Finite Group Databasebeta

← #13150101

Subgroups — #13150101

Order 1728.

854 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C2×#88532C211yesmaximal, characteristic, complemented
4#11720893C212yesmaximal, complemented
5#11720887C212yesmaximal, complemented
6#11720883C212yesmaximal, complemented
7#11720882C212yesmaximal, complemented
8#11720881C214yesmaximal, complemented
9#10594786—31nomaximal
10#10589925—31nomaximal
11C22×C3×C4×C9C2211yescharacteristic
12C2×C9×#67—21no—
13C2×C3×#611—21no—
14C9×#235—21no—
15C9×#235—21no—
16C9×#235C2212yes—
17C9×#235C2212yescomplemented
18C2×C9×D12C2211yescharacteristic, complemented
19C2×C4×D3×C9—21no—
20C3×#3697—21no—
21C3×#3697—21no—
22C3×#3697C2212yes—
23C3×#3697C2212yescomplemented
24C2×C3×D36C2211yescharacteristic, complemented
25C2×C3×C4×D9—21no—
26C2×C9×Dic6C2211yescharacteristic, complemented
27C2×C3×Dic18C2211yescharacteristic, complemented
28#88532C2212yescomplemented
29#88532C2212yescomplemented
30C2×C3⋊(C9⋊C8)C2212yescomplemented
31#88427C2214yescomplemented
32#88425C2214yescomplemented
33#88423C2214yescomplemented
34#88421C2214yescomplemented
35C2×C3×#235—31no—
36C2×C3×#235D311yescharacteristic
37C2×#3748—31no—
38#63827—32no—
39#63827—32no—
40#63823—32no—
41#63823—32no—
42#63822—34no—
43C2×C9×#41—31no—
44C2×#3697D311yescharacteristic, complemented
45#63513—34no—
46#63512—32no—
47#63511—32no—
48#63505—32no—
49#63495—32no—
50C2×#3667—31no—
51C2×#3667—61no—
52C23×C3×C9D411yescharacteristic
53C22×D3×C9—21no—
54C22×C3×D9—21no—
55C2×C3×C4×C9D411yescharacteristic
56C2×C3×C4×C9D411yescharacteristic
57C2×C3×C4×C9D411yescharacteristic
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×C4×C9C2311yescharacteristic
61C9×#67—22no—
62C9×#67—22no—
63C3×#611—22no—
64C3×#611—22no—
65C2×C9×Dic3—21no—
66C2×C3×Dic9—21no—
67C9×D12—21no—
68C9×D12C2312yes—
69C4×D3×C9—22no—
70C4×D3×C9—22no—
71C3×D36—21no—
72C3×D36C2312yes—
73C3×C4×D9—22no—
74C3×C4×D9—22no—
75C9×Dic6—21no—
76C9×Dic6C2312yes—
77C3×Dic18—21no—
78C3×Dic18C2312yes—
79C3⋊(C9⋊C8)C2314yes—
80#5822—91no—
81#5822—91no—
82C22×C32×C4C2×D311yescharacteristic
83C2×C3×#67—21no—
84C2×C3×#67—61no—
85C3×#235—21no—
86C3×#235—21no—
87C3×#235—32no—
88C3×#235—32no—
89C3×#235—61no—
90C3×#235—61no—
91C3×#235C2×D312yes—
92C3×#235C2×D312yes—
93C2×C3×D12—31no—
94C2×C3×D12C2×D311yescharacteristic
95C2×C3×C4×D3—21no—
96C2×C3×C4×D3—61no—
97C2×C3×Dic6—31no—
98C2×C3×Dic6C2×D311yescharacteristic
99#3748—32no—
100#3748—32no—
101C2×C3⋊(C3⋊C8)—32no—
102#3718—34no—
103#3715—34no—
104#3715—34no—
105#3713—34no—
106C9×#41—32no—
107C9×#41—32no—
108C9×#41—61no—
109C9×#41—61no—
110C2×Q8×C9—31no—
111C2×D4×C9—31no—
112C2×D4×C9—61no—
113C22×C4×C9—21no—
114C22×C4×C9—61no—
115C22×C4×C9C2×D311yescharacteristic
116C2×#611—21no—
117#3697—21no—
118#3697—21no—
119#3697C2×D312yes—
120#3697C2×D312yescomplemented
121C2×D36C2×D311yescharacteristic, complemented
122C2×C4×D9—21no—
123C2×Dic18C2×D311yescharacteristic, complemented
124#3675—34no—
125#3674—34no—
126#3673—34no—
127#3672—34no—
128#3667—32no—
129#3667—32no—
130#3667—62no—
131#3667—62no—
132C2×C9⋊C8—32no—
133C2×C9⋊C8—62no—
134C22×C3×C9—41no—
135C22×C3×C9C2×D411yescharacteristic
136C22×C3×C9C2×D411yescharacteristic
137C22×C3×C9C2×D411yescharacteristic
138C2×D3×C9—22no—
139C2×D3×C9—42no—
140C2×C3×D9—22no—
141C2×C3×D9—42no—
142C3×C4×C9C2×D411yescharacteristic
143C3×C4×C9C2×D411yescharacteristic
144C3×C4×C9C2×D411yescharacteristic
145C3×C4×C9C2411yescharacteristic
146C9×Dic3—22no—
147C3×Dic9—22no—
148C2×C3×#41—31no—
149C2×C3×#41—91no—
150C2×#235—31no—
151C2×#235D911yescharacteristic, complemented
152#951—92no—
153#951—92no—
154#950—94no—
155#950—94no—
156#949—92no—
157#949—92no—
158#942—92no—
159#942—92no—
160#932—92no—
161#932—92no—
162C2×#208—91no—
163C2×#208—91no—
164C2×#208—181no—
165C23×C32#6711yescharacteristic
166C22×C3×D3—21no—
167C22×C3×D3—61no—
168C2×C32×C4#6711yescharacteristic
169C2×C32×C4#6711yescharacteristic
170C2×C32×C4#6711yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C3×#67—22no—
175C3×#67—22no—
176C3×#67—62no—
177C3×#67—62no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—61no—
180C3×D12—21no—
181C3×D12—32no—
182C3×D12—61no—
183C3×D12C22×D312yes—
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×C4×D3—62no—
187C3×C4×D3—62no—
188C3×Dic6—21no—
189C3×Dic6—32no—
190C3×Dic6—61no—
191C3×Dic6C22×D312yes—
192C23×C9—21no—
193C23×C9—61no—
194C23×C9#6711yescharacteristic
195C22×D9—21no—
196C3⋊(C3⋊C8)—34no—
197Q8×C9—32no—
198Q8×C9—61no—
199D4×C9—32no—
200D4×C9—62no—
201D4×C9—62no—
202D4×C9—61no—
203C2×C4×C9—21no—
204C2×C4×C9—21no—
205C2×C4×C9—21no—
206C2×C4×C9—21no—
207C2×C4×C9—21no—
208C2×C4×C9—21no—
209C2×C4×C9—61no—
210C2×C4×C9—62no—
211C2×C4×C9—62no—
212C2×C4×C9#6711yescharacteristic
213C2×C4×C9#6711yescharacteristic
214C2×C4×C9#6711yescharacteristic
215C2×C4×C9C22×D311yescharacteristic
216C2×C4×C9C22×D311yescharacteristic
217C2×C4×C9C22×D311yescharacteristic
218#611—22no—
219#611—22no—
220C2×Dic9—21no—
221D36—21no—
222D36C22×D312yes—
223C4×D9—22no—
224C4×D9—22no—
225Dic18—21no—
226Dic18C22×D312yes—
227C9⋊C8—34no—
228C9⋊C8—64no—
229#575—271noSylow
230C2×C3×C9—21no—
231C2×C3×C9—21no—
232C2×C3×C9—21no—
233C2×C3×C9C22×D411yescharacteristic
234D3×C9—42no—
235C3×D9—42no—
236C3×#41—32no—
237C3×#41—32no—
238C3×#41—61no—
239C3×#41—61no—
240C3×#41—92no—
241C3×#41—92no—
242C3×#41—181no—
243C3×#41—181no—
244C2×C3×Q8—31no—
245C2×C3×Q8—91no—
246C2×C3×D4—31no—
247C2×C3×D4—61no—
248C2×C3×D4—91no—
249C2×C3×D4—181no—
250C22×C3×C4—21no—
251C22×C3×C4—61no—
252C22×C3×C4—181no—
253C22×C3×C4C2×D911yescharacteristic
254C22×C3×C4D3211yescharacteristic
255C2×#67—21no—
256C2×#67—61no—
257#235—21no—
258#235—21no—
259#235—32no—
260#235—32no—
261#235—61no—
262#235—61no—
263#235C2×D912yes—
264#235C2×D912yescomplemented
265C2×D12—31no—
266C2×D12C2×D911yescharacteristic, complemented
267C2×C4×D3—21no—
268C2×C4×D3—61no—
269C2×Dic6—31no—
270C2×Dic6C2×D911yescharacteristic, complemented
271#216—94no—
272#216—94no—
273#215—94no—
274#215—94no—
275#214—94no—
276#214—94no—
277#213—94no—
278#213—94no—
279#208—92no—
280#208—92no—
281#208—92no—
282#208—92no—
283#208—182no—
284#208—182no—
285C2×C3⋊C8—92no—
286C2×C3⋊C8—92no—
287C2×C3⋊C8—182no—
288C22×C32—41no—
289C22×C32C2×#6711yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C22×C32C2×#6711yescharacteristic
292C2×C3×D3—22no—
293C2×C3×D3—42no—
294C2×C3×D3—62no—
295C2×C3×D3—122no—
296C32×C4C2×#6711yescharacteristic
297C32×C4C2×#6711yescharacteristic
298C32×C4C2×#6711yescharacteristic
299C32×C4C23×D311yescharacteristic
300C3×Dic3—22no—
301C3×Dic3—62no—
302C22×C9—21no—
303C22×C9—21no—
304C22×C9—21no—
305C22×C9—41no—
306C22×C9—42no—
307C22×C9—62no—
308C22×C9—122no—
309C22×C9C2×#6711yescharacteristic
310C22×C9C2×#6711yescharacteristic
311C22×C9C2×#6711yescharacteristic
312C2×D9—22no—
313C2×D9—42no—
314C4×C9—21no—
315C4×C9—21no—
316C4×C9—21no—
317C4×C9—21no—
318C4×C9—62no—
319C4×C9C2×#6711yescharacteristic
320C4×C9C2×#6711yescharacteristic
321C4×C9C2×#6711yescharacteristic
322C4×C9C23×D311yescharacteristic
323Dic9—22no—
324C2×#41—31no—
325C2×#41—91no—
326#137—272no—
327#137—272no—
328AGL(1,ℤ/8)—272no—
329AGL(1,ℤ/8)—272no—
330#135—274no—
331C2×M16—271no—
332C3×C9#57511yescharacteristic, complemented, Sylow
333C23×C3—21no—
334C23×C3—61no—
335C23×C3—181no—
336C23×C3#61111yescharacteristic
337C23×C3#62511yescharacteristic
338C22×D3—21no—
339C22×D3—61no—
340C3×Q8—32no—
341C3×Q8—61no—
342C3×Q8—92no—
343C3×Q8—181no—
344C3×D4—32no—
345C3×D4—62no—
346C3×D4—62no—
347C3×D4—61no—
348C3×D4—92no—
349C3×D4—182no—
350C3×D4—182no—
351C3×D4—181no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—181no—
362C2×C3×C4—182no—
363C2×C3×C4—182no—
364C2×C3×C4#61111yescharacteristic
365C2×C3×C4#61111yescharacteristic
366C2×C3×C4#61111yescharacteristic
367C2×C3×C4C22×D911yescharacteristic
368C2×C3×C4C22×D911yescharacteristic
369C2×C3×C4C22×D911yescharacteristic
370C2×C3×C4#62511yescharacteristic
371C2×C3×C4#62511yescharacteristic
372C2×C3×C4#62511yescharacteristic
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376#67—22no—
377#67—22no—
378#67—62no—
379#67—62no—
380C2×Dic3—21no—
381C2×Dic3—61no—
382D12—21no—
383D12—32no—
384D12—61no—
385D12C22×D912yes—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—62no—
389C4×D3—62no—
390Dic6—21no—
391Dic6—32no—
392Dic6—61no—
393Dic6C22×D912yes—
394C3⋊C8—94no—
395C3⋊C8—94no—
396C3⋊C8—184no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32C22×#6711yescharacteristic
401C3×D3—42no—
402C3×D3—122no—
403C2×C9—22no—
404C2×C9—21no—
405C2×C9—21no—
406C2×C9—21no—
407C2×C9—21no—
408C2×C9—41no—
409C2×C9—41no—
410C2×C9—122no—
411C2×C9C22×#6711yescharacteristic
412D9—42no—
413#41—32no—
414#41—32no—
415#41—61no—
416#41—61no—
417#41—92no—
418#41—92no—
419#41—181no—
420#41—181no—
421C2×Q8—31no—
422C2×Q8—91no—
423C2×D4—31no—
424C2×D4—61no—
425C2×D4—91no—
426C2×D4—181no—
427C22×C4—61no—
428C22×C4—181no—
429C22×C4D3×D911yescharacteristic
430Q16—274no—
431SD16—274no—
432SD16—274no—
433D8—274no—
434M16—272no—
435M16—272no—
436C2×C8—272no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—42no—
443C22×C3—62no—
444C22×C3—122no—
445C22×C3—182no—
446C22×C3—362no—
447C22×C3C2×#61111yescharacteristic
448C22×C3C2×#61111yescharacteristic
449C22×C3C2×#61111yescharacteristic
450C22×C3C2×#62511yescharacteristic
451C22×C3C2×#62511yescharacteristic
452C22×C3C2×#62511yescharacteristic
453C2×D3—22no—
454C2×D3—42no—
455C2×D3—62no—
456C2×D3—122no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—62no—
462C3×C4—182no—
463C3×C4C2×#61111yescharacteristic
464C3×C4C2×#61111yescharacteristic
465C3×C4C2×#61111yescharacteristic
466C3×C4C23×D911yescharacteristic
467C3×C4C2×#62511yescharacteristic
468C3×C4C2×#62511yescharacteristic
469C3×C4C2×#62511yescharacteristic
470C3×C4C22×D3211yescharacteristic
471Dic3—22no—
472Dic3—62no—
473C32#582211yescharacteristic
474C9—21no—
475C9#582211yescharacteristic, complemented
476C23—61no—
477C23—181no—
478C23#619411yescharacteristic
479Q8—32no—
480Q8—61no—
481Q8—92no—
482Q8—181no—
483D4—32no—
484D4—62no—
485D4—62no—
486D4—61no—
487D4—92no—
488D4—182no—
489D4—182no—
490D4—181no—
491C2×C4—61no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—181no—
495C2×C4—182no—
496C2×C4—182no—
497C2×C4#619411yescharacteristic
498C2×C4#619411yescharacteristic
499C2×C4#619411yescharacteristic
500C2×C4C2×D3×D911yescharacteristic
501C2×C4C2×D3×D911yescharacteristic
502C2×C4C2×D3×D911yescharacteristic
503C8—274no—
504C2×C3—22no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—122no—
515C2×C3—362no—
516C2×C3C22×#61111yescharacteristic
517C2×C3C22×#62511yescharacteristic
518D3—42no—
519D3—122no—
520C22—41no—
521C22—62no—
522C22—122no—
523C22—182no—
524C22—362no—
525C22C2×#619411yescharacteristic
526C22C2×#619411yescharacteristic
527C22C2×#619411yescharacteristic
528C4—62no—
529C4—182no—
530C4C2×#619411yescharacteristic
531C4C2×#619411yescharacteristic
532C4C2×#619411yescharacteristic
533C4C22×D3×D911yescharacteristic
534C3—21no—
535C3#1058992511yescharacteristic, complemented
536C3#1059478611yescharacteristic
537C2—21no—
538C2—21no—
539C2—21no—
540C2—122no—
541C2—362no—
542C2C22×#619411yescharacteristic
subgroupG/H
centerC4C2×#6194
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC22×C3×C4×C9C22
socleC2×C32C22×#67
hypercenterC22×C4D3×D9
hypocenterC3×C9#575
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#235C2semidirect
2. C2×C3×#3697C2semidirect
3. C2×#88532C2semidirect
4. #11720893C2semidirect
5. #11720887C2semidirect
6. #11720883C2semidirect
7. #11720882C2semidirect
8. #11720881C2semidirect
17. C9×#235C22semidirect
18. C2×C9×D12C22semidirect
23. C3×#3697C22semidirect
24. C2×C3×D36C22semidirect
26. C2×C9×Dic6C22semidirect
27. C2×C3×Dic18C22semidirect
28. #88532C22semidirect
29. #88532C22semidirect
30. C2×C3⋊(C9⋊C8)C22semidirect
31. #88427C22semidirect
32. #88425C22semidirect
33. #88423C22semidirect
34. #88421C22semidirect
44. C2×#3697D3semidirect
120. #3697C2×D3semidirect
121. C2×D36C2×D3semidirect
123. C2×Dic18C2×D3semidirect
151. C2×#235D9semidirect
264. #235C2×D9semidirect
266. C2×D12C2×D9semidirect
270. C2×Dic6C2×D9semidirect
332. C3×C9#575semidirect
475. C9#5822semidirect
535. C3#10589925semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C2×C3×#36972
C2×#885322
#117208932
#117208872
#117208832
#117208822
#117208812
#105947863
#105899253