Finite Group Databasebeta

← #29836472

Subgroups — #29836472

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×#16840197C211yesmaximal, characteristic, complemented
2C22×#14519718C211yesmaximal, characteristic, complemented
3D4×D488C211yesmaximal, characteristic, complemented
4#21448086C211yesmaximal, characteristic, complemented
5#21447920C211yesmaximal, characteristic, complemented
6#21447917C211yesmaximal, characteristic, complemented
7#21447871C211yesmaximal, characteristic, complemented
8#21447854C211yesmaximal, characteristic, complemented
9#21447747C211yesmaximal, characteristic, complemented
10#21447742C211yesmaximal, characteristic, complemented
11#21447731C211yesmaximal, characteristic, complemented
12#21447730C211yesmaximal, characteristic, complemented
13#21447693C211yesmaximal, characteristic, complemented
14#21447645C211yesmaximal, characteristic, complemented
15C61×#3280C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C22×D488C2211yescharacteristic, complemented
18D4×D244—21no—
19#16840197—24no—
20C2×#14519720C2211yescharacteristic, complemented
21C2×#14519720C2211yescharacteristic, complemented
22C2×#14519720C2211yescharacteristic, complemented
23C2×#14519718—24no—
24C2×#14519718C2211yescharacteristic
25C2×#14519718C2211yescharacteristic
26C2×#14519718C2211yescharacteristic, complemented
27C2×#14519718C2211yescharacteristic, complemented
28C2×#14519718C2211yescharacteristic, complemented
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519717C2211yescharacteristic, complemented
31C22×#11758185C2211yescharacteristic, complemented
32#16840139C2211yescharacteristic, complemented
33#16839998C2211yescharacteristic
34#16839998C2211yescharacteristic, complemented
35#16839958—21no—
36#16839898—21no—
37#16839898—21no—
38C4×D488C2211yescharacteristic, complemented
39#16839589C2211yescharacteristic, complemented
40C4⋊#11758185C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839563C2211yescharacteristic, complemented
43#16839563C2211yescharacteristic, complemented
44#16839562C2211yescharacteristic, complemented
45#16839562C2211yescharacteristic, complemented
46#16839562C2211yescharacteristic, complemented
47C22⋊#11758185C2211yescharacteristic, complemented
48C22⋊#11758185C2211yescharacteristic, complemented
49#16839538C2211yescharacteristic, complemented
50C2×#135×C61C2211yescharacteristic, complemented
51C22×D8×C61C2211yescharacteristic, complemented
52C61×#546—21no—
53C61×#495C2211yescharacteristic, complemented
54C61×#467C2211yescharacteristic, complemented
55C61×#466C2211yescharacteristic, complemented
56C61×#463—21no—
57C61×#450—21no—
58C61×#449—21no—
59C4×D8×C61C2211yescharacteristic, complemented
60C8×D4×C61C2211yescharacteristic
61C22×D244—21no—
62C22×D244—21no—
63C2×D4×D61—41no—
64C2×D488—22no—
65C2×D488—22no—
66C2×D488—21no—
67C2×D488—21no—
68C2×D488—21no—
69C2×D488—41no—
70C2×D488—41no—
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73C2×D488C2311yescharacteristic
74#14519774—21no—
75#14519752—41no—
76#14519751—21no—
77#14519751—21no—
78#14519747—41no—
79#14519747—41no—
80C4×D244—21no—
81#14519720—22no—
82#14519720—22no—
83#14519720—41no—
84#14519718—22no—
85#14519718—22no—
86#14519718—22no—
87#14519718—22no—
88#14519718—41no—
89#14519718D414yescomplemented
90#14519717—22no—
91C2×#11758185—22no—
92C2×#11758185C2311yescharacteristic
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519699—21no—
98#14519699—21no—
99#14519699—21no—
100C2×#41×C61—21no—
101C22×D4×C61—21no—
102#135×C61—22no—
103#135×C61—41no—
104C2×Q16×C61C2311yescharacteristic
105C2×SD16×C61—21no—
106C2×SD16×C61—21no—
107C2×D8×C61—22no—
108C2×D8×C61—21no—
109C2×D8×C61—41no—
110C2×D8×C61C2311yescharacteristic
111C2×D8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114#124×C61—21no—
115#123×C61—41no—
116C22⋊Q8×C61—21no—
117#121×C61—21no—
118#121×C61—41no—
119#120×C61—41no—
120C4×D4×C61—21no—
121C4×D4×C61D411yescharacteristic
122C2×C22⋊C4×C61—41no—
123#107×C61C2311yescharacteristic
124C4⋊C8×C61D411yescharacteristic
125#103×C61—21no—
126#102×C61—21no—
127#102×C61—21no—
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C24×C61—41no—
132C23×D61—41no—
133C23×D61—41no—
134#41×C61—42no—
135#41×C61—81no—
136C2×Q8×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—41no—
143C2×D4×C61—81no—
144C2×D4×C61D811yescharacteristic
145C22×C4×C61—41no—
146C22×C4×C61—41no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C2×C22⋊D61—41no—
150C2×C22⋊D61—41no—
151D4×D61—84no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—42no—
156C2×D244—42no—
157C2×D244—41no—
158C2×D244—41no—
159C2×D244—41no—
160C2×D244—41no—
161C2×D244—81no—
162C2×D244—81no—
163C2×C4×D61—41no—
164Q16×C61—21no—
165Q16×C61—21no—
166SD16×C61—42no—
167SD16×C61—81no—
168D8×C61—21no—
169D8×C61—21no—
170D8×C61—21no—
171D8×C61—42no—
172D8×C61C2×D412yescomplemented
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2411yescharacteristic
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61D811yescharacteristic
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61D811yescharacteristic
190C22⋊C4×C61D811yescharacteristic
191C42×C61C2×D411yescharacteristic
192#11758198—41no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—21no—
201D488—42no—
202D488—42no—
203D488—81no—
204D488C2×D412yes—
205#11758185—21no—
206#11758185—21no—
207#11758185—21no—
208#11758185C2×D412yescomplemented
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—82no—
212C23×C61—82no—
213C23×C61—161no—
214C23×C61C2×D811yescharacteristic
215C23×C61C2×D811yescharacteristic
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—82no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61—82no—
223C22×D61—161no—
224C22×D61—161no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—82no—
233D4×C61—82no—
234D4×C61—161no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—82no—
242C2×C4×C61—161no—
243C2×C4×C61C2×D811yescharacteristic
244C2×C4×C61C2×D811yescharacteristic
245C2×C4×C61C2×D811yescharacteristic
246C2×C4×C61C2×D811yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C22⋊D61—82no—
249C22⋊D61—82no—
250C2×Dic61—41no—
251D244—22no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—41no—
257D244—82no—
258D244—82no—
259D244—82no—
260D244—161no—
261C4×D61—82no—
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—42no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—164no—
280C22×C61#50912yescomplemented
281C22×C61C22×D811yescharacteristic
282C2×D61—42no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—164no—
292C2×D61—164no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61D4211yescharacteristic
300C4×C61#57611yescharacteristic
301Dic61—81no—
302C2×#509—611no—
303C22×D16—611no—
304#3280D6111yescharacteristic, complemented
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#2168—611no—
317C2×C61—22no—
318C2×C61—41no—
319C2×C61—82no—
320C2×C61—161no—
321C2×C61D4×D811yescharacteristic
322C2×C61C2×#50911yescharacteristic
323C2×C61#341511yescharacteristic
324D61—82no—
325D61—161no—
326D61—161no—
327C2×#135C2×D6111yescharacteristic, complemented
328C22×D8—611no—
329C22×D8—611no—
330C22×D8C2×D6111yescharacteristic, complemented
331#546—21no—
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×M32—611no—
345C22×C16—611no—
346#495C2×D6111yescharacteristic, complemented
347#493—611no—
348#467C2×D6111yescharacteristic, complemented
349#466—611no—
350#466—611no—
351#466C2×D6111yescharacteristic, complemented
352#463—21no—
353#459—1221no—
354#450—21no—
355#449—21no—
356#447—1221no—
357#447—1221no—
358C4×D8—611no—
359C4×D8C2×D6111yescharacteristic, complemented
360C8×D4C2×D6111yescharacteristic
361#367—611no—
362C4⋊C16—611no—
363#358—611no—
364#357—611no—
365#357—611no—
366#357—611no—
367#357—611no—
368#357—611no—
369C22⋊C16—611no—
370C22⋊C16—611no—
371#346—611no—
372C61#3383011yescharacteristic, complemented, Sylow
373C2×#41—21no—
374C22×D4—21no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4—2441no—
378#135—22no—
379#135—41no—
380C2×Q16C22×D6111yescharacteristic
381C2×SD16—21no—
382C2×SD16—21no—
383C2×D8—22no—
384C2×D8—21no—
385C2×D8—41no—
386C2×D8—611no—
387C2×D8—611no—
388C2×D8—611no—
389C2×D8—1222no—
390C2×D8—1222no—
391C2×D8—1221no—
392C2×D8—1221no—
393C2×D8—1221no—
394C2×D8—2441no—
395C2×D8—2441no—
396C2×D8C22×D6111yescharacteristic
397C2×D8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400#127—1221no—
401#124—21no—
402#123—41no—
403C22⋊Q8—21no—
404#121—21no—
405#121—41no—
406#121—1221no—
407#121—1221no—
408#121—2441no—
409#120—41no—
410#120—2441no—
411#120—2441no—
412C4×D4—21no—
413C4×D4—1221no—
414C4×D4C22⋊D6111yescharacteristic
415C2×C22⋊C4—41no—
416SD32—1222no—
417SD32—1222no—
418SD32—2441no—
419D16—614no—
420D16—1222no—
421D16—1222no—
422D16—1222no—
423D16—1222no—
424D16—2441no—
425M32—1222no—
426C2×C16—611no—
427C2×C16—611no—
428C2×C16—611no—
429C2×C16—611no—
430C2×C16—1222no—
431#107—611no—
432#107C22×D6111yescharacteristic
433C4⋊C8C22⋊D6111yescharacteristic
434#103—21no—
435#102—21no—
436#102—21no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440C22⋊C8C22⋊D6111yescharacteristic
441C22⋊C8C22⋊D6111yescharacteristic
442C4×C8C22×D6111yescharacteristic
443C24—41no—
444C24—2441no—
445C24—2441no—
446#41—42no—
447#41—81no—
448C2×Q8—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—81no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—4881no—
460C2×D4—4881no—
461C2×D4—4884no—
462C2×D4—2442no—
463C2×D4—2442no—
464C2×D4—2441no—
465C2×D4—2441no—
466C2×D4—2441no—
467C2×D4—2441no—
468C2×D4—2441no—
469C2×D4—2441no—
470C2×D4#1175819911yescharacteristic
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—2441no—
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6111yescharacteristic
476Q16—21no—
477Q16—21no—
478SD16—42no—
479SD16—81no—
480D8—21no—
481D8—21no—
482D8—21no—
483D8—42no—
484D8—612no—
485D8—1221no—
486D8—1221no—
487D8—1221no—
488D8—1221no—
489D8—1221no—
490D8—4881no—
491D8—2442no—
492D8—2442no—
493D8D4×D6112yescomplemented
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8C2×C22⋊D6111yescharacteristic
499C2×C8C2×C22⋊D6111yescharacteristic
500C2×C8C2×C22⋊D6111yescharacteristic
501C2×C8C23×D6111yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—1221no—
505C4⋊C4#1175819911yescharacteristic
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C22⋊C4#1175819911yescharacteristic
515C22⋊C4#1175819911yescharacteristic
516C42C2×C22⋊D6111yescharacteristic
517C16—612no—
518C16—1221no—
519C16—1221no—
520C16—1221no—
521C23—41no—
522C23—41no—
523C23—82no—
524C23—82no—
525C23—161no—
526C23—9761no—
527C23—9761no—
528C23—4882no—
529C23—4882no—
530C23—4882no—
531C23—4882no—
532C23—2441no—
533C23—2441no—
534C23—2441no—
535C23C2×#1175819911yescharacteristic
536C23C2×#1175819911yescharacteristic
537Q8—41no—
538Q8—41no—
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—82no—
545D4—82no—
546D4—161no—
547D4—1222no—
548D4—9761no—
549D4—4882no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—4882no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—82no—
566C2×C4—161no—
567C2×C4—4882no—
568C2×C4—2441no—
569C2×C4C2×#1175819911yescharacteristic
570C2×C4C2×#1175819911yescharacteristic
571C2×C4C2×#1175819911yescharacteristic
572C2×C4C2×#1175819911yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C8—21no—
575C8—21no—
576C8—21no—
577C8C2×D4×D6111yescharacteristic
578C8#1451982211yescharacteristic
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—164no—
591C22—9764no—
592C22—9764no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22—2442no—
602C22#1684019712yescomplemented
603C22C22×#1175819911yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—81no—
609C4—81no—
610C4—4881no—
611C4D4×C22⋊D6111yescharacteristic
612C4#1684056511yescharacteristic
613C2—22no—
614C2—41no—
615C2—82no—
616C2—161no—
617C2—9761no—
618C2—9761no—
619C2—4882no—
620C2D4×#1175819911yescharacteristic
621C2C2×#1684019711yescharacteristic
622C2#2145895811yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3280C2
socleC22×C61C22×D8
hypercenter#3280D61
hypocenterC61#33830
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. C2×#16840197C2semidirect
2. C22×#14519718C2semidirect
3. D4×D488C2semidirect
4. #21448086C2semidirect
5. #21447920C2semidirect
6. #21447917C2semidirect
7. #21447871C2semidirect
8. #21447854C2semidirect
9. #21447747C2semidirect
10. #21447742C2semidirect
11. #21447731C2semidirect
12. #21447730C2semidirect
13. #21447693C2semidirect
14. #21447645C2semidirect
15. C61×#3280C2semidirect
17. C22×D488C22semidirect
20. C2×#14519720C22semidirect
21. C2×#14519720C22semidirect
22. C2×#14519720C22semidirect
26. C2×#14519718C22semidirect
27. C2×#14519718C22semidirect
28. C2×#14519718C22semidirect
29. C2×#14519718C22semidirect
30. C2×#14519717C22semidirect
31. C22×#11758185C22semidirect
32. #16840139C22semidirect
34. #16839998C22semidirect
38. C4×D488C22semidirect
39. #16839589C22semidirect
40. C4⋊#11758185C22semidirect
41. #16839566C22semidirect
42. #16839563C22semidirect
43. #16839563C22semidirect
44. #16839562C22semidirect
45. #16839562C22semidirect
46. #16839562C22semidirect
47. C22⋊#11758185C22semidirect
48. C22⋊#11758185C22semidirect
49. #16839538C22semidirect
50. C2×#135×C61C22semidirect
51. C22×D8×C61C22semidirect
53. C61×#495C22semidirect
54. C61×#467C22semidirect
55. C61×#466C22semidirect
59. C4×D8×C61C22semidirect
89. #14519718D4semidirect
172. D8×C61C2×D4semidirect
208. #11758185C2×D4semidirect
280. C22×C61#509semidirect
304. #3280D61semidirect
327. C2×#135C2×D61semidirect
330. C22×D8C2×D61semidirect
346. #495C2×D61semidirect
348. #467C2×D61semidirect
351. #466C2×D61semidirect
359. C4×D8C2×D61semidirect
372. C61#33830semidirect
493. D8D4×D61semidirect
602. C22#16840197semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
C22×#145197182
D4×D4882
#214480862
#214479202
#214479172
#214478712
#214478542
#214477472
#214477422
#214477312
#214477302
#214476932
#214476452
C61×#32802
#3383061