Finite Group Databasebeta

← #13176874

Subgroups — #13176874

Order 1728.

1844 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×C3×#3828C211yesmaximal, characteristic, complemented
2C2×C32×#235C211yesmaximal, characteristic, complemented
3C2×#88855C211yesmaximal, characteristic, complemented
4#11723375C212yesmaximal, complemented
5#11723373C212yesmaximal, complemented
6#11723365C212yesmaximal, complemented
7#11723364C212yesmaximal, complemented
8#11723363C214yesmaximal, complemented
9#10596293—31nomaximal
10#10594786—34nomaximal
11C22×C33×C4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×#3828—21no—
14C3×#3828—21no—
15C3×#3828C2212yes—
16C3×#3828C2212yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×C4×C3⋊D3—21no—
19C2×C3×#634C2211yescharacteristic, complemented
20C2×C32×#67—21no—
21C32×#235—21no—
22C32×#235—21no—
23C32×#235C2212yes—
24C32×#235C2212yescomplemented
25C2×C32×D12C2211yescharacteristic, complemented
26C2×C32×C4×D3—21no—
27C2×C32×Dic6C2211yescharacteristic, complemented
28#88855C2212yescomplemented
29#88855C2212yescomplemented
30C2×C3⋊(C3⋊(C3⋊C8))C2212yescomplemented
31#88798C2214yescomplemented
32#88793C2214yescomplemented
33#88792C2214yescomplemented
34#88789C2214yescomplemented
35C2×C32×#41—31no—
36C2×#3828D311yescharacteristic, complemented
37C2×C3×#235—34no—
38C2×C3×#235D314yescomplemented
39#64160—32no—
40#64159—34no—
41#64158—32no—
42#64151—32no—
43#64141—32no—
44C2×#3748—31no—
45C2×#3748—34no—
46C2×#3748—64no—
47#63827—38no—
48#63827—38no—
49#63823—38no—
50#63823—38no—
51#63822—316no—
52C23×C33D411yescharacteristic
53C22×C3×C3⋊D3—21no—
54C22×C32×D3—21no—
55C2×C33×C4D411yescharacteristic
56C2×C33×C4D411yescharacteristic
57C2×C33×C4D411yescharacteristic
58C2×C33×C4C2311yescharacteristic
59C2×C33×C4C2311yescharacteristic
60C2×C33×C4C2311yescharacteristic
61C3×#638—22no—
62C3×#638—22no—
63C2×C3×C3⋊Dic3—21no—
64C3×C4⋊(C3⋊D3)—21no—
65C3×C4⋊(C3⋊D3)C2312yes—
66C3×C4×C3⋊D3—22no—
67C3×C4×C3⋊D3—22no—
68C3×#634—21no—
69C3×#634C2312yes—
70C32×#67—22no—
71C32×#67—22no—
72C2×C32×Dic3—21no—
73C32×D12—21no—
74C32×D12C2312yes—
75C32×C4×D3—22no—
76C32×C4×D3—22no—
77C32×Dic6—21no—
78C32×Dic6C2312yes—
79C3⋊(C3⋊(C3⋊C8))C2314yes—
80#5822—94no—
81#5822—91no—
82C32×#41—32no—
83C32×#41—32no—
84C32×#41—61no—
85C32×#41—61no—
86C2×C32×Q8—31no—
87C2×C32×D4—31no—
88C2×C32×D4—61no—
89C22×C32×C4—24no—
90C22×C32×C4—61no—
91C22×C32×C4C2×D314yes—
92C22×C32×C4C2×D311yescharacteristic
93C2×#638—21no—
94#3828—21no—
95#3828—21no—
96#3828C2×D312yes—
97#3828C2×D312yescomplemented
98C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
99C2×C4×C3⋊D3—21no—
100C2×#634C2×D311yescharacteristic, complemented
101C2×C3×#67—24no—
102C2×C3×#67—64no—
103C3×#235—24no—
104C3×#235—24no—
105C3×#235—38no—
106C3×#235—38no—
107C3×#235—64no—
108C3×#235—64no—
109C3×#235C2×D318yes—
110C3×#235C2×D318yescomplemented
111C2×C3×D12—34no—
112C2×C3×D12C2×D314yescomplemented
113C2×C3×C4×D3—24no—
114C2×C3×C4×D3—64no—
115C2×C3×Dic6—34no—
116C2×C3×Dic6C2×D314yescomplemented
117#3756—34no—
118#3755—34no—
119#3754—34no—
120#3753—34no—
121#3748—32no—
122#3748—32no—
123#3748—38no—
124#3748—38no—
125#3748—68no—
126#3748—68no—
127C2×C3⋊(C3⋊C8)—32no—
128C2×C3⋊(C3⋊C8)—38no—
129C2×C3⋊(C3⋊C8)—68no—
130#3718—316no—
131#3715—316no—
132#3715—316no—
133#3713—316no—
134C22×C33—41no—
135C22×C33C2×D411yescharacteristic
136C22×C33C2×D411yescharacteristic
137C22×C33C2×D411yescharacteristic
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3—42no—
140C2×C32×D3—22no—
141C2×C32×D3—42no—
142C33×C4C2×D411yescharacteristic
143C33×C4C2×D411yescharacteristic
144C33×C4C2×D411yescharacteristic
145C33×C4C2411yescharacteristic
146C3×C3⋊Dic3—22no—
147C32×Dic3—22no—
148C2×C3×#41—34no—
149C2×C3×#41—91no—
150C2×#235—34no—
151C2×#235C3⋊D311yescharacteristic, complemented
152#951—98no—
153#951—92no—
154#950—916no—
155#950—94no—
156#949—98no—
157#949—92no—
158#942—98no—
159#942—92no—
160#932—98no—
161#932—92no—
162C2×#208—94no—
163C2×#208—91no—
164C2×#208—184no—
165C23×C32—24no—
166C23×C32—61no—
167C23×C32#6714yes—
168C23×C32#6711yescharacteristic
169C22×C3⋊D3—21no—
170C22×C3×D3—24no—
171C22×C3×D3—64no—
172C32×Q8—32no—
173C32×Q8—61no—
174C32×D4—32no—
175C32×D4—62no—
176C32×D4—62no—
177C32×D4—61no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—24no—
181C2×C32×C4—24no—
182C2×C32×C4—24no—
183C2×C32×C4—24no—
184C2×C32×C4—61no—
185C2×C32×C4—62no—
186C2×C32×C4—62no—
187C2×C32×C4#6714yes—
188C2×C32×C4#6714yes—
189C2×C32×C4#6714yes—
190C2×C32×C4#6711yescharacteristic
191C2×C32×C4#6711yescharacteristic
192C2×C32×C4#6711yescharacteristic
193C2×C32×C4C22×D314yes—
194C2×C32×C4C22×D314yes—
195C2×C32×C4C22×D314yes—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199#638—22no—
200#638—22no—
201C2×C3⋊Dic3—21no—
202C4⋊(C3⋊D3)—21no—
203C4⋊(C3⋊D3)C22×D312yes—
204C4×C3⋊D3—22no—
205C4×C3⋊D3—22no—
206#634—21no—
207#634C22×D312yes—
208C3×#67—28no—
209C3×#67—28no—
210C3×#67—68no—
211C3×#67—68no—
212C2×C3×Dic3—24no—
213C2×C3×Dic3—64no—
214C3×D12—24no—
215C3×D12—38no—
216C3×D12—64no—
217C3×D12C22×D318yes—
218C3×C4×D3—28no—
219C3×C4×D3—28no—
220C3×C4×D3—68no—
221C3×C4×D3—68no—
222C3×Dic6—24no—
223C3×Dic6—38no—
224C3×Dic6—64no—
225C3×Dic6C22×D318yes—
226C3⋊(C3⋊C8)—34no—
227C3⋊(C3⋊C8)—316no—
228C3⋊(C3⋊C8)—616no—
229#575—271noSylow
230C2×C33—21no—
231C2×C33—21no—
232C2×C33—21no—
233C2×C33C22×D411yescharacteristic
234C3×C3⋊D3—42no—
235C32×D3—42no—
236C3×#41—38no—
237C3×#41—38no—
238C3×#41—64no—
239C3×#41—64no—
240C3×#41—92no—
241C3×#41—92no—
242C3×#41—181no—
243C3×#41—181no—
244C2×C3×Q8—34no—
245C2×C3×Q8—91no—
246C2×C3×D4—34no—
247C2×C3×D4—64no—
248C2×C3×D4—91no—
249C2×C3×D4—181no—
250C22×C3×C4—24no—
251C22×C3×C4—64no—
252C22×C3×C4—181no—
253C22×C3×C4D3214yes—
254C22×C3×C4C2×C3⋊D311yescharacteristic
255C2×#67—21no—
256C2×#67—64no—
257#235—21no—
258#235—21no—
259#235—38no—
260#235—38no—
261#235—64no—
262#235—64no—
263#235C2×C3⋊D312yes—
264#235C2×C3⋊D312yescomplemented
265C2×D12—34no—
266C2×D12C2×C3⋊D311yescharacteristic, complemented
267C2×C4×D3—21no—
268C2×C4×D3—64no—
269C2×Dic6—34no—
270C2×Dic6C2×C3⋊D311yescharacteristic, complemented
271#216—916no—
272#216—94no—
273#215—916no—
274#215—94no—
275#214—916no—
276#214—94no—
277#213—916no—
278#213—94no—
279#208—98no—
280#208—98no—
281#208—92no—
282#208—92no—
283#208—188no—
284#208—188no—
285C2×C3⋊C8—98no—
286C2×C3⋊C8—92no—
287C2×C3⋊C8—188no—
288C22×C32—24no—
289C22×C32—24no—
290C22×C32—24no—
291C22×C32—41no—
292C22×C32—44no—
293C22×C32—48no—
294C22×C32—62no—
295C22×C32—122no—
296C22×C32C2×#6714yes—
297C22×C32C2×#6714yes—
298C22×C32C2×#6714yes—
299C22×C32C2×#6711yescharacteristic
300C22×C32C2×#6711yescharacteristic
301C22×C32C2×#6711yescharacteristic
302C2×C3⋊D3—22no—
303C2×C3⋊D3—42no—
304C2×C3×D3—28no—
305C2×C3×D3—48no—
306C2×C3×D3—68no—
307C2×C3×D3—128no—
308C32×C4—24no—
309C32×C4—24no—
310C32×C4—24no—
311C32×C4—24no—
312C32×C4—62no—
313C32×C4C2×#6714yes—
314C32×C4C2×#6714yes—
315C32×C4C2×#6714yes—
316C32×C4C2×#6711yescharacteristic
317C32×C4C2×#6711yescharacteristic
318C32×C4C2×#6711yescharacteristic
319C32×C4C23×D314yes—
320C32×C4C23×D311yescharacteristic
321C3⋊Dic3—22no—
322C3×Dic3—28no—
323C3×Dic3—68no—
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—
332C33#57511yescharacteristic, complemented, Sylow
333C23×C3—24no—
334C23×C3—64no—
335C23×C3—181no—
336C23×C3#62514yes—
337C23×C3#63811yescharacteristic
338C22×D3—21no—
339C22×D3—64no—
340C3×Q8—38no—
341C3×Q8—64no—
342C3×Q8—92no—
343C3×Q8—181no—
344C3×D4—38no—
345C3×D4—68no—
346C3×D4—68no—
347C3×D4—64no—
348C3×D4—92no—
349C3×D4—182no—
350C3×D4—182no—
351C3×D4—181no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—64no—
359C2×C3×C4—68no—
360C2×C3×C4—68no—
361C2×C3×C4—181no—
362C2×C3×C4—182no—
363C2×C3×C4—182no—
364C2×C3×C4#62514yes—
365C2×C3×C4#62514yes—
366C2×C3×C4#62514yes—
367C2×C3×C4#63811yescharacteristic
368C2×C3×C4#63811yescharacteristic
369C2×C3×C4#63811yescharacteristic
370C2×C3×C4C2×D3214yes—
371C2×C3×C4C2×D3214yes—
372C2×C3×C4C2×D3214yes—
373C2×C3×C4C22×C3⋊D311yescharacteristic
374C2×C3×C4C22×C3⋊D311yescharacteristic
375C2×C3×C4C22×C3⋊D311yescharacteristic
376#67—22no—
377#67—22no—
378#67—68no—
379#67—68no—
380C2×Dic3—21no—
381C2×Dic3—64no—
382D12—21no—
383D12—38no—
384D12—64no—
385D12C22×C3⋊D312yes—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—68no—
389C4×D3—68no—
390Dic6—21no—
391Dic6—38no—
392Dic6—64no—
393Dic6C22×C3⋊D312yes—
394C3⋊C8—916no—
395C3⋊C8—94no—
396C3⋊C8—1816no—
397C2×C32—28no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—24no—
401C2×C32—21no—
402C2×C32—24no—
403C2×C32—24no—
404C2×C32—24no—
405C2×C32—44no—
406C2×C32—44no—
407C2×C32—122no—
408C2×C32C22×#6714yes—
409C2×C32C22×#6711yescharacteristic
410C3⋊D3—42no—
411C3×D3—48no—
412C3×D3—128no—
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×C3⋊D311yescharacteristic
430Q16—274no—
431SD16—274no—
432SD16—274no—
433D8—274no—
434M16—272no—
435M16—272no—
436C2×C8—272no—
437C22×C3—24no—
438C22×C3—24no—
439C22×C3—24no—
440C22×C3—44no—
441C22×C3—41no—
442C22×C3—48no—
443C22×C3—68no—
444C22×C3—128no—
445C22×C3—182no—
446C22×C3—362no—
447C22×C3C2×#62514yes—
448C22×C3C2×#62514yes—
449C22×C3C2×#62514yes—
450C22×C3C2×#63811yescharacteristic
451C22×C3C2×#63811yescharacteristic
452C22×C3C2×#63811yescharacteristic
453C2×D3—22no—
454C2×D3—42no—
455C2×D3—68no—
456C2×D3—128no—
457C3×C4—24no—
458C3×C4—24no—
459C3×C4—24no—
460C3×C4—24no—
461C3×C4—68no—
462C3×C4—182no—
463C3×C4C2×#62514yes—
464C3×C4C2×#62514yes—
465C3×C4C2×#62514yes—
466C3×C4C2×#63811yescharacteristic
467C3×C4C2×#63811yescharacteristic
468C3×C4C2×#63811yescharacteristic
469C3×C4C22×D3214yes—
470C3×C4C23×C3⋊D311yescharacteristic
471Dic3—22no—
472Dic3—68no—
473C32—24no—
474C32#582214yescomplemented
475C32#582211yescharacteristic, complemented
476C23—61no—
477C23—181no—
478C23#629011yescharacteristic
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#629011yescharacteristic
498C2×C4#629011yescharacteristic
499C2×C4#629011yescharacteristic
500C2×C4C2×D3×C3⋊D311yescharacteristic
501C2×C4C2×D3×C3⋊D311yescharacteristic
502C2×C4C2×D3×C3⋊D311yescharacteristic
503C8—274no—
504C2×C3—28no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—24no—
509C2×C3—24no—
510C2×C3—24no—
511C2×C3—24no—
512C2×C3—44no—
513C2×C3—44no—
514C2×C3—128no—
515C2×C3—362no—
516C2×C3C22×#62514yes—
517C2×C3C22×#63811yescharacteristic
518D3—42no—
519D3—128no—
520C22—41no—
521C22—62no—
522C22—122no—
523C22—182no—
524C22—362no—
525C22C2×#629011yescharacteristic
526C22C2×#629011yescharacteristic
527C22C2×#629011yescharacteristic
528C4—62no—
529C4—182no—
530C4C2×#629011yescharacteristic
531C4C2×#629011yescharacteristic
532C4C2×#629011yescharacteristic
533C4C22×D3×C3⋊D311yescharacteristic
534C3—24no—
535C3#1059478614yescomplemented
536C3#1059629311yescharacteristic, complemented
537C2—21no—
538C2—21no—
539C2—21no—
540C2—122no—
541C2—362no—
542C2C22×#629011yescharacteristic
subgroupG/H
centerC4C2×#6290
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC2×C33C22×D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C2×C32×#235C2semidirect
3. C2×#88855C2semidirect
4. #11723375C2semidirect
5. #11723373C2semidirect
6. #11723365C2semidirect
7. #11723364C2semidirect
8. #11723363C2semidirect
16. C3×#3828C22semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C2×C3×#634C22semidirect
24. C32×#235C22semidirect
25. C2×C32×D12C22semidirect
27. C2×C32×Dic6C22semidirect
28. #88855C22semidirect
29. #88855C22semidirect
30. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
31. #88798C22semidirect
32. #88793C22semidirect
33. #88792C22semidirect
34. #88789C22semidirect
36. C2×#3828D3semidirect
38. C2×C3×#235D3semidirect
97. #3828C2×D3semidirect
98. C2×C4⋊(C3⋊D3)C2×D3semidirect
100. C2×#634C2×D3semidirect
110. C3×#235C2×D3semidirect
112. C2×C3×D12C2×D3semidirect
116. C2×C3×Dic6C2×D3semidirect
151. C2×#235C3⋊D3semidirect
264. #235C2×C3⋊D3semidirect
266. C2×D12C2×C3⋊D3semidirect
270. C2×Dic6C2×C3⋊D3semidirect
332. C33#575semidirect
474. C32#5822semidirect
475. C32#5822semidirect
535. C3#10594786semidirect
536. C3#10596293semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C32×#2352
C2×#888552
#117233752
#117233732
#117233652
#117233642
#117233632
#105962933
#105947863