Finite Group Databasebeta

← #29836477

Subgroups — #29836477

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×#16840199C211yesmaximal, characteristic, complemented
2C2×D16×D61C211yesmaximal, characteristic, complemented
3#21457503C211yesmaximal, characteristic, complemented
4D8×D244C211yesmaximal, characteristic, complemented
5#21448086C211yesmaximal, characteristic, complemented
6#21447921C211yesmaximal, characteristic, complemented
7#21447918C211yesmaximal, characteristic, complemented
8#21447855C211yesmaximal, characteristic, complemented
9#21447746C211yesmaximal, characteristic, complemented
10#21447745C211yesmaximal, characteristic, complemented
11#21447732C211yesmaximal, characteristic, complemented
12#21447729C211yesmaximal, characteristic, complemented
13#21447693C211yesmaximal, characteristic, complemented
14#21447643C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic, complemented
17C2×D8×D61C2211yescharacteristic
18C2×D8×D61C2211yescharacteristic, complemented
19C2×D8×D61C2211yescharacteristic, complemented
20#16840405—21no—
21D4×D244—21no—
22#16840199—24no—
23C2×#14519720C2211yescharacteristic, complemented
24C2×#14519720C2211yescharacteristic, complemented
25C2×D976C2211yescharacteristic, complemented
26C2×D976C2211yescharacteristic, complemented
27C2×#14519718C2211yescharacteristic
28C2×#14519718C2211yescharacteristic, complemented
29C2×#14519718C2211yescharacteristic, complemented
30D16×D61—24no—
31C2×#14519716C2211yescharacteristic, complemented
32C2×C16×D61C2211yescharacteristic, complemented
33#16840139C2211yescharacteristic, complemented
34#16840000C2211yescharacteristic, complemented
35#16839999C2211yescharacteristic
36#16839999C2211yescharacteristic, complemented
37#16839995C2211yescharacteristic, complemented
38#16839974—21no—
39#16839957—21no—
40#16839911—21no—
41#16839906—21no—
42#16839897—21no—
43#16839897—21no—
44C4×D488C2211yescharacteristic, complemented
45C8×D244C2211yescharacteristic
46#16839589C2211yescharacteristic, complemented
47#16839564C2211yescharacteristic, complemented
48#16839564C2211yescharacteristic, complemented
49#16839563C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic, complemented
51#16839541C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839538C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C2×D16×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C22×D244—41no—
63C2×D4×D61—21no—
64C2×D4×D61—21no—
65C2×D4×D61—21no—
66#14519800—22no—
67#14519800—41no—
68C2×#11758201—21no—
69C2×D488—21no—
70C2×D488C2311yescharacteristic
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76D8×D61—22no—
77D8×D61—22no—
78D8×D61—22no—
79D8×D61—41no—
80D8×D61—41no—
81D8×D61—41no—
82C2×C8×D61C2311yescharacteristic
83C2×C8×D61C2311yescharacteristic
84#14519774—21no—
85#14519759—41no—
86#14519753—21no—
87#14519752—21no—
88#14519752—21no—
89#14519751—41no—
90#14519750—41no—
91#14519748—21no—
92#14519747—41no—
93#14519747—41no—
94#14519747—41no—
95C4×D244—21no—
96C4×D244D411yescharacteristic
97C22⋊C4×D61—41no—
98#14519720—22no—
99#14519720—41no—
100D976—22no—
101D976—22no—
102#14519718—22no—
103#14519718—41no—
104#14519718D414yescomplemented
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519701—21no—
112#14519699—21no—
113#14519698—21no—
114#14519697—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×SD16×C61—21no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122#124×C61—21no—
123C4×D4×C61—21no—
124SD32×C61—22no—
125D16×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C23×D61—41no—
133C23×D61—41no—
134C2×Q8×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C22×C4×C61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142#11758226—42no—
143#11758226—81no—
144D4×D61—42no—
145D4×D61—42no—
146D4×D61—42no—
147D4×D61—81no—
148D4×D61—81no—
149D4×D61—81no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244—41no—
157C2×D244—41no—
158C2×D244—84no—
159C2×D244D811yescharacteristic
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164Q16×C61—21no—
165Q16×C61—21no—
166SD16×C61—81no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61—21no—
170D8×C61C2×D412yes—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2411yescharacteristic
173C4⋊C4×C61—21no—
174C22⋊C4×C61—41no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C42×C61C2×D411yescharacteristic
178C22⋊Dic61—41no—
179#11758201—42no—
180#11758199—42no—
181#11758199—42no—
182#11758199—42no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198D811yescharacteristic
188#11758198D811yescharacteristic
189C4⋊Dic61—21no—
190C4⋊Dic61D811yescharacteristic
191#11758196—41no—
192C2×#92418C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194D488—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488—81no—
200D488C2×D412yescomplemented
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61—21no—
207#11758185—21no—
208#11758185C2×D412yescomplemented
209C23×C61—41no—
210C23×C61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—161no—
221C22×D61—161no—
222C22×D61—161no—
223C22×D61C2×D811yescharacteristic
224C22×D61C2×D811yescharacteristic
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—22no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61C2×D811yescharacteristic
235C2×C4×C61C2×D811yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C22⋊D61—82no—
238C22⋊D61—82no—
239C22⋊D61—82no—
240C22⋊D61—161no—
241C2×Dic61—41no—
242C2×Dic61C2×D811yescharacteristic
243C2×Dic61C2×D811yescharacteristic
244D244—22no—
245D244—22no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—82no—
252D244—82no—
253D244—82no—
254D244—82no—
255D244—161no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—82no—
261C4×D61—161no—
262C8×C61—21no—
263C8×C61C22×D411yescharacteristic
264C8×C61H(5,𝔽2)11yescharacteristic
265#92418—21no—
266#92418—21no—
267#33830—611nomaximal, Sylow
268C22×C61—42no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61C22×D811yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—42no—
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—164no—
290C2×D61—164no—
291C2×D61—164no—
292C2×D61#50912yescomplemented
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61D4211yescharacteristic
298C4×C61#57611yescharacteristic
299Dic61—21no—
300Dic61—21no—
301Dic61—81no—
302C2×#509—611no—
303C22×D16—611no—
304#3280—611no—
305D4×D8—611no—
306#2250—611no—
307#2218—611no—
308#2217—611no—
309#2210D6111yescharacteristic, complemented
310#2206—611no—
311#2187—611no—
312#2186—611no—
313#2184—611no—
314#2184—611no—
315#2177—611no—
316#2168—611no—
317C2×C61—82no—
318C2×C61—161no—
319C2×C61D4×D811yescharacteristic
320C2×C61C2×#50911yescharacteristic
321C2×C61#341511yescharacteristic
322D61—22no—
323D61—41no—
324D61—82no—
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×SD32C2×D6111yescharacteristic, complemented
337C2×D16—611no—
338C2×D16—611no—
339C2×D16—611no—
340C2×D16—611no—
341C2×D16—611no—
342C2×D16—1224no—
343C2×D16C2×D6111yescharacteristic
344C2×M32—611no—
345C22×C16—611no—
346#495C2×D6111yescharacteristic, complemented
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×D8C2×D6111yescharacteristic, complemented
360C8×D4—611no—
361#367—611no—
362C4⋊C16C2×D6111yescharacteristic, complemented
363#358C2×D6111yescharacteristic, complemented
364#357—611no—
365#357—611no—
366#357—611no—
367#357—611no—
368#357C2×D6111yescharacteristic, complemented
369C22⋊C16—611no—
370C22⋊C16—611no—
371#346—611no—
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×Q16C22×D6111yescharacteristic
381C2×SD16—21no—
382C2×SD16—1221no—
383C2×D8—611no—
384C2×D8—611no—
385C2×D8—611no—
386C2×D8—1221no—
387C2×D8—1221no—
388C2×D8—1221no—
389C2×D8—1222no—
390C2×D8—1222no—
391C2×D8—1222no—
392C2×D8—1221no—
393C2×D8—2441no—
394C2×D8—2441no—
395C2×D8—2441no—
396C2×D8C22×D6111yescharacteristic
397C2×D8C22×D6111yescharacteristic
398C22×C8—611no—
399C22×C8—611no—
400#127—1221no—
401#124—21no—
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—21no—
413C4×D4—611no—
414C4×D4—1221no—
415C2×C22⋊C4—2441no—
416SD32—22no—
417SD32—1222no—
418SD32—2441no—
419D16—22no—
420D16—614no—
421D16—1222no—
422D16—1222no—
423D16—1222no—
424D16—2441no—
425M32—1222no—
426C2×C16—611no—
427C2×C16—611no—
428C2×C16—1222no—
429C2×C16C22×D6111yescharacteristic
430C2×C16C22×D6111yescharacteristic
431#107—611no—
432#107C22×D6111yescharacteristic
433C4⋊C8—611no—
434#103—1221no—
435#102—21no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440C22⋊C8—611no—
441C22⋊C8—611no—
442C4×C8C22×D6111yescharacteristic
443C24—2441no—
444C24—2441no—
445C24—2441no—
446#41—4881no—
447#41—2442no—
448C2×Q8—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—611no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—4884no—
456C2×D4—4881no—
457C2×D4—4881no—
458C2×D4—4881no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2442no—
466C2×D4—2442no—
467C2×D4—2442no—
468C2×D4—2441no—
469C2×D4—2441no—
470C2×D4—2441no—
471C22×C4—41no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—2441no—
475C22×C4—2441no—
476Q16—21no—
477Q16—21no—
478SD16—81no—
479SD16—2442no—
480D8—21no—
481D8—21no—
482D8—21no—
483D8—612no—
484D8—1221no—
485D8—1221no—
486D8—1221no—
487D8—1221no—
488D8—1221no—
489D8—4881no—
490D8—2442no—
491D8—2442no—
492D8—2442no—
493D8D4×D6112yes—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8D4×D6111yescharacteristic
501C2×C8C23×D6111yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4—2441no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—611no—
510C22⋊C4—611no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42D4×D6111yescharacteristic
517C16—21no—
518C16—21no—
519C16—612no—
520C16—1221no—
521C23—41no—
522C23—41no—
523C23—611no—
524C23—611no—
525C23—9761no—
526C23—9761no—
527C23—9761no—
528C23—4882no—
529C23—4882no—
530C23—4882no—
531C23—4882no—
532C23—4882no—
533C23—4882no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537Q8—41no—
538Q8—41no—
539D4—22no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—1222no—
544D4—1222no—
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—
558D4—2441no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—82no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—9761no—
569C2×C4—4882no—
570C2×C4—2441no—
571C2×C4D8×D6111yescharacteristic
572C2×C4D8×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C8—21no—
575C8—1221no—
576C8—1221no—
577C8C2×D4×D6111yescharacteristic
578C8#1451982311yescharacteristic
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—612no—
586C22—1221no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—9764no—
592C22—9764no—
593C22—9764no—
594C22—4881no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22—2442no—
603C22C2×D8×D6111yescharacteristic
604C4—21no—
605C4—21no—
606C4—81no—
607C4—81no—
608C4—1221no—
609C4—1221no—
610C4—4881no—
611C4#1684039611yescharacteristic
612C4#1684056611yescharacteristic
613C2—82no—
614C2—161no—
615C2—1222no—
616C2—9761no—
617C2—9761no—
618C2—4882no—
619C2—2441no—
620C2#2145749111yescharacteristic
621C2#509×D6111yescharacteristic
622C2#2145895711yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
hypercenter#2210D61
hypocenterC61#33830
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×D16×D61C2semidirect
3. #21457503C2semidirect
4. D8×D244C2semidirect
5. #21448086C2semidirect
6. #21447921C2semidirect
7. #21447918C2semidirect
8. #21447855C2semidirect
9. #21447746C2semidirect
10. #21447745C2semidirect
11. #21447732C2semidirect
12. #21447729C2semidirect
13. #21447693C2semidirect
14. #21447643C2semidirect
15. C61×#2210C2semidirect
16. C2×#14519800C22semidirect
18. C2×D8×D61C22semidirect
19. C2×D8×D61C22semidirect
23. C2×#14519720C22semidirect
24. C2×#14519720C22semidirect
25. C2×D976C22semidirect
26. C2×D976C22semidirect
28. C2×#14519718C22semidirect
29. C2×#14519718C22semidirect
31. C2×#14519716C22semidirect
32. C2×C16×D61C22semidirect
33. #16840139C22semidirect
34. #16840000C22semidirect
36. #16839999C22semidirect
37. #16839995C22semidirect
44. C4×D488C22semidirect
46. #16839589C22semidirect
47. #16839564C22semidirect
48. #16839564C22semidirect
49. #16839563C22semidirect
50. #16839562C22semidirect
51. #16839541C22semidirect
52. #16839541C22semidirect
53. #16839538C22semidirect
54. C2×SD32×C61C22semidirect
56. C61×#495C22semidirect
57. C4×D8×C61C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
104. #14519718D4semidirect
200. D488C2×D4semidirect
208. #11758185C2×D4semidirect
292. C2×D61#509semidirect
309. #2210D61semidirect
336. C2×SD32C2×D61semidirect
346. #495C2×D61semidirect
359. C4×D8C2×D61semidirect
362. C4⋊C16C2×D61semidirect
363. #358C2×D61semidirect
368. #357C2×D61semidirect
372. C61#33830semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×D16×D612
#214575032
D8×D2442
#214480862
#214479212
#214479182
#214478552
#214477462
#214477452
#214477322
#214477292
#214476932
#214476432
C61×#22102
#3383061