Finite Group Databasebeta

← #29670381

Subgroups — #29670381

Order 15616.

632 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458464C211yesmaximal, characteristic, complemented
2#21458055C211yesmaximal, characteristic
3#21457645C211yesmaximal, characteristic, complemented
4#21445477C211yesmaximal, characteristic, complemented
5#21445392C211yesmaximal, characteristic, complemented
6#21445237C211yesmaximal, characteristic, complemented
7#21445228C211yesmaximal, characteristic, complemented
8#21445191C211yesmaximal, characteristic, complemented
9#21444997C211yesmaximal, characteristic, complemented
10#21444964C211yesmaximal, characteristic, complemented
11#21444952C211yesmaximal, characteristic, complemented
12#21444929C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3340C211yesmaximal, characteristic, complemented
16#16840472C2211yescharacteristic
17D4×D244C2211yescharacteristic
18#16840141C2211yescharacteristic, complemented
19#16840116—21no—
20#16840112C2211yescharacteristic, complemented
21#16840110C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840086—21no—
25#16840082—21no—
26#16840069—21no—
27#16840052C2211yescharacteristic
28#16840052C2211yescharacteristic, complemented
29#16840049C2211yescharacteristic
30#16840013—21no—
31#16839994—21no—
32#16839974—21no—
33#16839964C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839911—21no—
37#16839895—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C4×#11758201—21no—
40C4×#11758199C2211yescharacteristic
41#16839840C2211yescharacteristic
42C8×D244C2211yescharacteristic
43#16839766—21no—
44#102×D61—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic, complemented
57C61×#555C2211yescharacteristic
58C61×#498C2211yescharacteristic, complemented
59C61×#485—21no—
60C61×#482—21no—
61C61×#477C2211yescharacteristic, complemented
62C61×#474C2211yescharacteristic
63C61×#465—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic, complemented
66C61×#427—21no—
67C22×D244—21no—
68C2×D4×D61—21no—
69C2×#11758202—41no—
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×#11758199—21no—
73C2×#11758199D411yescharacteristic
74C2×#11758194—21no—
75C2×C8×D61—21no—
76#14519777C2311yescharacteristic
77#14519774C2311yescharacteristic
78#14519769—21no—
79#14519755—41no—
80#14519753—21no—
81#14519752—21no—
82#14519751—21no—
83#14519747—41no—
84C4×D244—21no—
85C4×D244C2311yescharacteristic
86C4⋊C4×D61—21no—
87#14519710D411yescharacteristic
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519701—21no—
97#14519701—21no—
98#14519699—21no—
99#14519699—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698D411yescharacteristic
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—41no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×SD16×C61D411yescharacteristic
116C22×C8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—41no—
120#124×C61—21no—
121#124×C61—41no—
122C22⋊Q8×C61—21no—
123#121×C61—21no—
124#121×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61C2311yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131#117×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—41no—
149C2×D4×C61C2×D411yescharacteristic
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C2×C22⋊D61—41no—
154D4×D61—42no—
155D4×D61—81no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—42no—
160C2×D244—41no—
161C2×D244—81no—
162C2×D244C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166SD16×C61—41no—
167C2×C8×C61—21no—
168C2×C8×C61—42no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—41no—
188C42×C61—41no—
189C42×C61C2411yescharacteristic
190#11758202—82no—
191#11758201—42no—
192#11758201—81no—
193#11758199—22no—
194#11758199—81no—
195#11758198—21no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—41no—
202#11758196—41no—
203#11758194—81no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C8×D61—81no—
211C23×C61—21no—
212C23×C61—21no—
213C22×D61—21no—
214C22×D61—41no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—161no—
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—81no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—161no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C22⋊D61—82no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244D244—21no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—81no—
250D244—82no—
251D244#13512yes—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—161no—
255Dic122—81no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259#92418—21no—
260#92418—21no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#19823—611nomaximal, Sylow
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61D4211yescharacteristic
273C2×D61—22no—
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—164no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61C2×#13511yescharacteristic
288C4×C61#57511yescharacteristic
289C4×C61#57711yescharacteristic
290C4×C61#57711yescharacteristic
291Dic61—41no—
292Dic61—81no—
293#3360—611no—
294#3340D6111yescharacteristic, complemented
295#3325—611no—
296#3295—611no—
297#1720—611no—
298#1703—611no—
299#1678—611no—
300#1677—611no—
301#1671—611no—
302#1639—611no—
303#1634—611no—
304#1632—611no—
305#1628—611no—
306#1583—611no—
307#1580—611no—
308C2×C61—81no—
309C2×C61—81no—
310C2×C61#326711yescharacteristic
311C2×C61#328211yescharacteristic
312C2×C61#328811yescharacteristic
313D61—42no—
314D61—161no—
315#556C2×D6111yescharacteristic, complemented
316#555C2×D6111yescharacteristic
317#554—611no—
318D42—611no—
319#498C2×D6111yescharacteristic, complemented
320#494—611no—
321#488—1221no—
322#487—611no—
323#486—611no—
324#486—611no—
325#486—611no—
326#485—21no—
327#482—21no—
328#482—1221no—
329#481—1221no—
330#479—1221no—
331#477C2×D6111yescharacteristic, complemented
332#476—611no—
333#476—611no—
334#476—611no—
335#474C2×D6111yescharacteristic
336#469—1221no—
337#465—21no—
338#465—1221no—
339#463—1221no—
340#461—611no—
341#460—611no—
342#459—611no—
343#450—1221no—
344#447—1221no—
345C8×Q8—611no—
346C4×SD16—1221no—
347C4×SD16C2×D6111yescharacteristic
348C4×D8—611no—
349#436—611no—
350C8×D4—611no—
351C8×D4C2×D6111yescharacteristic, complemented
352#427—21no—
353#418—1221no—
354C2×#102—1221no—
355#334—611no—
356#334—611no—
357#333—611no—
358#333—611no—
359#332—611no—
360#331—611no—
361Q8⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365#322—611no—
366C61#1982311yescharacteristic, complemented, Sylow
367C22×D4—1221no—
368C22×D4—1221no—
369C2×Q16—2441no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16C22⋊D6111yescharacteristic
373C2×D8—611no—
374C2×D8—1221no—
375C2×M16—1221no—
376C22×C8—21no—
377C22×C8—1221no—
378C4⋊Q8—611no—
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381#127—611no—
382#126—41no—
383#125—1221no—
384#124—21no—
385#124—41no—
386C22⋊Q8—21no—
387C22⋊Q8—1221no—
388C22⋊Q8—2441no—
389#121—21no—
390#121—21no—
391#121—1221no—
392#121—1221no—
393#120—2441no—
394C4×Q8—21no—
395C4×Q8C22×D6111yescharacteristic
396C4×D4—611no—
397C4×D4—1221no—
398C4×D4C22×D6111yescharacteristic
399C4×D4C22×D6111yescharacteristic
400#117—21no—
401#117—21no—
402#117—21no—
403C2×C4⋊C4—1221no—
404#107—21no—
405#107—21no—
406#107—611no—
407#106—21no—
408#106—21no—
409#106—1221no—
410#106—1221no—
411#106—1221no—
412#106C22⋊D6111yescharacteristic
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8C22×D6111yescharacteristic
418#103—21no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103C22⋊D6111yescharacteristic
423#102—21no—
424#102—611no—
425#102—611no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—2441no—
436#102C22⋊D6111yescharacteristic
437C22⋊C8—21no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8C22×D6111yescharacteristic
444C24—2441no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—1221no—
448C2×Q8C2×C22⋊D6111yescharacteristic
449C2×D4—41no—
450C2×D4—611no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—4881no—
455C2×D4—4881no—
456C2×D4—2441no—
457C2×D4—2442no—
458C2×D4—2441no—
459C2×D4—2442no—
460C2×D4D4×D6111yescharacteristic
461C2×D4C2×C22⋊D6111yescharacteristic
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—1221no—
465C22×C4—2441no—
466Q16—4882no—
467SD16—41no—
468SD16—4881no—
469SD16—2442no—
470D8—1222no—
471D8—4881no—
472M16—4881no—
473C2×C8—21no—
474C2×C8—42no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—4881no—
482C2×C8C2×C22⋊D6111yescharacteristic
483C2×C8C2×C22⋊D6111yescharacteristic
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—41no—
490C4⋊C4—611no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—2441no—
494C4⋊C4—2441no—
495C4⋊C4D4×D6111yescharacteristic
496C4⋊C4C2×C22⋊D6111yescharacteristic
497C4⋊C4C2×C22⋊D6111yescharacteristic
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—1221no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C42—21no—
508C42—41no—
509C42—41no—
510C42C23×D6111yescharacteristic
511C23—21no—
512C23—21no—
513C23—1221no—
514C23—9761no—
515C23—4882no—
516C23—4882no—
517C23—2441no—
518Q8—21no—
519Q8—21no—
520Q8—41no—
521Q8—41no—
522Q8—81no—
523Q8—4881no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—21no—
528D4—161no—
529D4—612no—
530D4—1221no—
531D4—4881no—
532D4—4882no—
533D4—4882no—
534D4—2441no—
535D4—2441no—
536D4—2441no—
537D4—2441no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—1221no—
549C2×C4—9761no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C22×C22⋊D6111yescharacteristic
555C2×C4#1451982211yescharacteristic
556C8—21no—
557C8—21no—
558C8—41no—
559C8—1221no—
560C8—1221no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—1222no—
572C22—9764no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—2441no—
577C22—2441no—
578C22D4×C22⋊D6111yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—4881no—
587C4—2441no—
588C4#135×D6111yescharacteristic
589C4#1684056011yescharacteristic
590C4#1684056911yescharacteristic
591C4#1684057011yescharacteristic
592C2—81no—
593C2—81no—
594C2—9761no—
595C2—2442no—
596C2#2145734511yescharacteristic
597C2#2145752011yescharacteristic
598C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3340C2
socleC22×C61D42
prime radical#21458055C2
hypercenter#3340D61
hypocenterC61#19823
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3340

Complements

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

NH ≅ G/Ntype
1. #21458464C2semidirect
3. #21457645C2semidirect
4. #21445477C2semidirect
5. #21445392C2semidirect
6. #21445237C2semidirect
7. #21445228C2semidirect
8. #21445191C2semidirect
9. #21444997C2semidirect
10. #21444964C2semidirect
11. #21444952C2semidirect
12. #21444929C2semidirect
13. #21444663C2semidirect
14. #21444641C2semidirect
15. C61×#3340C2semidirect
18. #16840141C22semidirect
20. #16840112C22semidirect
28. #16840052C22semidirect
38. Q8×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839510C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839484C22semidirect
56. C61×#556C22semidirect
58. C61×#498C22semidirect
61. C61×#477C22semidirect
65. C8×D4×C61C22semidirect
294. #3340D61semidirect
315. #556C2×D61semidirect
319. #498C2×D61semidirect
331. #477C2×D61semidirect
351. C8×D4C2×D61semidirect
366. C61#19823semidirect

Maximal subgroups

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

subgroupindex
#214584642
#214580552
#214576452
#214454772
#214453922
#214452372
#214452282
#214451912
#214449972
#214449642
#214449522
#214449292
#214446632
#214446412
C61×#33402
#1982361