Finite Group Databasebeta

← #29836537

Subgroups — #29836537

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×#16840196C211yesmaximal, characteristic, complemented
2C2×#16840191C211yesmaximal, characteristic, complemented
3#21457577C211yesmaximal, characteristic, complemented
4#21457505C211yesmaximal, characteristic, complemented
5#21448083C211yesmaximal, characteristic, complemented
6#21447931C211yesmaximal, characteristic, complemented
7#21447904C211yesmaximal, characteristic, complemented
8#21447882C211yesmaximal, characteristic, complemented
9#21447760C211yesmaximal, characteristic, complemented
10#21447743C211yesmaximal, characteristic, complemented
11#21447739C211yesmaximal, characteristic, complemented
12#21447729C211yesmaximal, characteristic, complemented
13#21447683C211yesmaximal, characteristic, complemented
14#21447643C211yesmaximal, characteristic, complemented
15C61×#2206C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×#14519796C2211yescharacteristic
18C2×#14519796C2211yescharacteristic, complemented
19C2×D8×D61C2211yescharacteristic
20#16840432—21no—
21#16840407—21no—
22#16840196—24no—
23#16840191—24no—
24C2×Dic488C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic
26C2×#14519721C2211yescharacteristic
27C2×#14519721C2211yescharacteristic, complemented
28C2×#14519721C2211yescharacteristic, complemented
29C2×#14519718C2211yescharacteristic
30C2×#14519718C2211yescharacteristic
31C2×#14519716C2211yescharacteristic
32C2×C16×D61C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic
34#16840004C2211yescharacteristic
35#16839999C2211yescharacteristic
36#16839995C2211yescharacteristic
37#16839995C2211yescharacteristic, complemented
38#16839977—21no—
39#16839959—21no—
40#16839914—21no—
41#16839908—21no—
42#16839904—21no—
43#16839904—21no—
44C4×Dic244C2211yescharacteristic, complemented
45C8×D244C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839567C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic
50#16839563C2211yescharacteristic
51#16839541C2211yescharacteristic
52#16839541C2211yescharacteristic, complemented
53#16839538C2211yescharacteristic, complemented
54C2×D16×C61C2211yescharacteristic
55C2×D16×C61C2211yescharacteristic
56C61×#493C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×#11758224—21no—
63C2×#11758224—21no—
64C2×D4×D61—21no—
65#14519796—22no—
66#14519796—22no—
67#14519796—22no—
68#14519796—41no—
69#14519796—41no—
70#14519796—41no—
71C2×Dic244C2311yescharacteristic
72C2×Dic244C2311yescharacteristic
73C2×#11758190—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×D488C2311yescharacteristic
78C2×#11758199—21no—
79D8×D61—22no—
80D8×D61—41no—
81C2×C8×D61C2311yescharacteristic
82C2×C8×D61C2311yescharacteristic
83#14519773—41no—
84#14519765—21no—
85#14519761—41no—
86#14519757—21no—
87#14519752—21no—
88#14519751—41no—
89#14519749—41no—
90#14519749—41no—
91#14519748—21no—
92#14519748—21no—
93#14519747—41no—
94C4×Dic122—21no—
95C4×D244D411yescharacteristic
96D4×Dic61—41no—
97C2×#11758198—41no—
98Dic488—22no—
99#14519722—22no—
100#14519721—22no—
101#14519721—41no—
102#14519721D414yescomplemented
103#14519718—22no—
104#14519718—41no—
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519702—21no—
112#14519700—21no—
113#14519698—21no—
114#14519698—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×D8×C61—21no—
119C2×D8×C61C2311yescharacteristic
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122#127×C61—21no—
123C4×D4×C61—21no—
124D16×C61—22no—
125D16×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—41no—
136C2×D4×C61—41no—
137C22×C4×C61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C22×Dic61—41no—
145C22×Dic61—41no—
146C22×Dic61—41no—
147#11758224—42no—
148#11758224—42no—
149#11758224—42no—
150#11758224—81no—
151#11758224—81no—
152#11758224—81no—
153D4×D61—42no—
154D4×D61—81no—
155C2×D244—21no—
156C2×D244—41no—
157C2×D244D811yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×Dic122—21no—
161C2×Dic122—21no—
162D8×C61—21no—
163D8×C61—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—81no—
168D8×C61C2×D412yes—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2411yescharacteristic
171C4⋊C4×C61—21no—
172C22⋊C4×C61—41no—
173C42×C61C2×D411yescharacteristic
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176#11758200—42no—
177#11758200—42no—
178#11758200—42no—
179#11758199—42no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198D811yescharacteristic
185#11758198D811yescharacteristic
186C4⋊Dic61—21no—
187C4⋊Dic61—41no—
188C4⋊Dic61D811yescharacteristic
189#11758196—41no—
190C4×Dic61—41no—
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244—21no—
194Dic244—21no—
195Dic244—21no—
196Dic244C2×D412yes—
197D488—21no—
198D488—21no—
199#11758190—81no—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C22×D61—41no—
212C22×D61—82no—
213C22×D61—82no—
214C22×D61—161no—
215C22×D61C2×D811yescharacteristic
216C22×D61C2×D811yescharacteristic
217D4×C61—22no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—82no—
224D4×C61—161no—
225C2×C4×C61—41no—
226C2×C4×C61—161no—
227C2×C4×C61C2×D811yescharacteristic
228C2×C4×C61C2×D811yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C22⋊D61—82no—
231C22⋊D61—82no—
232C22⋊D61—82no—
233C22⋊D61—82no—
234C22⋊D61—161no—
235C22⋊D61—161no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—82no—
240C2×Dic61—82no—
241C2×Dic61—82no—
242C2×Dic61C2×D811yescharacteristic
243C2×Dic61C2×D811yescharacteristic
244D244—41no—
245D244—41no—
246D244—41no—
247D244—161no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252Dic122—22no—
253Dic122—41no—
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—42no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61C22×D811yescharacteristic
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—164no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—81no—
285C4×C61D4211yescharacteristic
286C4×C61#57611yescharacteristic
287Dic61—21no—
288Dic61—21no—
289Dic61—42no—
290Dic61—81no—
291Dic61—81no—
292C2×#509—611no—
293C2×#508—611no—
294#3287—611no—
295#3280—611no—
296#2250—611no—
297#2220—611no—
298#2215—611no—
299#2211—611no—
300#2206D6111yescharacteristic, complemented
301#2189—611no—
302#2186—611no—
303#2186—611no—
304#2184—611no—
305#2175—611no—
306#2168—611no—
307C2×C61—82no—
308C2×C61—161no—
309C2×C61—161no—
310C2×C61D4×D811yescharacteristic
311C2×C61#341411yescharacteristic
312C2×C61#341511yescharacteristic
313D61—41no—
314D61—41no—
315D61—161no—
316C2×#135—611no—
317C2×#135—611no—
318C2×#135—611no—
319C22×D8—611no—
320#550—1221no—
321#546—1221no—
322#509—1224no—
323#508—1224no—
324C2×Q32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×D16—611no—
330C2×D16—611no—
331C2×D16C2×D6111yescharacteristic
332C2×D16C2×D6111yescharacteristic
333C2×M32—611no—
334C22×C16—611no—
335#495—611no—
336#493C2×D6111yescharacteristic, complemented
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×D8C2×D6111yescharacteristic
349C8×D4—611no—
350#366—611no—
351C4⋊C16C2×D6111yescharacteristic, complemented
352#358—611no—
353#358—611no—
354#357—611no—
355#357—611no—
356#357C2×D6111yescharacteristic, complemented
357#357C2×D6111yescharacteristic, complemented
358C22⋊C16—611no—
359C22⋊C16—611no—
360#346—611no—
361C61#3383411yescharacteristic, complemented, Sylow
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C22×D4—1221no—
366#135—1222no—
367#135—1222no—
368#135—1222no—
369#135—2441no—
370#135—2441no—
371#135—2441no—
372C2×Q16—611no—
373C2×Q16—611no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—21no—
379C2×D8—611no—
380C2×D8—1221no—
381C2×D8—1222no—
382C2×D8—2441no—
383C2×D8C22×D6111yescharacteristic
384C2×D8C22×D6111yescharacteristic
385C2×D8C22×D6111yescharacteristic
386C22×C8—611no—
387C22×C8—611no—
388#127—21no—
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—21no—
402C4×D4—611no—
403C4×D4—2441no—
404C2×C22⋊C4—2441no—
405Q32—1222no—
406SD32—614no—
407SD32—1222no—
408SD32—1222no—
409SD32—2441no—
410D16—22no—
411D16—22no—
412D16—1222no—
413D16—2441no—
414M32—1222no—
415C2×C16—611no—
416C2×C16—611no—
417C2×C16—1222no—
418C2×C16C22×D6111yescharacteristic
419C2×C16C22×D6111yescharacteristic
420#107—611no—
421#107C22×D6111yescharacteristic
422C4⋊C8—611no—
423#103—1221no—
424#103—1221no—
425#102—21no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429C22⋊C8—611no—
430C22⋊C8—611no—
431C4×C8C22×D6111yescharacteristic
432C24—2441no—
433#41—4881no—
434#41—4881no—
435#41—4881no—
436#41—2442no—
437#41—2442no—
438#41—2442no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—611no—
447C2×D4—1221no—
448C2×D4—4881no—
449C2×D4—2441no—
450C2×D4—2441no—
451C2×D4—2442no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4—2441no—
457C22×C4—41no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4—2441no—
463Q16—612no—
464Q16—1221no—
465Q16—1221no—
466Q16—1221no—
467SD16—4881no—
468SD16—2442no—
469SD16—2442no—
470SD16—2442no—
471D8—21no—
472D8—21no—
473D8—21no—
474D8—21no—
475D8—21no—
476D8—81no—
477D8—1221no—
478D8—1221no—
479D8—2442no—
480D8D4×D6112yes—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8D4×D6111yescharacteristic
488C2×C8C23×D6111yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—611no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C22⋊C4—41no—
495C22⋊C4—611no—
496C22⋊C4—611no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C42—2441no—
504C42D4×D6111yescharacteristic
505C16—21no—
506C16—21no—
507C16—612no—
508C16—1221no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—611no—
513C23—611no—
514C23—9761no—
515C23—4882no—
516C23—4882no—
517C23—2441no—
518Q8—1222no—
519Q8—2441no—
520Q8—2441no—
521Q8—2441no—
522D4—22no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—82no—
529D4—161no—
530D4—9761no—
531D4—9761no—
532D4—9761no—
533D4—4882no—
534D4—4882no—
535D4—4882no—
536D4—4882no—
537D4—2441no—
538D4—2441no—
539D4—2441no—
540C2×C4—41no—
541C2×C4—161no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—4882no—
549C2×C4—4882no—
550C2×C4—4882no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4D8×D6111yescharacteristic
555C2×C4D8×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C8—21no—
558C8—1221no—
559C8—1221no—
560C8C2×D4×D6111yescharacteristic
561C8#1451982311yescharacteristic
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—1221no—
572C22—1221no—
573C22—1221no—
574C22—1221no—
575C22—1221no—
576C22—1221no—
577C22—9764no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22C2×D8×D6111yescharacteristic
582C4—21no—
583C4—21no—
584C4—81no—
585C4—1221no—
586C4—1221no—
587C4—4881no—
588C4—4881no—
589C4—2442no—
590C4#1684039611yescharacteristic
591C4#1684056611yescharacteristic
592C2—82no—
593C2—161no—
594C2—161no—
595C2—9761no—
596C2—2441no—
597C2—2441no—
598C2#2145749111yescharacteristic
599C2#2145895111yescharacteristic
600C2#2145895611yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2206C2
socleC22×C61C22×D8
hypercenter#2206D61
hypocenterC61#33834
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2206

Complements

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

NH ≅ G/Ntype
1. C2×#16840196C2semidirect
2. C2×#16840191C2semidirect
3. #21457577C2semidirect
4. #21457505C2semidirect
5. #21448083C2semidirect
6. #21447931C2semidirect
7. #21447904C2semidirect
8. #21447882C2semidirect
9. #21447760C2semidirect
10. #21447743C2semidirect
11. #21447739C2semidirect
12. #21447729C2semidirect
13. #21447683C2semidirect
14. #21447643C2semidirect
15. C61×#2206C2semidirect
18. C2×#14519796C22semidirect
24. C2×Dic488C22semidirect
27. C2×#14519721C22semidirect
28. C2×#14519721C22semidirect
32. C2×C16×D61C22semidirect
37. #16839995C22semidirect
44. C4×Dic244C22semidirect
46. #16839587C22semidirect
47. #16839567C22semidirect
48. #16839565C22semidirect
52. #16839541C22semidirect
53. #16839538C22semidirect
56. C61×#493C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#357C22semidirect
60. C61×#357C22semidirect
102. #14519721D4semidirect
300. #2206D61semidirect
336. #493C2×D61semidirect
351. C4⋊C16C2×D61semidirect
356. #357C2×D61semidirect
357. #357C2×D61semidirect
361. C61#33834semidirect

Maximal subgroups

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

subgroupindex
C2×#168401962
C2×#168401912
#214575772
#214575052
#214480832
#214479312
#214479042
#214478822
#214477602
#214477432
#214477392
#214477292
#214476832
#214476432
C61×#22062
#3383461