Finite Group Databasebeta

← D4×#6821 #14483062

Subgroups — D4×#6821 #14483062

Order 1920.

3296 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6821C212yesmaximal, complemented
2C3×C5×D4×Q8C211yesmaximal, characteristic, complemented
3#11756576C212yesmaximal, complemented
4#11756485C213yesmaximal, complemented
5#11756484C213yesmaximal, complemented
6#11756389C216yesmaximal, complemented
7#11756386C216yesmaximal, complemented
8#11756336C211yesmaximal, characteristic, complemented
9C4×#6821C211yesmaximal, characteristic, complemented
10D4×D60C213yesmaximal, complemented
11C4×D4×D15C213yesmaximal, complemented
12D4×#713—31nomaximal, Hall
13C22×C3×C5×Q8C2212yescomplemented
14C2×#6821—216no—
15C2×#6821C2211yescharacteristic
16C2×#6821C2212yescomplemented
17C2×#6821C2216yescomplemented
18C2×#6821C2216yescomplemented
19C2×D4×D15C2213yes—
20C22×D60C2216yescomplemented
21C22×C4×D15C2216yescomplemented
22C3×C5×C4⋊Q8C2213yescomplemented
23C3×C5×C22⋊Q8C2216yescomplemented
24C3×C4×C5×Q8C2211yescharacteristic, complemented
25C3×C4×C5×D4C2213yes—
26#91836C2216yescomplemented
27#91835C2216yescomplemented
28Q8×Dic15C2211yescharacteristic, complemented
29D4×Dic15C2211yescharacteristic
30#91819C2216yescomplemented
31C4×#1220C2216yescomplemented
32#91784C2216yescomplemented
33#91784C2216yescomplemented
34#91784C2216yescomplemented
35#91783C22112yescomplemented
36#91782C2213yescomplemented
37#91782C2213yescomplemented
38#91782C2216yescomplemented
39#91781C2213yescomplemented
40C4⋊C4×D15C2213yescomplemented
41#91771C22112yescomplemented
42#91770C2216yescomplemented
43C22⋊C4×D15C2216yescomplemented
44#91763C2213yescomplemented
45C4×D60C2213yescomplemented
46C4×D60C2213yescomplemented
47C42×D15C2213yescomplemented
48D4×#239—51nomaximal, Hall
49C22×#713—32no—
50C5×D4×Q8D311yescharacteristic, complemented
51#66151—32no—
52#66060—33no—
53#66059—33no—
54#65964—36no—
55#65961—36no—
56#65911—31no—
57C4×#713—31no—
58D4×D20—33no—
59C4×D4×D5—33no—
60C23×D15C2316yes—
61C2×C3×C5×Q8—28no—
62C2×C3×C5×Q8C2316yes—
63C2×C3×C5×Q8C2311yescharacteristic
64C2×C3×C5×D4C2311yescharacteristic
65C22×C3×C4×C5C2316yes—
66C2×#1220C2316yes—
67C22×Dic15C2312yes—
68#6821—28no—
69#6821—224no—
70#6821—224no—
71#6821D418yesdirect product factor
72D4×D15—212no—
73C2×D60—224no—
74C2×D60C2313yes—
75C2×D60C2316yes—
76C2×D60C23112yes—
77C2×D60C2316yes—
78C2×D60C2313yes—
79C2×C4×D15—224no—
80C2×C4×D15C2313yes—
81C2×C4×D15C2316yes—
82C2×C4×D15C2316yes—
83C2×C4×D15C2316yes—
84C2×C4×D15C2313yes—
85C2×C4×D15C2313yes—
86C3×C5×C4⋊C4C2313yes—
87C3×C5×C4⋊C4C2313yes—
88C3×C5×C4⋊C4C2316yes—
89C3×C5×C22⋊C4C2316yes—
90C3×C42×C5C2313yes—
91#6719C2312yes—
92#6714C2316yes—
93#6714C2316yes—
94#6714C2316yes—
95#6714C23112yes—
96C4⋊Dic15C2311yescharacteristic
97C4⋊Dic15C2313yes—
98#6712C2316yes—
99C4×Dic15C2311yescharacteristic
100C4×Dic15C2313yes—
101C4×Dic15C2313yes—
102C22×#239—52no—
103C3×D4×Q8D511yescharacteristic, complemented
104#5817—52no—
105#5726—53no—
106#5725—53no—
107#5630—56no—
108#5627—56no—
109#5577—51no—
110C4×#239—51no—
111D4×D12—53no—
112C4×D3×D4—53no—
113C22×C5×Q8C2×D312yescomplemented
114C2×#713—31no—
115C2×#713—36no—
116C2×#713—36no—
117C2×#713—32no—
118C2×#713—616no—
119C2×D4×D5—33no—
120C22×D20—36no—
121C22×C4×D5—36no—
122C5×C4⋊Q8C2×D313yescomplemented
123C5×C22⋊Q8C2×D316yescomplemented
124C4×C5×Q8C2×D311yescharacteristic, complemented
125C4×C5×D4C2×D313yes—
126#4093—36no—
127#4092—36no—
128Q8×Dic5—31no—
129D4×Dic5—31no—
130#4076—36no—
131C4×C22⋊D5—36no—
132#4041—36no—
133#4041—36no—
134#4041—36no—
135#4040—312no—
136#4039—33no—
137#4039—36no—
138#4039—33no—
139#4038—33no—
140D5×C4⋊C4—33no—
141#4028—312no—
142#4027—36no—
143D5×C22⋊C4—36no—
144#4020—33no—
145C4×D20—33no—
146C4×D20—33no—
147C42×D5—33no—
148D4×#41—151noSylow
149C23×C3×C5C2412yes—
150C22×D15—212no—
151C22×D15—212no—
152C22×D15—46no—
153C22×D15C2416yes—
154C22×D15C2413yes—
155C3×C5×Q8—212no—
156C3×C5×Q8C2×D414yescomplemented
157C3×C5×D4#4114yescomplemented
158C2×C3×C4×C5—212no—
159C2×C3×C4×C5C2413yes—
160C2×C3×C4×C5C2416yes—
161C2×C3×C4×C5C2413yes—
162C2×C3×C4×C5C2411yescharacteristic
163#1220—212no—
164C2×Dic15—24no—
165C2×Dic15C2411yescharacteristic
166C2×Dic15C2413yes—
167C2×Dic15C2412yes—
168C2×Dic15C2411yescharacteristic
169D60—224no—
170D60—212no—
171D60—26no—
172D60—46no—
173D60C2×D4112yescomplemented
174C4×D15—212no—
175C4×D15—212no—
176C4×D15—26no—
177C4×D15—212no—
178C4×D15—43no—
179C4×D15C2×D4112yescomplemented
180C22×C3×Q8C2×D512yescomplemented
181C2×#239—51no—
182C2×#239—56no—
183C2×#239—56no—
184C2×#239—52no—
185C2×#239—1016no—
186C2×D3×D4—53no—
187C22×D12—56no—
188C22×C4×D3—56no—
189C3×C4⋊Q8C2×D513yescomplemented
190C3×C22⋊Q8C2×D516yescomplemented
191C3×C4×Q8C2×D511yescharacteristic, complemented
192C3×C4×D4C2×D513yes—
193#947—56no—
194#946—56no—
195Q8×Dic3—51no—
196D4×Dic3—51no—
197#930—56no—
198C4×#67—56no—
199#895—56no—
200#895—56no—
201#895—56no—
202#894—512no—
203#893—53no—
204#893—56no—
205#893—53no—
206#892—53no—
207D3×C4⋊C4—53no—
208#882—512no—
209#67⋊C4—56no—
210D3×C22⋊C4—56no—
211#874—53no—
212C4×D12—53no—
213C4×D12—53no—
214C42×D3—53no—
215C23×D5—36no—
216C2×C5×Q8—28no—
217C2×C5×Q8C22×D316yes—
218C2×C5×Q8C22×D311yescharacteristic
219C2×C5×D4C22×D311yescharacteristic
220C22×C4×C5C22×D316yes—
221C2×C22⋊D5—36no—
222C22×Dic5—32no—
223#713—38no—
224#713—624no—
225#713—624no—
226#713—68no—
227D4×D5—612no—
228C2×D20—33no—
229C2×D20—36no—
230C2×D20—312no—
231C2×D20—33no—
232C2×D20—36no—
233C2×D20—624no—
234C2×C4×D5—33no—
235C2×C4×D5—36no—
236C2×C4×D5—36no—
237C2×C4×D5—36no—
238C2×C4×D5—33no—
239C2×C4×D5—33no—
240C2×C4×D5—624no—
241C5×C4⋊C4C22×D313yes—
242C5×C4⋊C4C22×D313yes—
243C5×C4⋊C4C22×D316yes—
244C5×C22⋊C4C22×D316yes—
245C42×C5C22×D313yes—
246C22⋊Dic5—32no—
247#685—36no—
248#685—36no—
249#685—36no—
250#685—312no—
251C4⋊Dic5—31no—
252C4⋊Dic5—33no—
253#683—36no—
254C4×Dic5—31no—
255C4×Dic5—33no—
256C4×Dic5—33no—
257C22×#41—152no—
258#550—151no—
259D4×Q8D1511yescharacteristic, complemented
260#548—152no—
261#547—153no—
262#546—156no—
263D42—153no—
264#532—153no—
265#525—156no—
266C4×#41—151no—
267C2×C4×D4—153no—
268C22×C3×C5—22no—
269C22×C3×C5—22no—
270C22×C3×C5C2×#4114yescomplemented
271C22×C3×C5C2511yescharacteristic
272C2×D15—23no—
273C2×D15—23no—
274C2×D15—26no—
275C2×D15—26no—
276C2×D15—26no—
277C2×D15—424no—
278C2×D15C22×D416yes—
279C3×C4×C5—26no—
280C3×C4×C5—23no—
281C3×C4×C5C22×D416yes—
282C3×C4×C5C2×#4112yescomplemented
283Dic15—22no—
284Dic15—23no—
285Dic15—21no—
286Dic15C22×D412yes—
287C23×D3—56no—
288C2×C3×Q8—28no—
289C2×C3×Q8C22×D516yes—
290C2×C3×Q8C22×D511yescharacteristic
291C2×C3×D4C22×D511yescharacteristic
292C22×C3×C4C22×D516yes—
293C2×#67—56no—
294C22×Dic3—52no—
295#239—58no—
296#239—1024no—
297#239—1024no—
298#239—108no—
299D3×D4—1012no—
300C2×D12—53no—
301C2×D12—53no—
302C2×D12—56no—
303C2×D12—512no—
304C2×D12—56no—
305C2×D12—1024no—
306C2×C4×D3—53no—
307C2×C4×D3—53no—
308C2×C4×D3—56no—
309C2×C4×D3—56no—
310C2×C4×D3—56no—
311C2×C4×D3—53no—
312C2×C4×D3—1024no—
313C3×C4⋊C4C22×D513yes—
314C3×C4⋊C4C22×D513yes—
315C3×C4⋊C4C22×D516yes—
316C3×C22⋊C4C22×D516yes—
317C3×C42C22×D513yes—
318#217—52no—
319(C2×D3)⋊C4—56no—
320(C2×D3)⋊C4—56no—
321(C2×D3)⋊C4—56no—
322(C2×D3)⋊C4—512no—
323C4⋊Dic3—51no—
324C4⋊Dic3—53no—
325Dic3⋊C4—56no—
326C4×Dic3—51no—
327C4×Dic3—53no—
328C4×Dic3—53no—
329C23×C5C23×D312yes—
330C22×D5—33no—
331C22×D5—36no—
332C22×D5—612no—
333C22×D5—612no—
334C22×D5—126no—
335C5×Q8—212no—
336C5×Q8D3×D414yescomplemented
337C5×D4#23914yescomplemented
338C2×C4×C5—212no—
339C2×C4×C5C23×D313yes—
340C2×C4×C5C23×D316yes—
341C2×C4×C5C23×D313yes—
342C2×C4×C5C23×D311yescharacteristic
343C22⋊D5—612no—
344C2×Dic5—31no—
345C2×Dic5—31no—
346C2×Dic5—33no—
347C2×Dic5—32no—
348C2×Dic5—64no—
349D20—312no—
350D20—624no—
351D20—612no—
352D20—66no—
353D20—126no—
354C4×D5—312no—
355C4×D5—612no—
356C4×D5—612no—
357C4×D5—612no—
358C4×D5—66no—
359C4×D5—123no—
360C2×#41—151no—
361C2×#41—156no—
362C2×#41—156no—
363C2×#41—152no—
364C2×#41—3016no—
365C22×Q8C2×D1512yescomplemented
366C22×D4—153no—
367C22×D4—156no—
368C23×C4—156no—
369C4⋊Q8C2×D1513yescomplemented
370#127—153no—
371#124—156no—
372#123—1512no—
373C22⋊Q8—156no—
374C22⋊Q8C2×D1516yescomplemented
375#121—156no—
376#121—156no—
377#121—156no—
378#121—156no—
379#120—1512no—
380C4×Q8—151no—
381C4×Q8C2×D1511yescharacteristic, complemented
382C4×D4—151no—
383C4×D4—156no—
384C4×D4—153no—
385C4×D4—153no—
386C4×D4—156no—
387C4×D4—156no—
388C4×D4—153no—
389C4×D4—153no—
390C4×D4C2×D1513yes—
391#117—153no—
392C2×C4⋊C4—153no—
393C2×C22⋊C4—156no—
394C2×C42—153no—
395C2×C3×C5—24no—
396C2×C3×C5C23×D411yescharacteristic
397C2×C3×C5C22×#4111yescharacteristic
398C2×C3×C5#58511yescharacteristic
399D15—26no—
400D15—46no—
401C23×C3C23×D512yes—
402C22×D3—53no—
403C22×D3—56no—
404C22×D3—1012no—
405C22×D3—1012no—
406C22×D3—206no—
407C3×Q8—212no—
408C3×Q8D4×D514yescomplemented
409C3×D4#71314yescomplemented
410C2×C3×C4—212no—
411C2×C3×C4C23×D513yes—
412C2×C3×C4C23×D516yes—
413C2×C3×C4C23×D513yes—
414C2×C3×C4C23×D511yescharacteristic
415#67—1012no—
416C2×Dic3—51no—
417C2×Dic3—51no—
418C2×Dic3—53no—
419C2×Dic3—52no—
420C2×Dic3—104no—
421D12—512no—
422D12—1024no—
423D12—1012no—
424D12—106no—
425D12—206no—
426C4×D3—512no—
427C4×D3—1012no—
428C4×D3—1012no—
429C4×D3—1012no—
430C4×D3—106no—
431C4×D3—203no—
432C22×C5—22no—
433C22×C5—22no—
434C22×C5C2×#23914yescomplemented
435C22×C5C24×D311yescharacteristic
436C2×D5—36no—
437C2×D5—63no—
438C2×D5—63no—
439C2×D5—66no—
440C2×D5—66no—
441C2×D5—66no—
442C2×D5—1224no—
443C4×C5—26no—
444C4×C5—23no—
445C4×C5C2×D3×D416yes—
446C4×C5C2×#23912yescomplemented
447Dic5—32no—
448Dic5—62no—
449Dic5—63no—
450Dic5—61no—
451C24—156no—
452#41—158no—
453#41—3024no—
454#41—3024no—
455#41—308no—
456C2×Q8—28no—
457C2×Q8C22×D1516yes—
458C2×Q8C22×D1511yescharacteristic
459C2×D4—156no—
460C2×D4—153no—
461C2×D4—153no—
462C2×D4—1512no—
463C2×D4—156no—
464C2×D4—156no—
465C2×D4—3024no—
466C2×D4—3012no—
467C2×D4C22×D1511yescharacteristic
468C22×C4—153no—
469C22×C4—152no—
470C22×C4—153no—
471C22×C4—156no—
472C22×C4—156no—
473C22×C4—156no—
474C22×C4—153no—
475C22×C4—3024no—
476C22×C4C22×D1516yes—
477C4⋊C4—151no—
478C4⋊C4—156no—
479C4⋊C4—153no—
480C4⋊C4C22×D1513yes—
481C4⋊C4C22×D1513yes—
482C4⋊C4C22×D1516yes—
483C22⋊C4—152no—
484C22⋊C4—156no—
485C22⋊C4—156no—
486C22⋊C4—156no—
487C22⋊C4—1512no—
488C22⋊C4C22×D1516yes—
489C42—151no—
490C42—153no—
491C42—153no—
492C42C22×D1513yes—
493C3×C5D4×#4111yescharacteristic, complemented, Hall
494C22×C3—22no—
495C22×C3—22no—
496C22×C3C2×#71314yescomplemented
497C22×C3C24×D511yescharacteristic
498C2×D3—56no—
499C2×D3—103no—
500C2×D3—103no—
501C2×D3—106no—
502C2×D3—106no—
503C2×D3—106no—
504C2×D3—2024no—
505C3×C4—26no—
506C3×C4—23no—
507C3×C4C2×D4×D516yes—
508C3×C4C2×#71312yescomplemented
509Dic3—52no—
510Dic3—102no—
511Dic3—103no—
512Dic3—101no—
513C2×C5—24no—
514C2×C5C22×D3×D411yescharacteristic
515C2×C5C22×#23911yescharacteristic
516C2×C5#596811yescharacteristic
517D5—66no—
518D5—126no—
519C23—153no—
520C23—156no—
521C23—3012no—
522C23—3012no—
523C23—606no—
524C23C23×D1512yes—
525Q8—212no—
526Q8D4×D1514yescomplemented
527D4—1512no—
528D4—3024no—
529D4—3012no—
530D4—3012no—
531D4—306no—
532D4—606no—
533D4#682114yesdirect product factor
534C2×C4—212no—
535C2×C4—151no—
536C2×C4—1512no—
537C2×C4—151no—
538C2×C4—153no—
539C2×C4—152no—
540C2×C4—3012no—
541C2×C4—3012no—
542C2×C4—304no—
543C2×C4—3012no—
544C2×C4—306no—
545C2×C4—603no—
546C2×C4C23×D1513yes—
547C2×C4C23×D1516yes—
548C2×C4C23×D1513yes—
549C2×C4C23×D1511yescharacteristic
550C2×C3—24no—
551C2×C3C22×D4×D511yescharacteristic
552C2×C3C22×#71311yescharacteristic
553C2×C3#6628711yescharacteristic
554D3—106no—
555D3—206no—
556C5D4×#23911yescharacteristic, complemented, Sylow
557C22—22no—
558C22—22no—
559C22—156no—
560C22—303no—
561C22—303no—
562C22—306no—
563C22—306no—
564C22—306no—
565C22—6024no—
566C22C2×#682114yescomplemented
567C22C24×D1511yescharacteristic
568C4—26no—
569C4—23no—
570C4—152no—
571C4—302no—
572C4—303no—
573C4—301no—
574C4C2×D4×D1516yes—
575C4C2×#682112yescomplemented
576C3D4×#71311yescharacteristic, complemented, Sylow
577C2—24no—
578C2—306no—
579C2—606no—
580C2C22×D4×D1511yescharacteristic
581C2C22×#682111yescharacteristic
582C2#1175715011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D15
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×D4×Q8C2
hypercenterD4×Q8D15
hypocenterC3×C5D4×#41
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#6821C2semidirect
2. C3×C5×D4×Q8C2semidirect
3. #11756576C2semidirect
4. #11756485C2semidirect
5. #11756484C2semidirect
6. #11756389C2semidirect
7. #11756386C2semidirect
8. #11756336C2semidirect
9. C4×#6821C2semidirect
10. D4×D60C2semidirect
11. C4×D4×D15C2semidirect
13. C22×C3×C5×Q8C22semidirect
16. C2×#6821C22semidirect
17. C2×#6821C22semidirect
18. C2×#6821C22semidirect
20. C22×D60C22semidirect
21. C22×C4×D15C22semidirect
22. C3×C5×C4⋊Q8C22semidirect
23. C3×C5×C22⋊Q8C22semidirect
24. C3×C4×C5×Q8C22semidirect
26. #91836C22semidirect
27. #91835C22semidirect
28. Q8×Dic15C22semidirect
30. #91819C22semidirect
31. C4×#1220C22semidirect
32. #91784C22semidirect
33. #91784C22semidirect
34. #91784C22semidirect
35. #91783C22semidirect
36. #91782C22semidirect
37. #91782C22semidirect
38. #91782C22semidirect
39. #91781C22semidirect
40. C4⋊C4×D15C22semidirect
41. #91771C22semidirect
42. #91770C22semidirect
43. C22⋊C4×D15C22semidirect
44. #91763C22semidirect
45. C4×D60C22semidirect
46. C4×D60C22semidirect
47. C42×D15C22semidirect
50. C5×D4×Q8D3semidirect
71. #6821D4direct (N × H)
103. C3×D4×Q8D5semidirect
113. C22×C5×Q8C2×D3semidirect
122. C5×C4⋊Q8C2×D3semidirect
123. C5×C22⋊Q8C2×D3semidirect
124. C4×C5×Q8C2×D3semidirect
156. C3×C5×Q8C2×D4semidirect
157. C3×C5×D4#41semidirect
173. D60C2×D4semidirect
179. C4×D15C2×D4semidirect
180. C22×C3×Q8C2×D5semidirect
189. C3×C4⋊Q8C2×D5semidirect
190. C3×C22⋊Q8C2×D5semidirect
191. C3×C4×Q8C2×D5semidirect
259. D4×Q8D15semidirect
270. C22×C3×C5C2×#41semidirect
282. C3×C4×C5C2×#41semidirect
336. C5×Q8D3×D4semidirect
337. C5×D4#239semidirect
365. C22×Q8C2×D15semidirect
369. C4⋊Q8C2×D15semidirect
374. C22⋊Q8C2×D15semidirect
381. C4×Q8C2×D15semidirect
408. C3×Q8D4×D5semidirect
409. C3×D4#713semidirect
434. C22×C5C2×#239semidirect
446. C4×C5C2×#239semidirect
493. C3×C5D4×#41semidirect
496. C22×C3C2×#713semidirect
508. C3×C4C2×#713semidirect
526. Q8D4×D15semidirect
533. D4#6821direct (N × H)
556. C5D4×#239semidirect
566. C22C2×#6821semidirect
575. C4C2×#6821semidirect
576. C3D4×#713semidirect

Maximal subgroups

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

subgroupindex
C22×#68212
C3×C5×D4×Q82
#117565762
#117564852
#117564842
#117563892
#117563862
#117563362
C4×#68212
D4×D602
C4×D4×D152
D4×#7133
D4×#2395