Finite Group Databasebeta

← #29836510

Subgroups — #29836510

Order 15616.

692 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840191C211yesmaximal, characteristic, complemented
2#21457535C211yesmaximal, characteristic, complemented
3#21457504C211yesmaximal, characteristic, complemented
4#21448089C211yesmaximal, characteristic, complemented
5#21447927C211yesmaximal, characteristic, complemented
6#21447912C211yesmaximal, characteristic, complemented
7#21447872C211yesmaximal, characteristic, complemented
8#21447862C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447756C211yesmaximal, characteristic, complemented
11#21447748C211yesmaximal, characteristic, complemented
12#21447739C211yesmaximal, characteristic, complemented
13#21447695C211yesmaximal, characteristic, complemented
14#21447642C211yesmaximal, characteristic, complemented
15C2×C61×#509C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×#14519796C2211yescharacteristic, complemented
18#16840411—21no—
19#16840406—21no—
20#16840191—24no—
21C2×Dic488C2211yescharacteristic, complemented
22C2×#14519723C2211yescharacteristic, complemented
23C2×#14519721C2211yescharacteristic
24C2×#14519721C2211yescharacteristic
25C2×#14519721C2211yescharacteristic, complemented
26C2×C16×D61C2211yescharacteristic, complemented
27#16840145C2211yescharacteristic
28#16840140C2211yescharacteristic, complemented
29#16840003C2211yescharacteristic
30#16840003C2211yescharacteristic
31#16840000C2211yescharacteristic, complemented
32#16839995C2211yescharacteristic
33#16839975—21no—
34#16839965—21no—
35#16839924—21no—
36#16839921—21no—
37#16839913—21no—
38#16839904—21no—
39D8×Dic61C2211yescharacteristic
40D8×Dic61C2211yescharacteristic, complemented
41C8×C22⋊D61C2211yescharacteristic
42#16839590C2211yescharacteristic, complemented
43#16839580C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839567C2211yescharacteristic, complemented
46#16839566C2211yescharacteristic, complemented
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839563C2211yescharacteristic, complemented
50C22⋊#11758185C2211yescharacteristic
51#16839541C2211yescharacteristic, complemented
52#16839537C2211yescharacteristic, complemented
53C2×#135×C61C2211yescharacteristic
54C22×D8×C61C2211yescharacteristic
55C61×#509—24no—
56C2×SD32×C61C2211yescharacteristic
57C2×SD32×C61C2211yescharacteristic
58C2×D16×C61C2211yescharacteristic
59C2×D16×C61C2211yescharacteristic
60C2×M32×C61C2211yescharacteristic
61C2×#11758224—21no—
62C2×#11758224—21no—
63#14519796—22no—
64#14519796—22no—
65#14519796—41no—
66#14519796—41no—
67C2×Dic244C2311yescharacteristic
68C2×Dic244C2311yescharacteristic
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×C8×D61C2311yescharacteristic
74#14519775—21no—
75#14519764—21no—
76#14519760—41no—
77#14519758—41no—
78#14519758—41no—
79#14519756—21no—
80#14519756—21no—
81#14519753—21no—
82#14519749—41no—
83#14519749—41no—
84#14519748—21no—
85#14519746—41no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88C4×C22⋊D61D411yescharacteristic
89C2×#11758196—41no—
90C2×C22⋊Dic61—41no—
91Dic488—22no—
92#14519723—41no—
93#14519721—22no—
94#14519721—22no—
95#14519721—41no—
96C2×#11758185C2311yescharacteristic
97C2×#11758185C2311yescharacteristic
98C16×D61—22no—
99#14519711C2311yescharacteristic
100#14519711C2311yescharacteristic
101#14519706D411yescharacteristic
102#14519702—21no—
103#14519702—21no—
104#14519701—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×C61—22no—
126D16×C61—22no—
127D16×C61D414yescomplemented
128M32×C61—22no—
129C2×C16×C61C2311yescharacteristic
130C2×C16×C61C2311yescharacteristic
131C24×C61—41no—
132#41×C61—42no—
133#41×C61—81no—
134C2×Q8×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—42no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C2×D4×C61—81no—
142C22×C4×C61—41no—
143C22×C4×C61C2×D411yescharacteristic
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61D811yescharacteristic
148C22×Dic61—41no—
149C22×Dic61—41no—
150C22×Dic61—41no—
151C22×Dic61—41no—
152#11758224—42no—
153#11758224—42no—
154#11758224—81no—
155#11758224—81no—
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161SD16×C61—42no—
162D8×C61—21no—
163D8×C61—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—42no—
167D8×C61C2×D412yes—
168D8×C61C2×D412yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2411yescharacteristic
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C22⋊Dic61—41no—
180C22⋊Dic61D811yescharacteristic
181#11758202—81no—
182#11758200—42no—
183#11758200—42no—
184#11758200—81no—
185#11758198—41no—
186#11758198D811yescharacteristic
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189#11758196—41no—
190#11758196—41no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194#11758196D811yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197C2×#92418C2×D411yescharacteristic
198Dic244—21no—
199Dic244—21no—
200Dic244—21no—
201Dic244—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—82no—
212C23×C61—82no—
213C23×C61—161no—
214C23×C61C2×D811yescharacteristic
215C22×D61C2×D811yescharacteristic
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—22no—
219D4×C61—22no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—82no—
225D4×C61—82no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—82no—
230C2×C4×C61C2×D811yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C22⋊D61—41no—
233C22⋊D61—82no—
234C22⋊D61—82no—
235C22⋊D61—161no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241C2×Dic61—82no—
242C2×Dic61—82no—
243C2×Dic61—161no—
244C2×Dic61C2×D811yescharacteristic
245C2×Dic61C2×D811yescharacteristic
246C2×Dic61C2×D811yescharacteristic
247C4×D61—21no—
248C4×D61—21no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—41no—
253C8×C61—21no—
254C8×C61C22×D411yescharacteristic
255C8×C61H(5,𝔽2)11yescharacteristic
256#92418—21no—
257#92418—21no—
258#33833—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—164no—
272C22×C61C22×D811yescharacteristic
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C4×C61—21no—
277C4×C61—81no—
278C4×C61D4211yescharacteristic
279C4×C61#57611yescharacteristic
280Dic61—21no—
281Dic61—21no—
282Dic61—21no—
283Dic61—81no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287C2×#509D6111yescharacteristic, complemented
288C2×#508—611no—
289#3283—611no—
290#3280—611no—
291#2251—611no—
292#2219—611no—
293#2217—611no—
294#2210—611no—
295#2208—611no—
296#2189—611no—
297#2189—611no—
298#2187—611no—
299#2186—611no—
300#2177—611no—
301#2168—611no—
302C2×C61—41no—
303C2×C61—82no—
304C2×C61—82no—
305C2×C61—161no—
306C2×C61D4×D811yescharacteristic
307C2×C61#341411yescharacteristic
308C2×C61#341611yescharacteristic
309D61—41no—
310C2×#135—611no—
311C2×#135—611no—
312C2×#135C2×D6111yescharacteristic
313C22×D8C2×D6111yescharacteristic
314#547—1221no—
315#546—1221no—
316#509—24no—
317#508—1224no—
318C2×Q32—611no—
319C2×Q32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32C2×D6111yescharacteristic
324C2×SD32C2×D6111yescharacteristic
325C2×D16C2×D6111yescharacteristic
326C2×D16C2×D6111yescharacteristic
327C2×M32C2×D6111yescharacteristic
328C22×C16—611no—
329#495—611no—
330#494—611no—
331#467—611no—
332#467—611no—
333#467—611no—
334#466—611no—
335#463—1221no—
336#461—1221no—
337#452—1221no—
338#452—1221no—
339#450—1221no—
340#449—1221no—
341C4×D8—611no—
342C4×D8—611no—
343C8×D4—611no—
344#367—611no—
345C4⋊C16—611no—
346#358—611no—
347#358—611no—
348#358—611no—
349#357—611no—
350#357—611no—
351#357—611no—
352C22⋊C16—611no—
353C22⋊C16—611no—
354#346—611no—
355C61#3383311yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×#41—1221no—
358C2×#41—1221no—
359C22×D4—21no—
360#135—22no—
361#135—41no—
362#135—1222no—
363#135—1222no—
364#135—2441no—
365#135—2441no—
366C2×Q16—611no—
367C2×Q16—611no—
368C2×Q16—1221no—
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—21no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—22no—
375C2×D8—21no—
376C2×D8—41no—
377C2×D8C22×D6111yescharacteristic
378C2×D8C22×D6111yescharacteristic
379C2×D8C22×D6111yescharacteristic
380C22×C8—611no—
381C22×C8C22×D6111yescharacteristic
382C4⋊Q8—1221no—
383#124—1221no—
384#123—2441no—
385#123—2441no—
386#123—2441no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8—1221no—
390#121—1221no—
391#121—2441no—
392#121—2441no—
393#120—2441no—
394C4×D4—611no—
395C4×D4—1221no—
396C4×D4—1221no—
397C2×C4⋊C4—2441no—
398C2×C22⋊C4—2441no—
399Q32—1222no—
400Q32—2441no—
401SD32—22no—
402SD32—22no—
403SD32—1222no—
404SD32—1222no—
405SD32—2441no—
406D16—22no—
407D16C22⋊D6114yescomplemented
408M32—22no—
409C2×C16—611no—
410C2×C16—611no—
411C2×C16—1222no—
412C2×C16C22×D6111yescharacteristic
413C2×C16C22×D6111yescharacteristic
414#107—611no—
415#107—611no—
416C4⋊C8—611no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—1221no—
421#102—1221no—
422#102—1221no—
423C22⋊C8—611no—
424C22⋊C8—611no—
425C4×C8—611no—
426C24—41no—
427#41—42no—
428#41—81no—
429#41—4881no—
430#41—4881no—
431#41—2442no—
432#41—2442no—
433C2×Q8—21no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—81no—
443C2×D4—611no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2441no—
447C22×C4—41no—
448C22×C4—611no—
449C22×C4—2441no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4D4×D6111yescharacteristic
454Q16—21no—
455Q16—21no—
456Q16—1221no—
457Q16—1221no—
458Q16—1221no—
459Q16—1221no—
460Q16—4881no—
461SD16—42no—
462SD16—4881no—
463SD16—2442no—
464SD16—2442no—
465D8—21no—
466D8—21no—
467D8—21no—
468D8—21no—
469D8—42no—
470D8C2×C22⋊D6112yes—
471D8C2×C22⋊D6112yes—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8D4×D6111yescharacteristic
479C2×C8C23×D6111yescharacteristic
480C4⋊C4—611no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C4⋊C4—2441no—
487C4⋊C4—2441no—
488C22⋊C4—611no—
489C22⋊C4—611no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C42—611no—
499C16—21no—
500C16—1221no—
501C16—1221no—
502C16C2×C22⋊D6112yes—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—82no—
507C23—82no—
508C23—161no—
509C23—611no—
510C23D8×D6111yescharacteristic
511Q8—41no—
512Q8—41no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517D4—22no—
518D4—22no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—82no—
524D4—82no—
525D4—9761no—
526D4—4882no—
527D4—4882no—
528D4—2441no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—9761no—
539C2×C4—4882no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4D8×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C8—21no—
549C8—1221no—
550C8—1221no—
551C8C22×C22⋊D6111yescharacteristic
552C8#1451982211yescharacteristic
553C22—21no—
554C22—21no—
555C22—21no—
556C22—42no—
557C22—42no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—164no—
566C22—1221no—
567C22—1221no—
568C22—1221no—
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—81no—
572C4—1221no—
573C4—1221no—
574C4—1221no—
575C4—4881no—
576C4—4881no—
577C4—4881no—
578C4—4881no—
579C4D4×C22⋊D6111yescharacteristic
580C4#1684056311yescharacteristic
581C2—41no—
582C2—82no—
583C2—82no—
584C2—161no—
585C2—2441no—
586C2D8×C22⋊D6111yescharacteristic
587C2#2145895111yescharacteristic
588C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33833
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×#16840191C2semidirect
2. #21457535C2semidirect
3. #21457504C2semidirect
4. #21448089C2semidirect
5. #21447927C2semidirect
6. #21447912C2semidirect
7. #21447872C2semidirect
8. #21447862C2semidirect
9. #21447759C2semidirect
10. #21447756C2semidirect
11. #21447748C2semidirect
12. #21447739C2semidirect
13. #21447695C2semidirect
14. #21447642C2semidirect
15. C2×C61×#509C2semidirect
17. C2×#14519796C22semidirect
21. C2×Dic488C22semidirect
22. C2×#14519723C22semidirect
25. C2×#14519721C22semidirect
26. C2×C16×D61C22semidirect
28. #16840140C22semidirect
31. #16840000C22semidirect
40. D8×Dic61C22semidirect
42. #16839590C22semidirect
43. #16839580C22semidirect
44. #16839567C22semidirect
45. #16839567C22semidirect
46. #16839566C22semidirect
49. #16839563C22semidirect
51. #16839541C22semidirect
52. #16839537C22semidirect
127. D16×C61D4semidirect
287. C2×#509D61semidirect
355. C61#33833semidirect
407. D16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401912
#214575352
#214575042
#214480892
#214479272
#214479122
#214478722
#214478622
#214477592
#214477562
#214477482
#214477392
#214476952
#214476422
C2×C61×#5092
#3383361