Finite Group Databasebeta

← #29836529

Subgroups — #29836529

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×#16840198C211yesmaximal, characteristic, complemented
2C2×#16840191C211yesmaximal, characteristic, complemented
3#21457576C211yesmaximal, characteristic, complemented
4#21457504C211yesmaximal, characteristic, complemented
5#21448082C211yesmaximal, characteristic, complemented
6#21447930C211yesmaximal, characteristic, complemented
7#21447904C211yesmaximal, characteristic, complemented
8#21447880C211yesmaximal, characteristic, complemented
9#21447852C211yesmaximal, characteristic, complemented
10#21447761C211yesmaximal, characteristic, complemented
11#21447743C211yesmaximal, characteristic, complemented
12#21447739C211yesmaximal, characteristic, complemented
13#21447680C211yesmaximal, characteristic, complemented
14#21447647C211yesmaximal, characteristic, complemented
15C61×#2184C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519796C2211yescharacteristic
18C2×#14519796C2211yescharacteristic, complemented
19#16840430—21no—
20#16840406—21no—
21#16840198—24no—
22#16840191—24no—
23C2×Dic488C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic, complemented
25C2×#14519721C2211yescharacteristic
26C2×#14519721C2211yescharacteristic
27C2×#14519721C2211yescharacteristic, complemented
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C2×#14519717C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic, complemented
32#16840145C2211yescharacteristic
33#16840137C2211yescharacteristic, complemented
34#16840003C2211yescharacteristic
35#16839999C2211yescharacteristic, complemented
36#16839995C2211yescharacteristic
37#16839975—21no—
38#16839961—21no—
39#16839913—21no—
40#16839908—21no—
41#16839905—21no—
42#16839904—21no—
43#16839856C2211yescharacteristic, complemented
44D8×Dic61C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839565C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic, complemented
51#16839563C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839537C2211yescharacteristic, complemented
54C22×D8×C61C2211yescharacteristic
55C2×D16×C61C2211yescharacteristic
56C2×D16×C61C2211yescharacteristic
57C61×#466C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C61×#357C2211yescharacteristic
60C61×C22⋊C16C2211yescharacteristic
61C2×#11758222—21no—
62C2×#11758224—21no—
63C2×#11758224—21no—
64#14519799—22no—
65#14519799—41no—
66#14519796—22no—
67#14519796—22no—
68#14519796—41no—
69#14519796—41no—
70C2×Dic244C2311yescharacteristic
71C2×Dic244C2311yescharacteristic
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78C2×C8×D61C2311yescharacteristic
79#14519773—21no—
80#14519773—41no—
81#14519764—21no—
82#14519758—41no—
83#14519756—21no—
84#14519752—21no—
85#14519750—41no—
86#14519749—41no—
87#14519749—41no—
88#14519748—21no—
89#14519746—41no—
90#14519742—21no—
91D4×Dic61—21no—
92C4×C22⋊D61D411yescharacteristic
93#14519737—41no—
94C2×C22⋊Dic61—41no—
95Dic488—22no—
96#14519722—41no—
97#14519721—22no—
98#14519721—22no—
99#14519721D414yescomplemented
100#14519718—22no—
101#14519718—22no—
102#14519717—22no—
103C2×#11758185C2311yescharacteristic
104C2×#11758185C2311yescharacteristic
105C16×D61—22no—
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519702—21no—
109#14519700—21no—
110#14519698—21no—
111#14519698—21no—
112#14519697—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C22×D4×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
124#121×C61—21no—
125D16×C61—22no—
126D16×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#102×C61—21no—
131C24×C61—41no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—42no—
136C2×D4×C61—41no—
137C2×D4×C61—41no—
138C2×D4×C61—81no—
139C22×C4×C61C2×D411yescharacteristic
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61D811yescharacteristic
147C22×Dic61—41no—
148C22×Dic61—41no—
149C22×Dic61—41no—
150#11758224—42no—
151#11758224—42no—
152#11758224—81no—
153#11758224—81no—
154#11758222—42no—
155#11758222—81no—
156C2×D244—21no—
157C2×D244—41no—
158C2×C4×D61—41no—
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—42no—
168D8×C61C2×D412yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2411yescharacteristic
173C4⋊C4×C61—21no—
174C22⋊C4×C61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C22⋊Dic61—41no—
180C22⋊Dic61D811yescharacteristic
181#11758200—42no—
182#11758200—42no—
183#11758200—81no—
184#11758199—81no—
185#11758198—41no—
186#11758198D811yescharacteristic
187C4⋊Dic61—21no—
188#11758196—41no—
189#11758196—41no—
190#11758196D811yescharacteristic
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195Dic244—21no—
196Dic244—21no—
197Dic244—21no—
198Dic244C2×D412yes—
199D488—21no—
200D488—21no—
201#11758190—42no—
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—
211C23×C61—82no—
212C23×C61—82no—
213C23×C61—161no—
214C23×C61C2×D811yescharacteristic
215C22×D61—41no—
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—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61C2×D811yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C22⋊D61—41no—
231C22⋊D61—82no—
232C22⋊D61—82no—
233C22⋊D61—82no—
234C22⋊D61—82no—
235C22⋊D61—161no—
236C22⋊D61—161no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241C2×Dic61—82no—
242C2×Dic61—161no—
243C2×Dic61C2×D811yescharacteristic
244C2×Dic61C2×D811yescharacteristic
245C2×Dic61C2×D811yescharacteristic
246D244—41no—
247D244—41no—
248D244—161no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—82no—
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—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—81no—
284C4×C61D4211yescharacteristic
285C4×C61#57611yescharacteristic
286Dic61—21no—
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#2206—611no—
301#2189—611no—
302#2186—611no—
303#2186—611no—
304#2184D6111yescharacteristic, complemented
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×#135—611no—
319C22×D8C2×D6111yescharacteristic
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#493—611no—
337#467—611no—
338#466—611no—
339#466—611no—
340#466C2×D6111yescharacteristic
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#357C2×D6111yescharacteristic
357#357C2×D6111yescharacteristic
358C22⋊C16—611no—
359C22⋊C16C2×D6111yescharacteristic
360#346—611no—
361C61#3383411yescharacteristic, complemented, Sylow
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C22×D4—21no—
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—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—21no—
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—614no—
407SD32—1222no—
408SD32—1222no—
409SD32—2441no—
410D16—22no—
411D16—41no—
412D16—1222no—
413D16—1222no—
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×C8—611no—
432C24—41no—
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—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—611no—
458C22×C4—2441no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4D4×D6111yescharacteristic
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—42no—
477D8—1221no—
478D8—1221no—
479D8—4881no—
480D8D4×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—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—611no—
504C42—2441no—
505C16—21no—
506C16—21no—
507C16—612no—
508C16—1221no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—82no—
513C23—82no—
514C23—161no—
515C23—611no—
516C23—2441no—
517C23D8×D6111yescharacteristic
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—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—9761no—
549C2×C4—4882no—
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—
560C8C2×D4×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—81no—
584C4—1221no—
585C4—1221no—
586C4—1221no—
587C4—4881no—
588C4—4881no—
589C4—2442no—
590C4#1684039411yescharacteristic
591C4#1684056611yescharacteristic
592C2—41no—
593C2—82no—
594C2—161no—
595C2—161no—
596C2—9761no—
597C2—2441no—
598C2#2145748911yescharacteristic
599C2#2145895111yescharacteristic
600C2#2145895611yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2184C2
socleC22×C61C22×D8
hypercenter#2184D61
hypocenterC61#33834
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

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

NH ≅ G/Ntype
1. C2×#16840198C2semidirect
2. C2×#16840191C2semidirect
3. #21457576C2semidirect
4. #21457504C2semidirect
5. #21448082C2semidirect
6. #21447930C2semidirect
7. #21447904C2semidirect
8. #21447880C2semidirect
9. #21447852C2semidirect
10. #21447761C2semidirect
11. #21447743C2semidirect
12. #21447739C2semidirect
13. #21447680C2semidirect
14. #21447647C2semidirect
15. C61×#2184C2semidirect
18. C2×#14519796C22semidirect
23. C2×Dic488C22semidirect
24. C2×#14519722C22semidirect
27. C2×#14519721C22semidirect
31. C2×C16×D61C22semidirect
33. #16840137C22semidirect
35. #16839999C22semidirect
43. #16839856C22semidirect
46. #16839587C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839565C22semidirect
50. #16839564C22semidirect
51. #16839563C22semidirect
52. #16839541C22semidirect
53. #16839537C22semidirect
99. #14519721D4semidirect
304. #2184D61semidirect
361. C61#33834semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
C2×#168401912
#214575762
#214575042
#214480822
#214479302
#214479042
#214478802
#214478522
#214477612
#214477432
#214477392
#214476802
#214476472
C61×#21842
#3383461