Finite Group Databasebeta

← #29836476

Subgroups — #29836476

Order 15616.

760 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840200C211yesmaximal, characteristic, complemented
2C22×D976C211yesmaximal, characteristic, complemented
3#21457506C211yesmaximal, characteristic, complemented
4D4×D488C211yesmaximal, characteristic, complemented
5#21448085C211yesmaximal, characteristic, complemented
6#21447922C211yesmaximal, characteristic, complemented
7#21447915C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447855C211yesmaximal, characteristic, complemented
10#21447750C211yesmaximal, characteristic, complemented
11#21447744C211yesmaximal, characteristic, complemented
12#21447733C211yesmaximal, characteristic, complemented
13#21447733C211yesmaximal, characteristic, complemented
14#21447696C211yesmaximal, characteristic, complemented
15C61×#2168C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic, complemented
17C22×D488C2211yescharacteristic
18C22×D488C2211yescharacteristic, complemented
19C22×D488C2211yescharacteristic, complemented
20#16840408—21no—
21D4×D244—21no—
22#16840200—24no—
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic, complemented
26C2×D976—24no—
27C2×D976C2211yescharacteristic
28C2×D976C2211yescharacteristic
29C2×D976C2211yescharacteristic, complemented
30C2×D976C2211yescharacteristic, complemented
31C2×D976C2211yescharacteristic, complemented
32C2×D976C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic, complemented
34#16840139C2211yescharacteristic, complemented
35#16840003C2211yescharacteristic, complemented
36#16839998C2211yescharacteristic
37#16839998C2211yescharacteristic, complemented
38#16839998C2211yescharacteristic, complemented
39#16839978—21no—
40#16839958—21no—
41#16839915—21no—
42#16839909—21no—
43#16839898—21no—
44#16839898—21no—
45C4×D488C2211yescharacteristic, complemented
46C4×D488C2211yescharacteristic, complemented
47#16839590C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic, complemented
51#16839564C2211yescharacteristic, complemented
52#16839564C2211yescharacteristic, complemented
53#16839564C2211yescharacteristic, complemented
54C2×M32×C61C2211yescharacteristic, complemented
55C22×C16×C61C2211yescharacteristic, complemented
56C8×D4×C61C2211yescharacteristic
57C61×C4⋊C16C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic, complemented
59C61×C22⋊C16C2211yescharacteristic, complemented
60C61×#346C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C22×D244—21no—
63C22×D244—21no—
64C22×D244—21no—
65C2×D4×D61—41no—
66#14519799—22no—
67#14519799—41no—
68C2×Dic244C2311yescharacteristic
69C2×#11758190—21no—
70C2×#11758190—21no—
71C2×D488—22no—
72C2×D488—22no—
73C2×D488—22no—
74C2×D488—21no—
75C2×D488—21no—
76C2×D488—21no—
77C2×D488—21no—
78C2×D488—41no—
79C2×D488—41no—
80C2×D488—41no—
81C2×D488C2311yescharacteristic
82C2×D488C2311yescharacteristic
83C2×D488C2311yescharacteristic
84C2×D488C2311yescharacteristic
85C2×D488C2311yescharacteristic
86#14519774—21no—
87#14519765—21no—
88#14519762—41no—
89#14519756—21no—
90#14519752—41no—
91#14519751—21no—
92#14519751—21no—
93#14519751—21no—
94#14519750—41no—
95#14519747—41no—
96#14519747—41no—
97#14519745—41no—
98C4×D244—21no—
99C4×D244—21no—
100C22⋊C4×D61—41no—
101#14519722—22no—
102#14519722—22no—
103#14519722—41no—
104D976—22no—
105D976—22no—
106D976—22no—
107D976—22no—
108D976—41no—
109D976D414yescomplemented
110#14519711C2311yescharacteristic
111#14519711C2311yescharacteristic
112#14519702—21no—
113#14519699—21no—
114#14519699—21no—
115#14519699—21no—
116#14519699—21no—
117#14519699—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61D411yescharacteristic
121M32×C61—22no—
122C2×C16×C61—22no—
123C2×C16×C61C2311yescharacteristic
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126C2×C16×C61C2311yescharacteristic
127C4⋊C8×C61D411yescharacteristic
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C23×D61—41no—
133C23×D61—41no—
134C2×D4×C61D811yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141D4×D61—84no—
142#11758222—42no—
143#11758222—81no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244—42no—
152C2×D244—41no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244—41no—
157C2×D244—81no—
158C2×D244—81no—
159C2×D244—81no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2411yescharacteristic
172C4⋊C4×C61D811yescharacteristic
173C22⋊C4×C61D811yescharacteristic
174C22⋊C4×C61D811yescharacteristic
175C42×C61C2×D411yescharacteristic
176C22⋊Dic61—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186#11758196—41no—
187Dic244—21no—
188Dic244—21no—
189D488—21no—
190D488—21no—
191D488—21no—
192D488—21no—
193D488—21no—
194D488—21no—
195D488—21no—
196D488—21no—
197D488—42no—
198D488—42no—
199D488—42no—
200D488—81no—
201D488C2×D412yes—
202D488C2×D412yescomplemented
203#11758190—42no—
204#11758190—81no—
205C16×C61—21no—
206C16×C61—21no—
207C16×C61—21no—
208C16×C61C2×D412yescomplemented
209C23×C61C2×D811yescharacteristic
210C23×C61C2×D811yescharacteristic
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61—161no—
223C22×D61—161no—
224C22×D61—161no—
225D4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61C2×D811yescharacteristic
231C2×C4×C61C2×D811yescharacteristic
232C2×C4×C61C2×D811yescharacteristic
233C2×C4×C61C2×D811yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C22⋊D61—82no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C22⋊D61—161no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242D244—22no—
243D244—22no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—82no—
253D244—82no—
254D244—82no—
255D244—82no—
256D244—161no—
257C4×D61—82no—
258C4×D61—82no—
259C4×D61—161no—
260Dic122—41no—
261Dic122—41no—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61C22×D411yescharacteristic
266C8×C61H(5,𝔽2)11yescharacteristic
267#33830—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61#50912yescomplemented
274C22×C61C22×D811yescharacteristic
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—164no—
291C2×D61—164no—
292C2×D61—164no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61D4211yescharacteristic
298C4×C61#57611yescharacteristic
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302C2×#509—611no—
303C22×D16—611no—
304#3280—611no—
305D4×D8—611no—
306#2250—611no—
307#2218—611no—
308#2217—611no—
309#2210—611no—
310#2206—611no—
311#2187—611no—
312#2186—611no—
313#2184—611no—
314#2184—611no—
315#2177—611no—
316#2168D6111yescharacteristic, complemented
317C2×C61—22no—
318C2×C61—41no—
319C2×C61D4×D811yescharacteristic
320C2×C61C2×#50911yescharacteristic
321C2×C61#341511yescharacteristic
322D61—82no—
323D61—82no—
324D61—161no—
325D61—161no—
326D61—161no—
327C2×#135—611no—
328C22×D8—611no—
329C22×D8—611no—
330C22×D8—611no—
331#546—1221no—
332D42—1221no—
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×D16—611no—
342C2×D16—611no—
343C2×D16—1224no—
344C2×M32C2×D6111yescharacteristic, complemented
345C22×C16C2×D6111yescharacteristic, complemented
346#495—611no—
347#493—611no—
348#467—611no—
349#466—611no—
350#466—611no—
351#466—611no—
352#463—1221no—
353#459—1221no—
354#450—1221no—
355#449—1221no—
356#447—1221no—
357#447—1221no—
358C4×D8—611no—
359C4×D8—611no—
360C8×D4C2×D6111yescharacteristic
361#367—611no—
362C4⋊C16C2×D6111yescharacteristic, complemented
363#358—611no—
364#357—611no—
365#357—611no—
366#357—611no—
367#357—611no—
368#357—611no—
369C22⋊C16C2×D6111yescharacteristic, complemented
370C22⋊C16C2×D6111yescharacteristic, complemented
371#346C2×D6111yescharacteristic, complemented
372C61#3383011yescharacteristic, complemented, Sylow
373C2×#41—1221no—
374C22×D4—1221no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4—2441no—
378#135—1222no—
379#135—2441no—
380C2×Q16—611no—
381C2×SD16—1221no—
382C2×SD16—1221no—
383C2×D8—611no—
384C2×D8—611no—
385C2×D8—611no—
386C2×D8—611no—
387C2×D8—611no—
388C2×D8—1222no—
389C2×D8—1222no—
390C2×D8—1222no—
391C2×D8—1221no—
392C2×D8—1221no—
393C2×D8—1221no—
394C2×D8—1221no—
395C2×D8—2441no—
396C2×D8—2441no—
397C2×D8—2441no—
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400#127—1221no—
401#124—1221no—
402#123—2441no—
403C22⋊Q8—1221no—
404#121—1221no—
405#121—1221no—
406#121—1221no—
407#121—2441no—
408#121—2441no—
409#120—2441no—
410#120—2441no—
411#120—2441no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4D24411yescharacteristic
415C2×C22⋊C4—2441no—
416SD32—1222no—
417SD32—1222no—
418SD32—2441no—
419D16—614no—
420D16—1222no—
421D16—1222no—
422D16—1222no—
423D16—1222no—
424D16—2441no—
425M32—22no—
426C2×C16—22no—
427C2×C16C22×D6111yescharacteristic
428C2×C16C22×D6111yescharacteristic
429C2×C16C22×D6111yescharacteristic
430C2×C16C22×D6111yescharacteristic
431#107—611no—
432#107—611no—
433C4⋊C8D24411yescharacteristic
434#103—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440C22⋊C8D24411yescharacteristic
441C22⋊C8D24411yescharacteristic
442C4×C8C22×D6111yescharacteristic
443C24—2441no—
444C24—2441no—
445C24—2441no—
446#41—4881no—
447#41—2442no—
448C2×Q8—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—4881no—
455C2×D4—4881no—
456C2×D4—4881no—
457C2×D4—4884no—
458C2×D4—2442no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2441no—
466C2×D4—2441no—
467C2×D4—2441no—
468C2×D4—2441no—
469C2×D4—2441no—
470C2×D4D48811yescharacteristic
471C22×C4—2441no—
472C22×C4—2441no—
473C22×C4—2441no—
474C22×C4C2×D24411yescharacteristic
475C22×C4C2×D24411yescharacteristic
476Q16—1221no—
477Q16—1221no—
478SD16—4881no—
479SD16—2442no—
480D8—612no—
481D8—612no—
482D8—1221no—
483D8—1221no—
484D8—1221no—
485D8—1221no—
486D8—1221no—
487D8—1221no—
488D8—1221no—
489D8—1221no—
490D8—4881no—
491D8—2442no—
492D8—2442no—
493D8—2442no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8C2×D24411yescharacteristic
499C2×C8C2×D24411yescharacteristic
500C2×C8C2×D24411yescharacteristic
501C2×C8C23×D6111yescharacteristic
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—2441no—
505C4⋊C4D48811yescharacteristic
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C22⋊C4—2441no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C22⋊C4D48811yescharacteristic
515C22⋊C4D48811yescharacteristic
516C42C2×D24411yescharacteristic
517C16—21no—
518C16—21no—
519C16—21no—
520C16D4×D6112yescomplemented
521C23—9761no—
522C23—9761no—
523C23—9761no—
524C23—4882no—
525C23—4882no—
526C23—4882no—
527C23—4882no—
528C23—4882no—
529C23—4882no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23—2441no—
535C23C2×D48811yescharacteristic
536C23C2×D48811yescharacteristic
537Q8—2441no—
538Q8—2441no—
539D4—22no—
540D4—1222no—
541D4—1222no—
542D4—9761no—
543D4—9761no—
544D4—4882no—
545D4—4882no—
546D4—4882no—
547D4—4882no—
548D4—4882no—
549D4—4882no—
550D4—4882no—
551D4—2441no—
552D4—2441no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—9761no—
564C2×C4—4882no—
565C2×C4—4882no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4C2×D48811yescharacteristic
570C2×C4C2×D48811yescharacteristic
571C2×C4C2×D48811yescharacteristic
572C2×C4C2×D48811yescharacteristic
573C2×C4C22×D24411yescharacteristic
574C8—21no—
575C8—21no—
576C8—21no—
577C8C2×D4×D6111yescharacteristic
578C8#1451982311yescharacteristic
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—9764no—
585C22—9764no—
586C22—9764no—
587C22—4881no—
588C22—4881no—
589C22—4881no—
590C22—4881no—
591C22—4881no—
592C22—4881no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—2442no—
601C22—2442no—
602C22#1684020012yescomplemented
603C22C22×D48811yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—4881no—
609C4—4881no—
610C4—4881no—
611C4D4×D24411yescharacteristic
612C4#1684056711yescharacteristic
613C2—22no—
614C2—41no—
615C2—9761no—
616C2—9761no—
617C2—9761no—
618C2—4882no—
619C2—4882no—
620C2D4×D48811yescharacteristic
621C2C2×#1684020011yescharacteristic
622C2#2145896011yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2168C2
socleC22×C61C22×D8
hypercenter#2168D61
hypocenterC61#33830
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. C2×#16840200C2semidirect
2. C22×D976C2semidirect
3. #21457506C2semidirect
4. D4×D488C2semidirect
5. #21448085C2semidirect
6. #21447922C2semidirect
7. #21447915C2semidirect
8. #21447875C2semidirect
9. #21447855C2semidirect
10. #21447750C2semidirect
11. #21447744C2semidirect
12. #21447733C2semidirect
13. #21447733C2semidirect
14. #21447696C2semidirect
15. C61×#2168C2semidirect
16. C2×#14519799C22semidirect
18. C22×D488C22semidirect
19. C22×D488C22semidirect
23. C2×#14519722C22semidirect
24. C2×#14519722C22semidirect
25. C2×#14519722C22semidirect
29. C2×D976C22semidirect
30. C2×D976C22semidirect
31. C2×D976C22semidirect
32. C2×D976C22semidirect
33. #16840146C22semidirect
34. #16840139C22semidirect
35. #16840003C22semidirect
37. #16839998C22semidirect
38. #16839998C22semidirect
45. C4×D488C22semidirect
46. C4×D488C22semidirect
47. #16839590C22semidirect
48. #16839567C22semidirect
49. #16839564C22semidirect
50. #16839564C22semidirect
51. #16839564C22semidirect
52. #16839564C22semidirect
53. #16839564C22semidirect
54. C2×M32×C61C22semidirect
55. C22×C16×C61C22semidirect
57. C61×C4⋊C16C22semidirect
58. C61×C22⋊C16C22semidirect
59. C61×C22⋊C16C22semidirect
60. C61×#346C22semidirect
109. D976D4semidirect
202. D488C2×D4semidirect
208. C16×C61C2×D4semidirect
273. C22×C61#509semidirect
316. #2168D61semidirect
344. C2×M32C2×D61semidirect
345. C22×C16C2×D61semidirect
362. C4⋊C16C2×D61semidirect
369. C22⋊C16C2×D61semidirect
370. C22⋊C16C2×D61semidirect
371. #346C2×D61semidirect
372. C61#33830semidirect
520. C16D4×D61semidirect
602. C22#16840200semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
C22×D9762
#214575062
D4×D4882
#214480852
#214479222
#214479152
#214478752
#214478552
#214477502
#214477442
#214477332
#214476962
C61×#21682
#3383061