722no—
510C2×D3—721no—
511C2×D3—721no—
512C2×D3—722no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—41no—
518C3×C4—61no—
519C3×C4—92no—
520C3×C4—182no—
521C3×C4D4×D911yescharacteristic
522C3×C4C23×D911yescharacteristic
523C3×C4D3×#6711yescharacteristic
524C3×C4C22×D3211yescharacteristic
525Dic3—21no—
526Dic3—32no—
527Dic3—62no—
528C32#577011yescharacteristic
529C9—21no—
530C9#583711yescharacteristic, complemented
531C23—542no—
532C23—541no—
533Q8—31no—
534Q8—91no—
535Q8—92no—
536Q8—182no—
537Q8—181no—
538Q8C2×D3×D911yescharacteristic
539D4—21no—
540D4—31no—
541D4—61no—
542D4—92no—
543D4—91no—
544D4—182no—
545D4—272no—
546D4—542no—
547D4—542no—
548D4—541no—
549D4C2×D3×D911yescharacteristic
550C2×C4—21no—
551C2×C4—61no—
552C2×C4—92no—
553C2×C4—182no—
554C2×C4—182no—
555C2×C4—181no—
556C2×C4—542no—
557C2×C4C2×D3×D911yescharacteristic
558C8—31no—
559C8—31no—
560C8—272no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—41no—
565C2×C3—41no—
566C2×C3—41no—
567C2×C3—81no—
568C2×C3—121no—
569C2×C3—182no—
570C2×C3C2×D4×D911yescharacteristic
571C2×C3C2×D3×#6711yescharacteristic
572D3—41no—
573D3—62no—
574D3—362no—
575D3—362no—
576D3—722no—
577C22—21no—
578C22—61no—
579C22—92no—
580C22—542no—
581C22—2161no—
582C22—2162no—
583C22D9×#6711yescharacteristic
584C4—21no—
585C4—61no—
586C4—92no—
587C4—182no—
588C4D9×#6711yescharacteristic
589C4C22×D3×D911yescharacteristic
590C3—21no—
591C3#1058987211yescharacteristic, complemented
592C3#1059545711yescharacteristic
593C2—21no—
594C2—41no—
595C2—121no—
596C2—182no—
597C2—1082no—
598C2C2×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×#63716C2semidirect
2. #11722478C2semidirect
3. #11721867C2semidirect
4. C9×#950C2semidirect
5. #11721222C2semidirect
6. D9×#215C2semidirect
7. #11721185C2semidirect
8. #11721173C2semidirect
9. #11720887C2semidirect
10. C2×#88416C2semidirect
11. #11720858C2semidirect
14. C3×C9×#41C22semidirect
16. C3×Q8×D9C22semidirect
18. C3×#3699C22semidirect
23. C2×C3×Dic18C22semidirect
29. #88548C22semidirect
30. #88544C22semidirect
31. #88532C22semidirect
32. C9×#216C22semidirect
33. C9×#215C22semidirect
34. C9×#214C22semidirect
35. C9×#213C22semidirect
36. C2×C9×C3⋊C8C22semidirect
37. #88427C22semidirect
38. #88423C22semidirect
39. #88416C22semidirect
40. #88416C22semidirect
41. #88413C22semidirect
42. D9×C3⋊C8C22semidirect
54. #63716D3semidirect
129. C9×#41C2×D3semidirect
131. Q8×D9C2×D3semidirect
133. #3699C2×D3semidirect
143. C2×Dic18C2×D3semidirect
181. #950D9semidirect
279. C3×#41C2×D9semidirect
312. #216C2×D9semidirect
319. #215C2×D9semidirect
321. #214C2×D9semidirect
325. #213C2×D9semidirect
331. C2×C3⋊C8C2×D9semidirect
380. C3×C9#577semidirect
473. #41D3×D9semidirect
530. C9#5837semidirect
591. C3#10589872semidirect

Maximal subgroups

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

subgroupindex
C3×#637162
#117224782
#117218672
C9×#9502
#117212222
D9×#2152
#117211852
#117211732
#117208872
C2×#884162
#117208582
#105954573
#105898723