Finite Group Databasebeta

← #29836605

Subgroups — #29836605

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×#16840199C211yesmaximal, characteristic, complemented
2#21457573C211yesmaximal, characteristic, complemented
3D8×C22⋊D61C211yesmaximal, characteristic, complemented
4#21448067C211yesmaximal, characteristic, complemented
5#21447922C211yesmaximal, characteristic, complemented
6#21447919C211yesmaximal, characteristic, complemented
7#21447857C211yesmaximal, characteristic, complemented
8#21447852C211yesmaximal, characteristic, complemented
9#21447742C211yesmaximal, characteristic, complemented
10#21447741C211yesmaximal, characteristic, complemented
11#21447731C211yesmaximal, characteristic, complemented
12#21447729C211yesmaximal, characteristic, complemented
13SD32×Dic61C211yesmaximal, characteristic, complemented
14#21447649C211yesmaximal, characteristic, complemented
15C2×C61×#509C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic, complemented
17C2×D8×D61C2211yescharacteristic
18#16840431—21no—
19D4×C22⋊D61—21no—
20#16840199—24no—
21C2×#14519720C2211yescharacteristic, complemented
22C2×D976C2211yescharacteristic, complemented
23C2×#14519718C2211yescharacteristic
24C2×#14519718C2211yescharacteristic, complemented
25C2×#14519718C2211yescharacteristic, complemented
26C2×#14519716C2211yescharacteristic, complemented
27#16840137C2211yescharacteristic
28#16840137C2211yescharacteristic, complemented
29#16839998C2211yescharacteristic
30#16839998C2211yescharacteristic, complemented
31#16839995C2211yescharacteristic
32#16839995C2211yescharacteristic, complemented
33#16839960—21no—
34#16839955—21no—
35#16839907—21no—
36#16839906—21no—
37#16839896—21no—
38#16839894—21no—
39Q16×Dic61C2211yescharacteristic, complemented
40D8×Dic61C2211yescharacteristic
41C8×C22⋊D61C2211yescharacteristic
42#16839590C2211yescharacteristic, complemented
43#16839580C2211yescharacteristic, complemented
44#16839566C2211yescharacteristic, complemented
45#16839564C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic, complemented
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic, complemented
49#16839563C2211yescharacteristic, complemented
50C22⋊#11758185C2211yescharacteristic, complemented
51#16839541C2211yescharacteristic, complemented
52C16×Dic61C2211yescharacteristic, complemented
53C2×#135×C61C2211yescharacteristic, complemented
54C22×D8×C61C2211yescharacteristic
55C61×#509—24no—
56C2×SD32×C61C2211yescharacteristic
57C2×SD32×C61C2211yescharacteristic, complemented
58C2×D16×C61C2211yescharacteristic
59C2×D16×C61C2211yescharacteristic, complemented
60C2×M32×C61C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C22×C22⋊D61—41no—
63C2×D4×D61—21no—
64#14519800—22no—
65#14519800—41no—
66C2×#11758201—21no—
67C2×D488C2311yescharacteristic
68C2×D488C2311yescharacteristic
69C2×#11758199—21no—
70C2×#11758199—21no—
71C2×#11758199—21no—
72D8×D61—22no—
73D8×D61—41no—
74C2×C8×D61C2311yescharacteristic
75#14519773—21no—
76#14519773—21no—
77#14519773—41no—
78#14519751—21no—
79#14519751—21no—
80#14519751—41no—
81#14519749—41no—
82#14519748—21no—
83#14519748—21no—
84#14519748—41no—
85#14519746—41no—
86#14519745—41no—
87Q8×Dic61—21no—
88D4×Dic61—21no—
89C4×C22⋊D61—41no—
90C4×C22⋊D61D411yescharacteristic
91#14519720—22no—
92D976—22no—
93#14519718—22no—
94#14519718—41no—
95#14519718—41no—
96#14519716—22no—
97C2×#11758185C2311yescharacteristic
98C2×#11758185C2311yescharacteristic
99#14519711C2311yescharacteristic
100#14519711C2311yescharacteristic
101#14519706D411yescharacteristic
102#14519701—21no—
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108C22⋊#92418D411yescharacteristic
109#14519689D411yescharacteristic
110C8×Dic61C2311yescharacteristic
111C2×#41×C61—21no—
112C22×D4×C61—21no—
113#135×C61—22no—
114#135×C61—41no—
115C2×Q16×C61C2311yescharacteristic
116C2×SD16×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
124SD32×C61—22no—
125SD32×C61D414yescomplemented
126D16×C61—22no—
127D16×C61—22no—
128M32×C61—22no—
129C2×C16×C61C2311yescharacteristic
130C2×C16×C61C2311yescharacteristic
131C24×C61—41no—
132C23×D61—41no—
133#41×C61—42no—
134#41×C61—81no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—42no—
140C2×D4×C61—41no—
141C2×D4×C61—41no—
142C2×D4×C61—81no—
143C22×C4×C61—41no—
144C22×C4×C61C2×D411yescharacteristic
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61—41no—
148C2×C22⋊D61—41no—
149C2×C22⋊D61—41no—
150C2×C22⋊D61—41no—
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—84no—
155C2×C22⋊D61D811yescharacteristic
156C22×Dic61—41no—
157#11758226—42no—
158#11758226—81no—
159D4×D61—42no—
160D4×D61—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—41no—
164C2×C4×D61—41no—
165C2×C4×D61C2×D411yescharacteristic
166Q16×C61—21no—
167Q16×C61C2×D412yescomplemented
168SD16×C61—42no—
169D8×C61—21no—
170D8×C61—21no—
171D8×C61—21no—
172D8×C61—21no—
173D8×C61—21no—
174D8×C61—42no—
175D8×C61C2×D412yes—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2411yescharacteristic
180C22⋊Dic61—41no—
181C22⋊Dic61—41no—
182C22⋊Dic61—41no—
183C22⋊Dic61D811yescharacteristic
184#11758201—42no—
185#11758199—42no—
186#11758199—81no—
187#11758199—81no—
188#11758198—41no—
189#11758198—41no—
190#11758198D811yescharacteristic
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—41no—
194#11758196D811yescharacteristic
195C4×Dic61—41no—
196C4×Dic61C2×D411yescharacteristic
197C2×#92418C2×D411yescharacteristic
198C2×#92418C2×D411yescharacteristic
199D488—21no—
200D488—21no—
201D488—21no—
202D488—21no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61C2×D412yescomplemented
207#11758185—21no—
208#11758185—21no—
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
222Q8×C61—22no—
223Q8×C61—41no—
224D4×C61—22no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—82no—
231D4×C61—82no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—82no—
236C2×C4×C61C2×D811yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C22⋊D61—22no—
239C22⋊D61—82no—
240C22⋊D61—82no—
241C22⋊D61—82no—
242C22⋊D61—82no—
243C22⋊D61—82no—
244C22⋊D61—161no—
245C22⋊D61—161no—
246C22⋊D61—161no—
247C22⋊D61—161no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—82no—
251C2×Dic61C2×D811yescharacteristic
252C2×Dic61C2×D811yescharacteristic
253C2×Dic61C2×D811yescharacteristic
254D244—41no—
255D244—41no—
256D244—41no—
257D244—41no—
258D244—82no—
259C4×D61—21no—
260C4×D61—21no—
261C4×D61—82no—
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—42no—
294C4×C61D4211yescharacteristic
295C4×C61#57611yescharacteristic
296Dic61—21no—
297Dic61—21no—
298Dic61—81no—
299Dic61—81no—
300Dic61#50912yescomplemented
301C2×#509—611no—
302C2×#509D6111yescharacteristic, complemented
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#2184—611no—
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×#135C2×D6111yescharacteristic, complemented
328C22×D8—611no—
329C22×D8C2×D6111yescharacteristic
330#550—1221no—
331D42—1221no—
332#509—24no—
333#509—1224no—
334C2×SD32—611no—
335C2×SD32C2×D6111yescharacteristic
336C2×SD32C2×D6111yescharacteristic, complemented
337C2×D16—611no—
338C2×D16—611no—
339C2×D16—611no—
340C2×D16—611no—
341C2×D16C2×D6111yescharacteristic
342C2×D16C2×D6111yescharacteristic, complemented
343C2×M32—611no—
344C2×M32C2×D6111yescharacteristic, complemented
345#493—611no—
346#493—611no—
347#466—611no—
348#466—611no—
349#466—611no—
350#466—611no—
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#357—611no—
367#357—611no—
368C22⋊C16—611no—
369C22⋊C16—611no—
370C4×C16—611no—
371C61#3384011yescharacteristic, complemented, Sylow
372C2×#41—21no—
373C2×#41—1221no—
374C22×D4—21no—
375C22×D4—1221no—
376C22×D4—2441no—
377#135—22no—
378#135—41no—
379#135—1222no—
380#135—2441no—
381C2×Q16C22×D6111yescharacteristic
382C2×SD16—21no—
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—1221no—
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—22no—
416SD32—1222no—
417SD32C22⋊D6114yescomplemented
418D16—22no—
419D16—22no—
420D16—1222no—
421D16—1222no—
422D16—2441no—
423D16—2441no—
424M32—22no—
425M32—1222no—
426C2×C16—611no—
427C2×C16—611no—
428C2×C16C22×D6111yescharacteristic
429C2×C16C22×D6111yescharacteristic
430#107—611no—
431#107—611no—
432C4⋊C8—611no—
433#103—1221no—
434#102—1221no—
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—42no—
445#41—81no—
446#41—4881no—
447#41—2442no—
448C2×Q8—21no—
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—41no—
473C22×C4—611no—
474C22×C4—2441no—
475C22×C4—2441no—
476C22×C4D4×D6111yescharacteristic
477Q16—21no—
478Q16C2×C22⋊D6112yescomplemented
479SD16—42no—
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—
494D8C2×C22⋊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—611no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—2441no—
507C22⋊C4—611no—
508C22⋊C4—611no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C42—611no—
515C42—2441no—
516C16—21no—
517C16—1221no—
518C16—1221no—
519C16C2×C22⋊D6112yescomplemented
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—22no—
534Q8—41no—
535D4—22no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—82no—
542D4—82no—
543D4—1222no—
544D4—9761no—
545D4—9761no—
546D4—9761no—
547D4—9761no—
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—82no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—1221no—
566C2×C4—1221no—
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—
576C8C22×C22⋊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—42no—
604C4—612no—
605C4—1221no—
606C4—1221no—
607C4—4881no—
608C4—4881no—
609C4D4×C22⋊D6111yescharacteristic
610C4#1684056311yescharacteristic
611C2—41no—
612C2—82no—
613C2—161no—
614C2—161no—
615C2—9761no—
616C2—9761no—
617C2—2441no—
618C2D8×C22⋊D6111yescharacteristic
619C2#509×D6111yescharacteristic
620C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33840
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. #21457573C2semidirect
3. D8×C22⋊D61C2semidirect
4. #21448067C2semidirect
5. #21447922C2semidirect
6. #21447919C2semidirect
7. #21447857C2semidirect
8. #21447852C2semidirect
9. #21447742C2semidirect
10. #21447741C2semidirect
11. #21447731C2semidirect
12. #21447729C2semidirect
13. SD32×Dic61C2semidirect
14. #21447649C2semidirect
15. C2×C61×#509C2semidirect
16. C2×#14519800C22semidirect
21. C2×#14519720C22semidirect
22. C2×D976C22semidirect
24. C2×#14519718C22semidirect
25. C2×#14519718C22semidirect
26. C2×#14519716C22semidirect
28. #16840137C22semidirect
30. #16839998C22semidirect
32. #16839995C22semidirect
39. Q16×Dic61C22semidirect
42. #16839590C22semidirect
43. #16839580C22semidirect
44. #16839566C22semidirect
45. #16839564C22semidirect
46. #16839564C22semidirect
48. #16839563C22semidirect
49. #16839563C22semidirect
50. C22⋊#11758185C22semidirect
51. #16839541C22semidirect
52. C16×Dic61C22semidirect
53. C2×#135×C61C22semidirect
57. C2×SD32×C61C22semidirect
59. C2×D16×C61C22semidirect
60. C2×M32×C61C22semidirect
125. SD32×C61D4semidirect
167. Q16×C61C2×D4semidirect
206. C16×C61C2×D4semidirect
300. Dic61#509semidirect
302. C2×#509D61semidirect
327. C2×#135C2×D61semidirect
336. C2×SD32C2×D61semidirect
342. C2×D16C2×D61semidirect
344. C2×M32C2×D61semidirect
371. C61#33840semidirect
417. SD32C22⋊D61semidirect
478. Q16C2×C22⋊D61semidirect
519. C16C2×C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
#214575732
D8×C22⋊D612
#214480672
#214479222
#214479192
#214478572
#214478522
#214477422
#214477412
#214477312
#214477292
SD32×Dic612
#214476492
C2×C61×#5092
#3384061