Finite Group Databasebeta

← #29669646

Subgroups — #29669646

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458129C211yesmaximal, characteristic, complemented
2#21457637C211yesmaximal, characteristic, complemented
3#21457634C211yesmaximal, characteristic
4#21445475C211yesmaximal, characteristic, complemented
5#21445354C211yesmaximal, characteristic, complemented
6#21445272C211yesmaximal, characteristic, complemented
7#21445263C211yesmaximal, characteristic, complemented
8#21445215C211yesmaximal, characteristic, complemented
9#21445033C211yesmaximal, characteristic, complemented
10#21444969C211yesmaximal, characteristic, complemented
11#21444934C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15C61×#3380C211yesmaximal, characteristic, complemented
16#16840417C2211yescharacteristic
17D4×D244C2211yescharacteristic
18#16840136C2211yescharacteristic
19#16840114C2211yescharacteristic, complemented
20#16840112C2211yescharacteristic, complemented
21#16840110C2211yescharacteristic
22#16840109C2211yescharacteristic
23#16840092—21no—
24#16840080—21no—
25#16840077—21no—
26#16840049C2211yescharacteristic
27#16840049C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840018—21no—
30#16839997—21no—
31#16839993—21no—
32#16839972C2211yescharacteristic, complemented
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839922—21no—
36#16839912—21no—
37#16839894—21no—
38C4×#11758201C2211yescharacteristic
39#16839840C2211yescharacteristic, complemented
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42#16839761—21no—
43C2×#14519697—21no—
44#102×D61—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839511C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic
57C61×#554C2211yescharacteristic, complemented
58C61×#500C2211yescharacteristic, complemented
59C61×#499—21no—
60C61×#478—21no—
61C61×#478—21no—
62C61×#476C2211yescharacteristic
63C61×#476C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×#11758202—21no—
71C2×#11758201—21no—
72C2×#11758201—41no—
73C2×#11758201D411yescharacteristic
74C2×#11758199—21no—
75C2×#11758194—21no—
76C22×#92418—21no—
77C2×C8×D61—21no—
78#14519777C2311yescharacteristic
79#14519774C2311yescharacteristic
80#14519759—41no—
81#14519756—21no—
82#14519752—21no—
83#14519751—21no—
84#14519748—21no—
85#14519747—41no—
86C4×D244C2311yescharacteristic
87C4⋊C4×D61—21no—
88#14519710—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708D411yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—41no—
96#14519701D411yescharacteristic
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519699—21no—
101#14519699—21no—
102#14519698—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×D8×C61D411yescharacteristic
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#125×C61—21no—
119#125×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61—41no—
123C22⋊Q8×C61—41no—
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61C2311yescharacteristic
128C4×D4×C61C2311yescharacteristic
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131#107×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—41no—
153C22×C4×C61—41no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—41no—
156D4×D61—42no—
157D4×D61—81no—
158#11758222—42no—
159#11758222—81no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—41no—
165C2×D244—81no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170D8×C61—22no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C42×C61—21no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193#11758202—81no—
194#11758201—22no—
195#11758201—81no—
196#11758201—82no—
197#11758199—81no—
198#11758198—21no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—41no—
204#11758194—81no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—81no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C8×D61—81no—
213C23×C61—21no—
214C23×C61—21no—
215C22×D61—21no—
216C22×D61—41no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—161no—
220Q8×C61—21no—
221Q8×C61—21no—
222Q8×C61—81no—
223Q8×C61—81no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61#13712yes—
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×C61—161no—
239C2×C4×C61—161no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C22⋊D61—82no—
244C22⋊D61—82no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247D244—21no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—81no—
253D244—82no—
254D244#13512yes—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—82no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—41no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#19774—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61D4211yescharacteristic
277C2×D61—22no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—164no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C2×#13511yescharacteristic
292C4×C61C2×#13711yescharacteristic
293C4×C61#57611yescharacteristic
294C4×C61#57711yescharacteristic
295Dic61—41no—
296Dic61—81no—
297#3380D6111yescharacteristic, complemented
298#3331—611no—
299#3295—611no—
300#3294—611no—
301#1719—611no—
302#1697—611no—
303#1683—611no—
304#1682—611no—
305#1675—611no—
306#1644—611no—
307#1635—611no—
308#1629—611no—
309#1622—611no—
310#1581—611no—
311SD16⋊C8—611no—
312C2×C61—42no—
313C2×C61—81no—
314C2×C61#326611yescharacteristic
315C2×C61#328011yescharacteristic
316C2×C61#328811yescharacteristic
317D61—42no—
318D61—161no—
319#554C2×D6111yescharacteristic
320#554C2×D6111yescharacteristic, complemented
321#547—611no—
322D42—611no—
323#500C2×D6111yescharacteristic, complemented
324#499—21no—
325#492—611no—
326#487—611no—
327#487—611no—
328#486—611no—
329#486—611no—
330#483—1221no—
331#481—1221no—
332#480—1221no—
333#478—21no—
334#478—21no—
335#476—611no—
336#476—611no—
337#476C2×D6111yescharacteristic
338#476C2×D6111yescharacteristic, complemented
339#474—611no—
340#470—1221no—
341#466—1221no—
342#465—1221no—
343#462—611no—
344#460—611no—
345#459—611no—
346#452—1221no—
347#450—1221no—
348#447—1221no—
349C8×Q8C2×D6111yescharacteristic, complemented
350C4×SD16—611no—
351C4×D8—21no—
352C4×D8C2×D6111yescharacteristic
353#436—611no—
354C8×D4—611no—
355C8×D4—611no—
356#417—1221no—
357C2×#102—1221no—
358C2×#102—1221no—
359#335—611no—
360#335—611no—
361#333—611no—
362#333—611no—
363#332—611no—
364#331—611no—
365Q8⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369#322—611no—
370C61#1977411yescharacteristic, complemented, Sylow
371C2×#41—1221no—
372C22×D4—1221no—
373C22×D4—1221no—
374C2×Q16—1221no—
375C2×SD16—611no—
376C2×SD16—1221no—
377C2×SD16—2441no—
378C2×D8—1221no—
379C2×D8C22⋊D6111yescharacteristic
380C2×M16—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383C4⋊Q8—611no—
384C4⋊Q8C22×D6111yescharacteristic
385C4⋊Q8C22×D6111yescharacteristic
386#127—611no—
387#125—21no—
388#125—21no—
389#123—2441no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—41no—
393C22⋊Q8—41no—
394C22⋊Q8—1221no—
395#121—1221no—
396#121—1221no—
397#121—1221no—
398#120—2441no—
399C4×Q8C22×D6111yescharacteristic
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—611no—
403C4×D4C22×D6111yescharacteristic
404C4×D4C22×D6111yescharacteristic
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—1221no—
408#107—21no—
409#107—21no—
410#107—1221no—
411#107C22⋊D6111yescharacteristic
412#106—21no—
413#106—21no—
414#106—611no—
415#106—1221no—
416#106—1221no—
417C4⋊C8—21no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8C22×D6111yescharacteristic
422#103—611no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—2441no—
427#102—21no—
428#102—21no—
429#102—611no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102C22⋊D6111yescharacteristic
440#102C22⋊D6111yescharacteristic
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C4×C8—21no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8C22×D6111yescharacteristic
448C24—2441no—
449#41—4881no—
450#41—2442no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—1221no—
454C2×Q8D4×D6111yescharacteristic
455C2×D4—611no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—4881no—
460C2×D4—4881no—
461C2×D4—2441no—
462C2×D4—2442no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2442no—
466C2×D4C2×C22⋊D6111yescharacteristic
467C2×D4C2×C22⋊D6111yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—1221no—
473C22×C4—2441no—
474Q16—4881no—
475SD16—1222no—
476SD16—4882no—
477SD16—4881no—
478D8—22no—
479D8—4881no—
480M16—4881no—
481C2×C8—21no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—4881no—
489C2×C8—4881no—
490C2×C8C2×C22⋊D6111yescharacteristic
491C2×C8C2×C22⋊D6111yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—611no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—2441no—
505C4⋊C4D4×D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C4⋊C4C2×C22⋊D6111yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—1221no—
513C22⋊C4—2441no—
514C22⋊C4—2441no—
515C42—21no—
516C42C23×D6111yescharacteristic
517C23—21no—
518C23—21no—
519C23—1221no—
520C23—9761no—
521C23—4882no—
522C23—4882no—
523C23—2441no—
524Q8—21no—
525Q8—21no—
526Q8—81no—
527Q8—81no—
528Q8—2441no—
529Q8—2441no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—612no—
534D4—1221no—
535D4—4881no—
536D4—4882no—
537D4—4882no—
538D4—4882no—
539D4—2441no—
540D4—2441no—
541D4—2441no—
542D4—2441no—
543D4#1451980812yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—161no—
555C2×C4—161no—
556C2×C4—1221no—
557C2×C4—4882no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4#1451982211yescharacteristic
564C8—21no—
565C8—21no—
566C8—41no—
567C8—1221no—
568C8—1221no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C8—2441no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—1222no—
580C22—9764no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—2441no—
585C22—2441no—
586C22D4×C22⋊D6111yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—81no—
594C4—4881no—
595C4—2441no—
596C4#135×D6111yescharacteristic
597C4C2×#1451980811yescharacteristic
598C4#1684056411yescharacteristic
599C4#1684057011yescharacteristic
600C2—42no—
601C2—81no—
602C2—9761no—
603C2—2442no—
604C2#2145733011yescharacteristic
605C2#2145749811yescharacteristic
606C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3380C2
socleC22×C61D42
prime radical#21457634C2
hypercenter#3380D61
hypocenterC61#19774
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #21458129C2semidirect
2. #21457637C2semidirect
4. #21445475C2semidirect
5. #21445354C2semidirect
6. #21445272C2semidirect
7. #21445263C2semidirect
8. #21445215C2semidirect
9. #21445033C2semidirect
10. #21444969C2semidirect
11. #21444934C2semidirect
12. #21444891C2semidirect
13. #21444649C2semidirect
14. #21444629C2semidirect
15. C61×#3380C2semidirect
19. #16840114C22semidirect
20. #16840112C22semidirect
32. #16839972C22semidirect
39. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839511C22semidirect
48. #16839511C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839484C22semidirect
57. C61×#554C22semidirect
58. C61×#500C22semidirect
63. C61×#476C22semidirect
64. C8×Q8×C61C22semidirect
297. #3380D61semidirect
320. #554C2×D61semidirect
323. #500C2×D61semidirect
338. #476C2×D61semidirect
349. C8×Q8C2×D61semidirect
370. C61#19774semidirect

Maximal subgroups

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

subgroupindex
#214581292
#214576372
#214576342
#214454752
#214453542
#214452722
#214452632
#214452152
#214450332
#214449692
#214449342
#214448912
#214446492
#214446292
C61×#33802
#1977461