Finite Group Databasebeta

← #29836525

Subgroups — #29836525

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840193C211yesmaximal, characteristic, complemented
2#21457573C211yesmaximal, characteristic, complemented
3#21457504C211yesmaximal, characteristic, complemented
4#21448082C211yesmaximal, characteristic, complemented
5#21447932C211yesmaximal, characteristic, complemented
6#21447902C211yesmaximal, characteristic, complemented
7#21447879C211yesmaximal, characteristic, complemented
8#21447852C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447741C211yesmaximal, characteristic, complemented
11#21447739C211yesmaximal, characteristic, complemented
12#21447731C211yesmaximal, characteristic, complemented
13#21447684C211yesmaximal, characteristic, complemented
14#21447642C211yesmaximal, characteristic, complemented
15C2×C61×#509C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic, complemented
17C2×#14519796C2211yescharacteristic
18#16840431—21no—
19#16840406—21no—
20#16840193—24no—
21C2×#14519723C2211yescharacteristic, complemented
22C2×#14519722C2211yescharacteristic, complemented
23C2×#14519721C2211yescharacteristic, complemented
24C2×#14519718C2211yescharacteristic
25C2×#14519718C2211yescharacteristic
26C2×C16×D61C2211yescharacteristic, complemented
27#16840145C2211yescharacteristic
28#16840137C2211yescharacteristic, complemented
29#16840003C2211yescharacteristic
30#16839998C2211yescharacteristic
31#16839995C2211yescharacteristic
32#16839995C2211yescharacteristic, complemented
33#16839975—21no—
34#16839960—21no—
35#16839913—21no—
36#16839907—21no—
37#16839906—21no—
38#16839904—21no—
39Q16×Dic61C2211yescharacteristic, complemented
40D8×Dic61C2211yescharacteristic
41C8×C22⋊D61C2211yescharacteristic
42#16839588C2211yescharacteristic, complemented
43#16839580C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839566C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic, 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×#11758226—21no—
62C2×#11758224—21no—
63#14519800—22no—
64#14519800—41no—
65#14519796—22no—
66#14519796—41no—
67C2×Dic244C2311yescharacteristic
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758201—21no—
71C2×D488C2311yescharacteristic
72C2×#11758199—21no—
73C2×C8×D61C2311yescharacteristic
74#14519773—21no—
75#14519773—41no—
76#14519764—21no—
77#14519758—41no—
78#14519756—21no—
79#14519751—21no—
80#14519751—41no—
81#14519749—41no—
82#14519748—21no—
83#14519748—21no—
84#14519748—41no—
85#14519746—41no—
86Q8×Dic61—21no—
87D4×Dic61—21no—
88C4×C22⋊D61—41no—
89C4×C22⋊D61D411yescharacteristic
90C2×C22⋊Dic61—41no—
91#14519723—22no—
92#14519722—22no—
93#14519721—41no—
94#14519718—22no—
95#14519718—41no—
96C2×#11758185C2311yescharacteristic
97C2×#11758185C2311yescharacteristic
98C16×D61—22no—
99#14519711C2311yescharacteristic
100#14519711C2311yescharacteristic
101#14519706D411yescharacteristic
102#14519702—21no—
103#14519701—21no—
104#14519699—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×C61D414yescomplemented
126D16×C61—22no—
127D16×C61—22no—
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⋊D61—41no—
148C2×C22⋊D61—41no—
149C2×C22⋊D61—41no—
150C2×C22⋊D61D811yescharacteristic
151C22×Dic61—41no—
152C22×Dic61—41no—
153#11758226—42no—
154#11758226—81no—
155#11758224—42no—
156#11758224—81no—
157C2×D244—21no—
158C2×D244—41no—
159C2×C4×D61—41no—
160C2×C4×D61C2×D411yescharacteristic
161C2×Dic122—21no—
162Q16×C61—21no—
163Q16×C61C2×D412yes—
164SD16×C61—42no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61—21no—
170D8×C61—42no—
171D8×C61C2×D412yes—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2411yescharacteristic
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C22⋊Dic61—41no—
180C22⋊Dic61—41no—
181C22⋊Dic61—41no—
182C22⋊Dic61D811yescharacteristic
183#11758201—42no—
184#11758200—42no—
185#11758200—81no—
186#11758199—81no—
187#11758198—41no—
188#11758198D811yescharacteristic
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192#11758196—41no—
193#11758196D811yescharacteristic
194C4×Dic61—41no—
195C4×Dic61C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197C2×#92418C2×D411yescharacteristic
198Dic244—21no—
199Dic244—21no—
200D488—21no—
201D488—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×D61—41no—
216C22×D61C2×D811yescharacteristic
217Q8×C61—22no—
218Q8×C61—41no—
219D4×C61—22no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—82no—
226D4×C61—82no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—82no—
231C2×C4×C61C2×D811yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C22⋊D61—41no—
234C22⋊D61—82no—
235C22⋊D61—82no—
236C22⋊D61—161no—
237C22⋊D61—161no—
238C22⋊D61—161no—
239C22⋊D61—161no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—82no—
244C2×Dic61—161no—
245C2×Dic61C2×D811yescharacteristic
246C2×Dic61C2×D811yescharacteristic
247C2×Dic61C2×D811yescharacteristic
248D244—41no—
249D244—41no—
250D244—82no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—82no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61C22×D411yescharacteristic
258C8×C61H(5,𝔽2)11yescharacteristic
259#92418—21no—
260#92418—21no—
261#33834—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—42no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—164no—
275C22×C61C22×D811yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C4×C61—21no—
283C4×C61—42no—
284C4×C61D4211yescharacteristic
285C4×C61#57611yescharacteristic
286Dic61—21no—
287Dic61—21no—
288Dic61—21no—
289Dic61—81no—
290Dic61—81no—
291Dic61—81no—
292C2×#509D6111yescharacteristic, complemented
293C2×#508—611no—
294#3287—611no—
295#3280—611no—
296#2250—611no—
297#2220—611no—
298#2215—611no—
299#2211—611no—
300#2206—611no—
301#2189—611no—
302#2186—611no—
303#2186—611no—
304#2184—611no—
305#2175—611no—
306#2168—611no—
307C2×C61—41no—
308C2×C61—82no—
309C2×C61—161no—
310C2×C61—161no—
311C2×C61D4×D811yescharacteristic
312C2×C61#341411yescharacteristic
313C2×C61#341511yescharacteristic
314D61—41no—
315D61—161no—
316C2×#135—611no—
317C2×#135—611no—
318C2×#135C2×D6111yescharacteristic
319C22×D8C2×D6111yescharacteristic
320#550—1221no—
321#546—1221no—
322#509—24no—
323#508—1224no—
324C2×Q32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32C2×D6111yescharacteristic
328C2×SD32C2×D6111yescharacteristic
329C2×D16—611no—
330C2×D16—611no—
331C2×D16C2×D6111yescharacteristic
332C2×D16C2×D6111yescharacteristic
333C2×M32C2×D6111yescharacteristic
334C22×C16—611no—
335#495—611no—
336#493—611no—
337#467—611no—
338#466—611no—
339#466—611no—
340#466—611no—
341#463—1221no—
342#460—1221no—
343#450—1221no—
344#449—1221no—
345#449—1221no—
346#449—1221no—
347C4×Q16—611no—
348C4×D8—611no—
349C8×D4—611no—
350#366—611no—
351C4⋊C16—611no—
352#358—611no—
353#358—611no—
354#357—611no—
355#357—611no—
356#357—611no—
357#357—611no—
358C22⋊C16—611no—
359C22⋊C16—611no—
360#346—611no—
361C61#3383411yescharacteristic, complemented, Sylow
362C2×#41—21no—
363C2×#41—1221no—
364C2×#41—1221no—
365C22×D4—21no—
366#135—22no—
367#135—41no—
368#135—1222no—
369#135—1222no—
370#135—2441no—
371#135—2441no—
372C2×Q16—611no—
373C2×Q16C22×D6111yescharacteristic
374C2×SD16—21no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—22no—
379C2×D8—21no—
380C2×D8—41no—
381C2×D8—611no—
382C2×D8—1221no—
383C2×D8C22×D6111yescharacteristic
384C2×D8C22×D6111yescharacteristic
385C2×D8C22×D6111yescharacteristic
386C22×C8—611no—
387C22×C8C22×D6111yescharacteristic
388#127—1221no—
389#127—2441no—
390#124—1221no—
391#123—2441no—
392C22⋊Q8—1221no—
393#121—1221no—
394#121—1221no—
395#121—1221no—
396#121—2441no—
397#121—2441no—
398#121—2441no—
399#120—2441no—
400C4×Q8—1221no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—2441no—
404C2×C22⋊C4—2441no—
405Q32—1222no—
406SD32—22no—
407SD32—1222no—
408SD32—2441no—
409SD32C22⋊D6114yescomplemented
410D16—22no—
411D16—22no—
412D16—1222no—
413D16—2441no—
414M32—22no—
415C2×C16—611no—
416C2×C16—611no—
417C2×C16—1222no—
418C2×C16C22×D6111yescharacteristic
419C2×C16C22×D6111yescharacteristic
420#107—611no—
421#107—611no—
422C4⋊C8—611no—
423#103—1221no—
424#103—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429C22⋊C8—611no—
430C22⋊C8—611no—
431C4×C8—611no—
432C24—41no—
433#41—42no—
434#41—81no—
435#41—4881no—
436#41—4881no—
437#41—2442no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—1221no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—42no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—81no—
448C2×D4—611no—
449C2×D4—1221no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4—2441no—
457C22×C4—41no—
458C22×C4—611no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4D4×D6111yescharacteristic
463Q16—21no—
464Q16—1221no—
465Q16—1221no—
466Q16C2×C22⋊D6112yes—
467SD16—42no—
468SD16—4881no—
469SD16—2442no—
470SD16—2442no—
471D8—21no—
472D8—21no—
473D8—21no—
474D8—21no—
475D8—21no—
476D8—42no—
477D8—1221no—
478D8—1221no—
479D8—4881no—
480D8C2×C22⋊D6112yes—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8D4×D6111yescharacteristic
488C2×C8C23×D6111yescharacteristic
489C4⋊C4—611no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C22⋊C4—611no—
495C22⋊C4—611no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C42—611no—
504C42—2441no—
505C16—21no—
506C16—1221no—
507C16—1221no—
508C16C2×C22⋊D6112yes—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—82no—
513C23—82no—
514C23—161no—
515C23—611no—
516C23—2441no—
517C23D8×D6111yescharacteristic
518Q8—22no—
519Q8—41no—
520Q8—2441no—
521Q8—2441no—
522D4—22no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—82no—
529D4—82no—
530D4—9761no—
531D4—9761no—
532D4—9761no—
533D4—9761no—
534D4—4882no—
535D4—4882no—
536D4—4882no—
537D4—2441no—
538D4—2441no—
539D4—2441no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—82no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—9761no—
550C2×C4—4882no—
551C2×C4—4882no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4D8×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C8—21no—
558C8—1221no—
559C8—1221no—
560C8C22×C22⋊D6111yescharacteristic
561C8#1451982211yescharacteristic
562C22—21no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—164no—
575C22—1221no—
576C22—1221no—
577C22—1221no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22C2×D8×D6111yescharacteristic
582C4—21no—
583C4—42no—
584C4—1221no—
585C4—1221no—
586C4—1221no—
587C4—4881no—
588C4—4881no—
589C4—4881no—
590C4D4×C22⋊D6111yescharacteristic
591C4#1684056311yescharacteristic
592C2—41no—
593C2—82no—
594C2—161no—
595C2—161no—
596C2—9761no—
597C2—2441no—
598C2D8×C22⋊D6111yescharacteristic
599C2#2145895111yescharacteristic
600C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33834
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×#16840193C2semidirect
2. #21457573C2semidirect
3. #21457504C2semidirect
4. #21448082C2semidirect
5. #21447932C2semidirect
6. #21447902C2semidirect
7. #21447879C2semidirect
8. #21447852C2semidirect
9. #21447759C2semidirect
10. #21447741C2semidirect
11. #21447739C2semidirect
12. #21447731C2semidirect
13. #21447684C2semidirect
14. #21447642C2semidirect
15. C2×C61×#509C2semidirect
16. C2×#14519800C22semidirect
21. C2×#14519723C22semidirect
22. C2×#14519722C22semidirect
23. C2×#14519721C22semidirect
26. C2×C16×D61C22semidirect
28. #16840137C22semidirect
32. #16839995C22semidirect
39. Q16×Dic61C22semidirect
42. #16839588C22semidirect
43. #16839580C22semidirect
44. #16839567C22semidirect
45. #16839566C22semidirect
46. #16839564C22semidirect
49. #16839563C22semidirect
51. #16839541C22semidirect
52. #16839537C22semidirect
125. SD32×C61D4semidirect
292. C2×#509D61semidirect
361. C61#33834semidirect
409. SD32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401932
#214575732
#214575042
#214480822
#214479322
#214479022
#214478792
#214478522
#214477592
#214477412
#214477392
#214477312
#214476842
#214476422
C2×C61×#5092
#3383461