Finite Group Databasebeta

← #29836609

Subgroups — #29836609

Order 15616.

748 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840198C211yesmaximal, characteristic, complemented
2C2×#16840196C211yesmaximal, characteristic, complemented
3#21457576C211yesmaximal, characteristic, complemented
4D8×C22⋊D61C211yesmaximal, characteristic, complemented
5#21448067C211yesmaximal, characteristic, complemented
6#21447921C211yesmaximal, characteristic, complemented
7#21447920C211yesmaximal, characteristic, complemented
8#21447858C211yesmaximal, characteristic, complemented
9#21447852C211yesmaximal, characteristic, complemented
10#21447744C211yesmaximal, characteristic, complemented
11#21447739C211yesmaximal, characteristic, complemented
12#21447732C211yesmaximal, characteristic, complemented
13#21447674C211yesmaximal, characteristic, complemented
14#21447648C211yesmaximal, characteristic, complemented
15C61×#2184C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic, complemented
17C2×#14519796C2211yescharacteristic, complemented
18C2×D8×D61C2211yescharacteristic
19#16840430—21no—
20D4×C22⋊D61—21no—
21#16840198—24no—
22#16840196—24no—
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519721C2211yescharacteristic
25C2×#14519721C2211yescharacteristic, complemented
26C2×D976C2211yescharacteristic, complemented
27C2×#14519718C2211yescharacteristic
28C2×#14519718C2211yescharacteristic, complemented
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519717C2211yescharacteristic, complemented
31C2×#14519716C2211yescharacteristic, complemented
32#16840137C2211yescharacteristic
33#16840137C2211yescharacteristic, complemented
34#16839999C2211yescharacteristic, complemented
35#16839998C2211yescharacteristic
36#16839995C2211yescharacteristic
37#16839961—21no—
38#16839955—21no—
39#16839908—21no—
40#16839905—21no—
41#16839896—21no—
42#16839894—21no—
43#16839856C2211yescharacteristic, complemented
44D8×Dic61C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839589C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839563C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53C16×Dic61C2211yescharacteristic, complemented
54C22×D8×C61C2211yescharacteristic
55C2×D16×C61C2211yescharacteristic
56C2×D16×C61C2211yescharacteristic, complemented
57C61×#466C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic
59C61×#357C2211yescharacteristic, complemented
60C61×C22⋊C16C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C2×#11758224—21no—
63C22×C22⋊D61—41no—
64C2×D4×D61—21no—
65#14519799—22no—
66#14519799—41no—
67#14519796—22no—
68#14519796—41no—
69C2×Dic244C2311yescharacteristic
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×D488C2311yescharacteristic
73C2×D488C2311yescharacteristic
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77D8×D61—22no—
78D8×D61—41no—
79C2×C8×D61C2311yescharacteristic
80#14519773—21no—
81#14519773—21no—
82#14519773—41no—
83#14519752—21no—
84#14519751—21no—
85#14519750—41no—
86#14519749—41no—
87#14519749—41no—
88#14519748—21no—
89#14519746—41no—
90#14519745—41no—
91#14519742—21no—
92D4×Dic61—21no—
93C4×C22⋊D61D411yescharacteristic
94#14519737—41no—
95#14519722—22no—
96#14519721—22no—
97#14519721D414yescomplemented
98D976—41no—
99#14519718—22no—
100#14519718—22no—
101#14519718—22no—
102#14519717—22no—
103#14519716—22no—
104C2×#11758185C2311yescharacteristic
105C2×#11758185C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519700—21no—
109#14519699—21no—
110#14519698—21no—
111#14519698—21no—
112#14519697—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C22×D4×C61—21no—
117C2×D8×C61—22no—
118C2×D8×C61—21no—
119C2×D8×C61—41no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124#121×C61—21no—
125D16×C61—22no—
126D16×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#102×C61—21no—
131C24×C61—41no—
132C23×D61—41no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—42no—
137C2×D4×C61—41no—
138C2×D4×C61—41no—
139C2×D4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61—41no—
148C2×C22⋊D61—41no—
149C2×C22⋊D61—41no—
150C2×C22⋊D61—84no—
151C2×C22⋊D61D811yescharacteristic
152C22×Dic61—41no—
153C22×Dic61—41no—
154#11758224—42no—
155#11758224—81no—
156D4×D61—42no—
157D4×D61—81no—
158#11758222—42no—
159#11758222—81no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—41no—
163C2×C4×D61—41no—
164C2×C4×D61C2×D411yescharacteristic
165C2×Dic122—21no—
166D8×C61—21no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61—21no—
170D8×C61—21no—
171D8×C61—42no—
172D8×C61C2×D412yes—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2411yescharacteristic
177C4⋊C4×C61—21no—
178C22⋊C4×C61—41no—
179C22⋊Dic61—41no—
180C22⋊Dic61—41no—
181C22⋊Dic61D811yescharacteristic
182#11758200—42no—
183#11758199—42no—
184#11758199—81no—
185#11758199—81no—
186#11758198—41no—
187#11758198—41no—
188#11758198D811yescharacteristic
189C4⋊Dic61—21no—
190#11758196—41no—
191#11758196D811yescharacteristic
192C4×Dic61—41no—
193C4×Dic61C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195C2×#92418C2×D411yescharacteristic
196Dic244—21no—
197Dic244C2×D412yescomplemented
198D488—21no—
199D488—21no—
200D488—21no—
201D488—21no—
202#11758190—42no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61—21no—
207#11758185—21no—
208#11758185C2×D412yescomplemented
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—82no—
213C23×C61—82no—
214C23×C61—161no—
215C23×C61C2×D811yescharacteristic
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—161no—
221C22×D61C2×D811yescharacteristic
222D4×C61—22no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—82no—
229D4×C61—161no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61C2×D811yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C22⋊D61—22no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C22⋊D61—82no—
239C22⋊D61—82no—
240C22⋊D61—82no—
241C22⋊D61—82no—
242C22⋊D61—82no—
243C22⋊D61—161no—
244C22⋊D61—161no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—82no—
248C2×Dic61—82no—
249C2×Dic61C2×D811yescharacteristic
250C2×Dic61C2×D811yescharacteristic
251C2×Dic61C2×D811yescharacteristic
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—161no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—82no—
260Dic122—22no—
261Dic122—41no—
262C8×C61—21no—
263C8×C61C22×D411yescharacteristic
264C8×C61H(5,𝔽2)11yescharacteristic
265#92418—21no—
266#92418—21no—
267#33840—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—42no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—164no—
281C22×C61C22×D811yescharacteristic
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—164no—
292C4×C61—21no—
293C4×C61—81no—
294C4×C61D4211yescharacteristic
295C4×C61#57611yescharacteristic
296Dic61—21no—
297Dic61—21no—
298Dic61—42no—
299Dic61—81no—
300Dic61#50912yescomplemented
301C2×#509—611no—
302C2×#509—611no—
303#3287—611no—
304D4×D8—611no—
305#2246—611no—
306#2218—611no—
307#2218—611no—
308#2207—611no—
309#2206—611no—
310#2186—611no—
311#2186—611no—
312#2184—611no—
313#2184D6111yescharacteristic, complemented
314C4×SD32—611no—
315#2169—611no—
316C2×C61—41no—
317C2×C61—82no—
318C2×C61—161no—
319C2×C61—161no—
320C2×C61D4×D811yescharacteristic
321C2×C61C2×#50911yescharacteristic
322C2×C61#341511yescharacteristic
323D61—41no—
324D61—161no—
325D61—161no—
326C2×#135—611no—
327C2×#135—611no—
328C22×D8—611no—
329C22×D8C2×D6111yescharacteristic
330#550—1221no—
331D42—1221no—
332#509—1224no—
333#509—1224no—
334C2×SD32—611no—
335C2×SD32—611no—
336C2×SD32—611no—
337C2×D16—611no—
338C2×D16—611no—
339C2×D16—611no—
340C2×D16—611no—
341C2×D16C2×D6111yescharacteristic
342C2×D16C2×D6111yescharacteristic, complemented
343C2×M32—611no—
344C2×M32—611no—
345#493—611no—
346#493—611no—
347#466—611no—
348#466—611no—
349#466—611no—
350#466C2×D6111yescharacteristic, complemented
351#460—1221no—
352#459—1221no—
353#449—1221no—
354#449—1221no—
355#447—1221no—
356#447—1221no—
357C4×Q16—611no—
358C4×D8—611no—
359C8×D4—611no—
360#367—611no—
361C4⋊C16—611no—
362#358—611no—
363#357—611no—
364#357—611no—
365#357—611no—
366#357C2×D6111yescharacteristic
367#357C2×D6111yescharacteristic, complemented
368C22⋊C16—611no—
369C22⋊C16C2×D6111yescharacteristic, complemented
370C4×C16—611no—
371C61#3384011yescharacteristic, complemented, Sylow
372C2×#41—1221no—
373C2×#41—1221no—
374C22×D4—21no—
375C22×D4—1221no—
376C22×D4—2441no—
377#135—1222no—
378#135—1222no—
379#135—2441no—
380#135—2441no—
381C2×Q16—611no—
382C2×SD16—1221no—
383C2×SD16—1221no—
384C2×D8—22no—
385C2×D8—21no—
386C2×D8—41no—
387C2×D8—611no—
388C2×D8—611no—
389C2×D8—1221no—
390C2×D8—1221no—
391C2×D8—1221no—
392C2×D8—1222no—
393C2×D8—2441no—
394C2×D8C22×D6111yescharacteristic
395C2×D8C22×D6111yescharacteristic
396C2×D8C22×D6111yescharacteristic
397C22×C8—611no—
398C22×C8C22×D6111yescharacteristic
399#127—1221no—
400#127—1221no—
401#127—2441no—
402#121—21no—
403#121—1221no—
404#121—1221no—
405#121—1221no—
406#121—2441no—
407#121—2441no—
408#121—2441no—
409#120—2441no—
410#120—2441no—
411C4×Q8—1221no—
412C4×D4—611no—
413C4×D4—1221no—
414C4×D4—2441no—
415SD32—614no—
416SD32—1222no—
417SD32—1222no—
418D16—22no—
419D16—41no—
420D16—1222no—
421D16—1222no—
422D16—1222no—
423D16—2441no—
424M32—1222no—
425M32—1222no—
426C2×C16—611no—
427C2×C16—611no—
428C2×C16C22×D6111yescharacteristic
429C2×C16C22×D6111yescharacteristic
430#107—611no—
431#107C22×D6111yescharacteristic
432C4⋊C8—611no—
433#103—1221no—
434#102—21no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439C22⋊C8—611no—
440C22⋊C8—611no—
441C4×C8—611no—
442C24—41no—
443C24—2441no—
444#41—4881no—
445#41—4881no—
446#41—2442no—
447#41—2442no—
448C2×Q8—1221no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—81no—
456C2×D4—611no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—4881no—
460C2×D4—4884no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2441no—
466C2×D4—2441no—
467C2×D4—2442no—
468C2×D4—2441no—
469C2×D4—2441no—
470C2×D4—2441no—
471C2×D4—2441no—
472C22×C4—611no—
473C22×C4—2441no—
474C22×C4—2441no—
475C22×C4—2441no—
476C22×C4D4×D6111yescharacteristic
477Q16—612no—
478Q16—1221no—
479SD16—2442no—
480SD16—2442no—
481D8—21no—
482D8—21no—
483D8—21no—
484D8—21no—
485D8—21no—
486D8—42no—
487D8—1221no—
488D8—1221no—
489D8—1221no—
490D8—1221no—
491D8—4881no—
492D8—4881no—
493D8—2442no—
494D8D4×D6112yes—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8D4×D6111yescharacteristic
502C2×C8C23×D6111yescharacteristic
503C4⋊C4—21no—
504C4⋊C4—611no—
505C4⋊C4—1221no—
506C4⋊C4—2441no—
507C22⋊C4—41no—
508C22⋊C4—611no—
509C22⋊C4—611no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C42—611no—
515C42—2441no—
516C16—21no—
517C16—21no—
518C16—612no—
519C16—1221no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—82no—
524C23—82no—
525C23—161no—
526C23—611no—
527C23—9761no—
528C23—4882no—
529C23—4882no—
530C23—2441no—
531C23—2441no—
532C23D8×D6111yescharacteristic
533Q8—1222no—
534Q8—2441no—
535D4—22no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—82no—
542D4—161no—
543D4—1222no—
544D4—9761no—
545D4—9761no—
546D4—9761no—
547D4—4882no—
548D4—4882no—
549D4—4882no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—4882no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—4882no—
567C2×C4—4882no—
568C2×C4—4882no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4D8×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C8—21no—
574C8—1221no—
575C8—1221no—
576C8C2×D4×D6111yescharacteristic
577C8#1451982211yescharacteristic
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—164no—
591C22—1221no—
592C22—1221no—
593C22—1221no—
594C22—9764no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22C2×D8×D6111yescharacteristic
602C4—21no—
603C4—81no—
604C4—612no—
605C4—1221no—
606C4—1221no—
607C4—4881no—
608C4—2442no—
609C4#1684039411yescharacteristic
610C4#1684056611yescharacteristic
611C2—41no—
612C2—82no—
613C2—161no—
614C2—161no—
615C2—9761no—
616C2—9761no—
617C2—2441no—
618C2#2145748911yescharacteristic
619C2#509×D6111yescharacteristic
620C2#2145895611yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2184C2
socleC22×C61C22×D8
hypercenter#2184D61
hypocenterC61#33840
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

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

NH ≅ G/Ntype
1. C2×#16840198C2semidirect
2. C2×#16840196C2semidirect
3. #21457576C2semidirect
4. D8×C22⋊D61C2semidirect
5. #21448067C2semidirect
6. #21447921C2semidirect
7. #21447920C2semidirect
8. #21447858C2semidirect
9. #21447852C2semidirect
10. #21447744C2semidirect
11. #21447739C2semidirect
12. #21447732C2semidirect
13. #21447674C2semidirect
14. #21447648C2semidirect
15. C61×#2184C2semidirect
16. C2×#14519799C22semidirect
17. C2×#14519796C22semidirect
23. C2×#14519722C22semidirect
25. C2×#14519721C22semidirect
26. C2×D976C22semidirect
28. C2×#14519718C22semidirect
29. C2×#14519718C22semidirect
30. C2×#14519717C22semidirect
31. C2×#14519716C22semidirect
33. #16840137C22semidirect
34. #16839999C22semidirect
43. #16839856C22semidirect
46. #16839589C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839564C22semidirect
50. #16839563C22semidirect
51. #16839562C22semidirect
52. #16839541C22semidirect
53. C16×Dic61C22semidirect
56. C2×D16×C61C22semidirect
57. C61×#466C22semidirect
59. C61×#357C22semidirect
60. C61×C22⋊C16C22semidirect
97. #14519721D4semidirect
197. Dic244C2×D4semidirect
208. #11758185C2×D4semidirect
300. Dic61#509semidirect
313. #2184D61semidirect
342. C2×D16C2×D61semidirect
350. #466C2×D61semidirect
367. #357C2×D61semidirect
369. C22⋊C16C2×D61semidirect
371. C61#33840semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
C2×#168401962
#214575762
D8×C22⋊D612
#214480672
#214479212
#214479202
#214478582
#214478522
#214477442
#214477392
#214477322
#214476742
#214476482
C61×#21842
#3384061