Finite Group Databasebeta

← #10780308

Subgroups — #10780308

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3252C211yesmaximal, characteristic
2#83337C211yesmaximal, characteristic, complemented
3#83129C211yesmaximal, characteristic
4#83124C211yesmaximal, characteristic
5#71832C211yesmaximal, characteristic, complemented
6#71155C211yesmaximal, characteristic, complemented
7#71151C211yesmaximal, characteristic, complemented
8#71139C211yesmaximal, characteristic, complemented
9#71135C211yesmaximal, characteristic, complemented
10#71058C211yesmaximal, characteristic, complemented
11#71054C211yesmaximal, characteristic, complemented
12#71042C211yesmaximal, characteristic, complemented
13#71038C211yesmaximal, characteristic, complemented
14#70508C211yesmaximal, characteristic, complemented
15#70492C211yesmaximal, characteristic, complemented
16#19381—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×#529C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#479—21no—
21C3×#477C2211yescharacteristic
22C3×#476C2211yescharacteristic
23C3×#470—21no—
24C3×#468—21no—
25C3×#464—21no—
26C3×#462C2211yescharacteristic
27C3×#461C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5093C2211yescharacteristic
30#5093C2211yescharacteristic
31#5093C2211yescharacteristic
32#5093C2211yescharacteristic
33#5069—21no—
34#5069—21no—
35#5064—21no—
36#5055—21no—
37#5044—21no—
38#5034—21no—
39#5024C2211yescharacteristic
40Q8×C3⋊C8C2211yescharacteristic
41#5020C2211yescharacteristic
42#5020C2211yescharacteristic
43#5012—21no—
44#5012C2211yescharacteristic
45#5011—21no—
46D4×C3⋊C8C2211yescharacteristic
47#5007C2211yescharacteristic
48#5007C2211yescharacteristic
49#5004—21no—
50#5004—21no—
51#5000—21no—
52#4997—21no—
53#4926C2211yescharacteristic
54#4925C2211yescharacteristic
55C4×#208C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4546C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4486C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3252D311yescharacteristic
69#3239—31no—
70#3139—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1700—31no—
75#1699—31no—
76#1684—31no—
77#1683—31no—
78#1680—31no—
79#1679—31no—
80#1569—31no—
81#1554—31no—
82#1549—31no—
83C22×Dic6—21no—
84C2×C3×Q16—21no—
85C2×C3×SD16—21no—
86C2×C3×M16—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×#126—21no—
90C3×#125—21no—
91C3×#125D411yescharacteristic
92C3×#124D411yescharacteristic
93C3×#123—21no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8D411yescharacteristic
97C3×#121D411yescharacteristic
98C3×C4×Q8—21no—
99C3×C4×Q8—21no—
100C3×C4×Q8C2311yescharacteristic
101C3×C4×D4—21no—
102C3×C4×D4C2311yescharacteristic
103C3×#117—21no—
104C3×#117—21no—
105C2×C3×C42C2311yescharacteristic
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×#208—21no—
109C2×#208D411yescharacteristic
110C2×#208D411yescharacteristic
111C22×C3⋊C8—21no—
112#869—21no—
113#869—21no—
114#869—21no—
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C3×#107—21no—
118C3×#106—21no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×#103—21no—
123C3×#103—21no—
124C3×#103—21no—
125C3×#102—21no—
126#835—21no—
127#835—21no—
128#832—21no—
129#832—21no—
130#830—21no—
131#830—21no—
132#830—21no—
133#830—21no—
134#810—21no—
135#810—21no—
136#810—21no—
137#810—21no—
138#810—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#808—21no—
143#808—21no—
144#807—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152#803—21no—
153#803—21no—
154#803D411yescharacteristic
155#803D411yescharacteristic
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533C2×D311yescharacteristic
161C2×C4⋊Q8—31no—
162#529C2×D311yescharacteristic
163#501—61no—
164#491—61no—
165#491—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#487—31no—
170#487—31no—
171#484—61no—
172#481—61no—
173#479—21no—
174#477C2×D311yescharacteristic
175#476C2×D311yescharacteristic
176#475—31no—
177#475—31no—
178#474—31no—
179#474—31no—
180#470—21no—
181#468—21no—
182#464—21no—
183#462C2×D311yescharacteristic
184#461C2×D311yescharacteristic
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432—31no—
193#431—31no—
194#423C2×D311yescharacteristic
195#419—61no—
196#419—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4#4111yescharacteristic
220C22×Dic3—41no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—42no—
224C2×Dic6—42no—
225C2×Dic6—41no—
226C3×Q16—81no—
227C3×SD16—81no—
228C3×M16—81no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4C2×D411yescharacteristic
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C22⋊C4—21no—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—41no—
249C3×C42—21no—
250C3×C42—41no—
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42#4111yescharacteristic
254C3×C42C2411yescharacteristic
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—41no—
260C4⋊Dic3—41no—
261#208—41no—
262#208—41no—
263#208—42no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—42no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C22×Q8—61no—
274C2×Q16—21no—
275C2×SD16—21no—
276C2×M16—21no—
277C2×M16—31no—
278C2×M16—31no—
279C2×M16—61no—
280C22×C8—61no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283C4⋊Q8—61no—
284C4⋊Q8—61no—
285C4⋊Q8—61no—
286C4⋊Q8C22×D311yescharacteristic
287C4⋊Q8C22×D311yescharacteristic
288#126—21no—
289#125—21no—
290#125#6711yescharacteristic
291#124#6711yescharacteristic
292#123—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8#6711yescharacteristic
296#121#6711yescharacteristic
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8C22×D311yescharacteristic
300C4×D4—21no—
301C4×D4C22×D311yescharacteristic
302#117—21no—
303#117—21no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C42C22×D311yescharacteristic
307#107—21no—
308#107—61no—
309#107—61no—
310#106—21no—
311#106—61no—
312#106—61no—
313C4⋊C8—21no—
314C4⋊C8—31no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8C22×D311yescharacteristic
321C4⋊C8C22×D311yescharacteristic
322#103—21no—
323#103—21no—
324#103—21no—
325#103—61no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#102—21no—
336#102—61no—
337#102—61no—
338C22⋊C8—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342#97—31no—
343#97—31no—
344#97—61no—
345#97—61no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C23×C3—21no—
351C23×C3#12111yescharacteristic
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—81no—
357C3×D4—21no—
358C3×D4—21no—
359C3×D4—81no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4#12111yescharacteristic
373C2×C3×C4#12111yescharacteristic
374C2×C3×C4#12111yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—82no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—82no—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—21no—
389C3⋊C8—21no—
390C3⋊C8—21no—
391C3⋊C8—21no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—122no—
401C2×Q8—122no—
402C2×Q8—121no—
403C2×Q8C2×#6711yescharacteristic
404C2×D4—21no—
405C2×D4C2×#6711yescharacteristic
406C22×C4—21no—
407C22×C4—121no—
408C22×C4D3×D411yescharacteristic
409C22×C4#23711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411Q16—81no—
412SD16—81no—
413M16—81no—
414M16—121no—
415M16—121no—
416M16—122no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—31no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—31no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—122no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—121no—
443C4⋊C4—121no—
444C4⋊C4C2×#6711yescharacteristic
445C4⋊C4C2×#6711yescharacteristic
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—41no—
452C42—21no—
453C42—41no—
454C42D3×D411yescharacteristic
455C42#23711yescharacteristic
456C42C2×#6711yescharacteristic
457C42C23×D311yescharacteristic
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3C2×#12111yescharacteristic
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4#57511yescharacteristic
475C3×C4#57511yescharacteristic
476C3×C4#57711yescharacteristic
477C3×C4#57811yescharacteristic
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23#93911yescharacteristic
482Q8—21no—
483Q8—21no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—121no—
488Q8—121no—
489Q8—121no—
490Q8—121no—
491Q8—242no—
492D4—21no—
493D4—21no—
494D4—81no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—242no—
510C2×C4#93911yescharacteristic
511C2×C4#93911yescharacteristic
512C2×C4#93911yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×#23711yescharacteristic
515C2×C4C22×#6711yescharacteristic
516C8—41no—
517C8—41no—
518C8—61no—
519C8—61no—
520C8—61no—
521C8—61no—
522C8—121no—
523C8—121no—
524C8—121no—
525C8—121no—
526C2×C3—41no—
527C2×C3—81no—
528C2×C3#304011yescharacteristic
529C2×C3#306311yescharacteristic
530C2×C3#306911yescharacteristic
531C22—21no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22C2×#93911yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—241no—
548C4—241no—
549C4#577711yescharacteristic
550C4#577711yescharacteristic
551C4#583611yescharacteristic
552C4#583811yescharacteristic
553C3#1938111yescharacteristic, complemented, Sylow
554C2—41no—
555C2—81no—
556C2#8342311yescharacteristic
557C2#8396811yescharacteristic
558C2#8400311yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3252C2
socleC22×C3C2×#121
prime radicalC3×#121D4
hypercenter#3252D3
hypocenterC3#19381
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3252

Complements

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

NH ≅ G/Ntype
2. #83337C2semidirect
5. #71832C2semidirect
6. #71155C2semidirect
7. #71151C2semidirect
8. #71139C2semidirect
9. #71135C2semidirect
10. #71058C2semidirect
11. #71054C2semidirect
12. #71042C2semidirect
13. #71038C2semidirect
14. #70508C2semidirect
15. #70492C2semidirect
553. C3#19381semidirect

Maximal subgroups

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

subgroupindex
C3×#32522
#833372
#831292
#831242
#718322
#711552
#711512
#711392
#711352
#710582
#710542
#710422
#710382
#705082
#704922
#193813