Finite Group Databasebeta

← #29836694

Subgroups — #29836694

Order 15616.

696 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×#16840192C211yesmaximal, characteristic, complemented
2C22×#14519721C211yesmaximal, characteristic, complemented
3#21457562C211yesmaximal, characteristic, complemented
4#21448078C211yesmaximal, characteristic, complemented
5#21447917C211yesmaximal, characteristic, complemented
6#21447913C211yesmaximal, characteristic, complemented
7#21447878C211yesmaximal, characteristic, complemented
8#21447874C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447752C211yesmaximal, characteristic, complemented
11#21447740C211yesmaximal, characteristic, complemented
12#21447731C211yesmaximal, characteristic, complemented
13C4×#14519721C211yesmaximal, characteristic, complemented
14D4×#11758185C211yesmaximal, characteristic, complemented
15C61×#3280C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic, complemented
17C22×Dic244C2211yescharacteristic
18#16840421—21no—
19#16840192—24no—
20C2×#14519723C2211yescharacteristic, complemented
21C2×#14519723C2211yescharacteristic, complemented
22C2×#14519721—24no—
23C2×#14519721C2211yescharacteristic
24C2×#14519721C2211yescharacteristic
25C2×#14519721C2211yescharacteristic
26C2×#14519721C2211yescharacteristic
27C2×#14519720C2211yescharacteristic, complemented
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C22×#11758185C2211yescharacteristic
31C22×#11758185C2211yescharacteristic, complemented
32#16840146C2211yescharacteristic, complemented
33#16840003C2211yescharacteristic, complemented
34#16839998C2211yescharacteristic
35#16839985—21no—
36#16839926—21no—
37#16839903—21no—
38C4×Dic244C2211yescharacteristic, complemented
39#16839589C2211yescharacteristic, complemented
40C4⋊#11758185C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839565C2211yescharacteristic, complemented
43#16839565C2211yescharacteristic, complemented
44#16839563C2211yescharacteristic
45#16839563C2211yescharacteristic
46#16839562C2211yescharacteristic, complemented
47C22⋊#11758185C2211yescharacteristic
48C22⋊#11758185C2211yescharacteristic, complemented
49C4×#11758185C2211yescharacteristic, complemented
50C2×#135×C61C2211yescharacteristic
51C22×D8×C61C2211yescharacteristic
52C61×#546—21no—
53C61×#495C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#463—21no—
57C61×#450—21no—
58C61×#449—21no—
59C4×D8×C61C2211yescharacteristic
60C8×D4×C61C2211yescharacteristic
61C2×#11758222—21no—
62C22×Dic122—21no—
63#14519799—22no—
64#14519799—41no—
65C2×Dic244—22no—
66C2×Dic244—21no—
67C2×Dic244—41no—
68C2×Dic244C2311yescharacteristic
69C2×Dic244C2311yescharacteristic
70C2×#11758190—21no—
71C2×#11758190—21no—
72C2×D488C2311yescharacteristic
73#14519765—21no—
74#14519763—41no—
75#14519756—21no—
76#14519755—41no—
77#14519751—21no—
78#14519749—41no—
79C4×Dic122—21no—
80#14519737—41no—
81#14519723—22no—
82#14519723—41no—
83#14519721—22no—
84#14519721—22no—
85#14519721—22no—
86#14519721D414yescomplemented
87#14519720—22no—
88#14519718—22no—
89#14519718—41no—
90C2×#11758185—22no—
91C2×#11758185—22no—
92C2×#11758185C2311yescharacteristic
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519702—21no—
98#14519702—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—
132#41×C61—42no—
133#41×C61—81no—
134C2×Q8×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—42no—
138C2×D4×C61—41no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C2×D4×C61—81no—
142C2×D4×C61D811yescharacteristic
143C22×C4×C61—41no—
144C22×C4×C61—41no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—41no—
148C2×C22⋊D61—41no—
149C22×Dic61—41no—
150#11758222—42no—
151#11758222—81no—
152C2×D244—21no—
153C2×C4×D61—41no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—42no—
157C2×Dic122—41no—
158C2×Dic122—81no—
159Q16×C61—21no—
160Q16×C61—21no—
161SD16×C61—42no—
162SD16×C61—81no—
163D8×C61—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—42no—
167D8×C61C2×D412yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61D811yescharacteristic
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61D811yescharacteristic
185C22⋊C4×C61D811yescharacteristic
186C42×C61C2×D411yescharacteristic
187C22⋊Dic61—41no—
188#11758198—41no—
189#11758198—41no—
190#11758198—41no—
191C4⋊Dic61—21no—
192#11758196—41no—
193#11758196—41no—
194C4×Dic61—41no—
195Dic244—21no—
196Dic244—21no—
197Dic244—21no—
198Dic244—42no—
199Dic244C2×D412yes—
200D488—21no—
201D488—21no—
202#11758190—42no—
203#11758190—81no—
204#11758185—21no—
205#11758185—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—82no—
211C23×C61—82no—
212C23×C61—161no—
213C23×C61C2×D811yescharacteristic
214C23×C61C2×D811yescharacteristic
215C22×D61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—22no—
219D4×C61—22no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—82no—
224D4×C61—82no—
225D4×C61—161no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—82no—
233C2×C4×C61—161no—
234C2×C4×C61C2×D811yescharacteristic
235C2×C4×C61C2×D811yescharacteristic
236C2×C4×C61C2×D811yescharacteristic
237C2×C4×C61C2×D811yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—82no—
240C22⋊D61—82no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—82no—
245D244—41no—
246D244—41no—
247C4×D61—82no—
248Dic122—22no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—82no—
253C8×C61—21no—
254C8×C61—21no—
255C8×C61—21no—
256C8×C61C22×D411yescharacteristic
257C8×C61H(5,𝔽2)11yescharacteristic
258#33848—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—42no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—164no—
271C22×C61#50812yescomplemented
272C22×C61C22×D811yescharacteristic
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61D4211yescharacteristic
283C4×C61#57611yescharacteristic
284Dic61—42no—
285Dic61—81no—
286Dic61—81no—
287C2×#508—611no—
288C22×SD32—611no—
289#3285—611no—
290#3280D6111yescharacteristic, complemented
291#2249—611no—
292#2217—611no—
293#2217—611no—
294#2211—611no—
295#2210—611no—
296#2189—611no—
297#2188—611no—
298#2186—611no—
299#2184—611no—
300C4×SD32—611no—
301D4×C16—611no—
302C2×C61—22no—
303C2×C61—41no—
304C2×C61—82no—
305C2×C61—161no—
306C2×C61D4×D811yescharacteristic
307C2×C61C2×#50811yescharacteristic
308C2×C61#341611yescharacteristic
309D61—161no—
310C2×#135—611no—
311C2×#135C2×D6111yescharacteristic
312C22×Q16—611no—
313C22×D8C2×D6111yescharacteristic
314#548—1221no—
315#546—21no—
316#508—1224no—
317C2×Q32—611no—
318C2×Q32—611no—
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32—611no—
324C2×SD32—1224no—
325C2×D16—611no—
326C2×D16—611no—
327C22×C16—611no—
328C22×C16—611no—
329#495—611no—
330#495C2×D6111yescharacteristic
331#467—611no—
332#467C2×D6111yescharacteristic
333#466—611no—
334#466C2×D6111yescharacteristic
335#464—1221no—
336#463—21no—
337#452—1221no—
338#450—21no—
339#449—21no—
340#448—1221no—
341C4×Q16—611no—
342C4×D8C2×D6111yescharacteristic
343C8×D4C2×D6111yescharacteristic
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—
354C4×C16—611no—
355C61#3384811yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×#41—1221no—
358C22×Q8—1221no—
359C22×D4—21no—
360#135—22no—
361#135—41no—
362#135—1222no—
363#135—2441no—
364C2×Q16—611no—
365C2×Q16—611no—
366C2×Q16—1222no—
367C2×Q16—1221no—
368C2×Q16—2441no—
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—21no—
371C2×SD16—21no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—22no—
375C2×D8—21no—
376C2×D8—41no—
377C2×D8—611no—
378C2×D8C22×D6111yescharacteristic
379C2×D8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6111yescharacteristic
382#124—21no—
383#124—1221no—
384#124—2441no—
385#123—41no—
386C22⋊Q8—21no—
387C22⋊Q8—1221no—
388C22⋊Q8—2441no—
389#121—21no—
390#121—41no—
391#121—1221no—
392#121—2441no—
393#120—41no—
394C4×Q8—1221no—
395C4×D4—21no—
396C4×D4—2441no—
397C4×D4C22⋊D6111yescharacteristic
398C2×C22⋊C4—41no—
399Q32—1222no—
400Q32—2441no—
401SD32—614no—
402SD32—1222no—
403SD32—1222no—
404SD32—1222no—
405SD32—1222no—
406D16—1222no—
407D16—2441no—
408C2×C16—611no—
409C2×C16—611no—
410C2×C16—611no—
411C2×C16—611no—
412C2×C16—1222no—
413C2×C16—1222no—
414#107—611no—
415#107C22×D6111yescharacteristic
416C4⋊C8C22⋊D6111yescharacteristic
417#103—21no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—21no—
422#102—1221no—
423C22⋊C8C22⋊D6111yescharacteristic
424C22⋊C8C22⋊D6111yescharacteristic
425C4×C8C22×D6111yescharacteristic
426C24—41no—
427#41—42no—
428#41—81no—
429#41—4881no—
430#41—2442no—
431C2×Q8—21no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—4881no—
435C2×Q8—2442no—
436C2×Q8—2441no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—81no—
444C2×D4—1221no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4#1175819911yescharacteristic
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454Q16—21no—
455Q16—21no—
456Q16—612no—
457Q16—1221no—
458Q16—1221no—
459Q16—1221no—
460Q16—2442no—
461SD16—42no—
462SD16—81no—
463SD16—4881no—
464SD16—2442no—
465D8—21no—
466D8—21no—
467D8—21no—
468D8—42no—
469D8—1221no—
470D8—1221no—
471D8D4×D6112yes—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8C2×C22⋊D6111yescharacteristic
477C2×C8C2×C22⋊D6111yescharacteristic
478C2×C8C2×C22⋊D6111yescharacteristic
479C2×C8C23×D6111yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—1221no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4#1175819911yescharacteristic
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4#1175819911yescharacteristic
496C22⋊C4#1175819911yescharacteristic
497C42—2441no—
498C42C2×C22⋊D6111yescharacteristic
499C16—612no—
500C16—1221no—
501C16—1221no—
502C16—1221no—
503C23—41no—
504C23—41no—
505C23—82no—
506C23—82no—
507C23—161no—
508C23—2441no—
509C23C2×#1175819911yescharacteristic
510C23C2×#1175819911yescharacteristic
511Q8—41no—
512Q8—41no—
513Q8—1222no—
514Q8—4882no—
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—82no—
524D4—82no—
525D4—161no—
526D4—4882no—
527D4—4882no—
528D4—2441no—
529D4—2441no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—161no—
538C2×C4—4882no—
539C2×C4—4882no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4C2×#1175819911yescharacteristic
544C2×C4C2×#1175819911yescharacteristic
545C2×C4C2×#1175819911yescharacteristic
546C2×C4C2×#1175819911yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8C2×D4×D6111yescharacteristic
552C8#1451982211yescharacteristic
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—164no—
565C22—4881no—
566C22—4881no—
567C22—4881no—
568C22#1684019212yescomplemented
569C22C22×#1175819911yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—81no—
575C4—81no—
576C4—4881no—
577C4—4881no—
578C4—2442no—
579C4D4×C22⋊D6111yescharacteristic
580C4#1684056511yescharacteristic
581C2—22no—
582C2—41no—
583C2—82no—
584C2—161no—
585C2—9761no—
586C2D4×#1175819911yescharacteristic
587C2C2×#1684019211yescharacteristic
588C2#2145896611yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3280C2
socleC22×C61C22×D8
hypercenter#3280D61
hypocenterC61#33848
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×#16840192C2semidirect
2. C22×#14519721C2semidirect
3. #21457562C2semidirect
4. #21448078C2semidirect
5. #21447917C2semidirect
6. #21447913C2semidirect
7. #21447878C2semidirect
8. #21447874C2semidirect
9. #21447759C2semidirect
10. #21447752C2semidirect
11. #21447740C2semidirect
12. #21447731C2semidirect
13. C4×#14519721C2semidirect
14. D4×#11758185C2semidirect
15. C61×#3280C2semidirect
16. C2×#14519799C22semidirect
20. C2×#14519723C22semidirect
21. C2×#14519723C22semidirect
27. C2×#14519720C22semidirect
31. C22×#11758185C22semidirect
32. #16840146C22semidirect
33. #16840003C22semidirect
38. C4×Dic244C22semidirect
39. #16839589C22semidirect
40. C4⋊#11758185C22semidirect
41. #16839566C22semidirect
42. #16839565C22semidirect
43. #16839565C22semidirect
46. #16839562C22semidirect
48. C22⋊#11758185C22semidirect
49. C4×#11758185C22semidirect
86. #14519721D4semidirect
271. C22×C61#508semidirect
290. #3280D61semidirect
355. C61#33848semidirect
568. C22#16840192semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
C22×#145197212
#214575622
#214480782
#214479172
#214479132
#214478782
#214478742
#214477592
#214477522
#214477402
#214477312
C4×#145197212
D4×#117581852
C61×#32802
#3384861