Finite Group Databasebeta

← #29671365

Subgroups — #29671365

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458717C211yesmaximal, characteristic, complemented
2#21457678C211yesmaximal, characteristic, complemented
3#21457640C211yesmaximal, characteristic, complemented
4#21457419C211yesmaximal, characteristic
5#21445422C211yesmaximal, characteristic, complemented
6#21445320C211yesmaximal, characteristic, complemented
7#21445294C211yesmaximal, characteristic, complemented
8#21445245C211yesmaximal, characteristic, complemented
9#21445230C211yesmaximal, characteristic, complemented
10#21445023C211yesmaximal, characteristic, complemented
11#21445000C211yesmaximal, characteristic, complemented
12#21444925C211yesmaximal, characteristic, complemented
13#21444712C211yesmaximal, characteristic, complemented
14#21444611C211yesmaximal, characteristic, complemented
15C61×#1635C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17#16840473C2211yescharacteristic
18D4×Dic122C2211yescharacteristic, complemented
19D4×D244C2211yescharacteristic
20#16840169—21no—
21#16840158C2211yescharacteristic, complemented
22#16840136C2211yescharacteristic
23#16840114C2211yescharacteristic, complemented
24#16840110C2211yescharacteristic
25#16840109C2211yescharacteristic
26#16840092—21no—
27#16840082—21no—
28#16840060—21no—
29#16840051—21no—
30#16840044C2211yescharacteristic, complemented
31#16840043C2211yescharacteristic
32#16840010—21no—
33#16839992—21no—
34#16839989—21no—
35#16839967C2211yescharacteristic
36#16839964C2211yescharacteristic
37#16839963C2211yescharacteristic
38#16839957C2211yescharacteristic
39#16839914—21no—
40#16839912—21no—
41#16839907—21no—
42#16839900—21no—
43#16839889C2211yescharacteristic, complemented
44#16839871—21no—
45C4×#11758200C2211yescharacteristic
46C4×#11758199C2211yescharacteristic
47#16839840C2211yescharacteristic
48D4×#92418C2211yescharacteristic, complemented
49#16839827C2211yescharacteristic
50C2×#14519700—21no—
51#102×D61—21no—
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839505C2211yescharacteristic
57#16839494C2211yescharacteristic, complemented
58#16839491C2211yescharacteristic
59C8×#92418C2211yescharacteristic, complemented
60C61×#547C2211yescharacteristic
61C61×#487C2211yescharacteristic, complemented
62C61×#476C2211yescharacteristic
63C61×#476C2211yescharacteristic, complemented
64C61×#332C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×D4⋊C8C2211yescharacteristic, complemented
67C22×Dic122—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519802—44no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758200D411yescharacteristic
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758199—21no—
78C2×#11758199D411yescharacteristic
79C2×#11758194—21no—
80C22×#92418—21no—
81C2×C8×D61—21no—
82#14519777C2311yescharacteristic
83#14519776—21no—
84#14519774C2311yescharacteristic
85#14519768—21no—
86#14519757—21no—
87#14519756—21no—
88#14519754—41no—
89#14519753—41no—
90#14519752—21no—
91#14519751—21no—
92#14519747—41no—
93C4×Dic122C2311yescharacteristic
94C4×D244C2311yescharacteristic
95D4×Dic61—21no—
96#14519739—21no—
97C4⋊C4×D61—21no—
98#14519711—21no—
99#14519710D411yescharacteristic
100#14519708—21no—
101#14519708—21no—
102#14519708—21no—
103#14519708D411yescharacteristic
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106#14519706—21no—
107#14519701—21no—
108#14519700—21no—
109#14519700—21no—
110#14519700—21no—
111#14519700—21no—
112#14519700D411yescharacteristic
113#14519699—21no—
114#14519699—21no—
115#14519699—41no—
116#14519698D411yescharacteristic
117#14519698D411yescharacteristic
118#14519697—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697D411yescharacteristic
122C22⋊#92418—21no—
123C22⋊#92418—21no—
124#14519689—21no—
125#14519687—21no—
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C2×#41×C61—21no—
129C4⋊Q8×C61C2311yescharacteristic
130C4⋊Q8×C61C2311yescharacteristic
131#123×C61—41no—
132C22⋊Q8×C61—21no—
133#121×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61—21no—
138#106×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—41no—
142#102×C61—21no—
143#102×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×D4×C61—41no—
151C2×D4×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C22×Dic61—41no—
158D4×D61—42no—
159D4×D61—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—
170C2×Dic122—42no—
171C2×Dic122—41no—
172C2×Dic122—81no—
173C2×Dic122C2×D411yescharacteristic
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758201—42no—
190#11758201—42no—
191#11758201—81no—
192#11758200—41no—
193#11758200—81no—
194#11758199—42no—
195#11758199SD1614yescomplemented
196#11758198—21no—
197#11758198—41no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—21no—
205#11758196—41no—
206#11758196—41no—
207C4×Dic61—21no—
208#11758194—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—81no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216#11758190—81no—
217C8×D61—81no—
218C23×C61—21no—
219C23×C61—21no—
220C22×D61—21no—
221C22×D61—41no—
222C22×D61—82no—
223C22×D61—82no—
224C22×D61—161no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—81no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—82no—
232D4×C61C2×SD1612yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—82no—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C22⋊D61—82no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—82no—
251D244—21no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—81no—
257D244—82no—
258D244C2×SD1612yes—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—161no—
262Dic122—21no—
263Dic122—21no—
264Dic122—41no—
265Dic122—41no—
266Dic122—82no—
267C8×C61—41no—
268C8×C61—41no—
269C8×C61—41no—
270#92418—21no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418C2×SD1612yes—
276#19891—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61D4211yescharacteristic
284C2×D61—22no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—164no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C22×SD1611yescharacteristic
297C4×C61#57511yescharacteristic
298C4×C61#57611yescharacteristic
299C4×C61#57711yescharacteristic
300Dic61—41no—
301Dic61—41no—
302Dic61—41no—
303Dic61—81no—
304#3384—611no—
305#3298—611no—
306#3295—611no—
307#3272—611no—
308#1708—611no—
309#1691—611no—
310#1686—611no—
311#1679—611no—
312#1677—611no—
313#1643—611no—
314#1639—611no—
315#1635D6111yescharacteristic, complemented
316#1627—611no—
317#1590—611no—
318C8×D8—611no—
319C2×C61—42no—
320C2×C61—81no—
321C2×C61#327311yescharacteristic
322C2×C61D4×SD1611yescharacteristic
323C2×C61#329011yescharacteristic
324D61—42no—
325D61—161no—
326C2×AGL(1,ℤ/8)—1221no—
327#554—611no—
328D4×Q8—611no—
329#547C2×D6111yescharacteristic
330D42—611no—
331#501—1221no—
332#498—611no—
333#492—611no—
334#487—611no—
335#487C2×D6111yescharacteristic, complemented
336#486—611no—
337#486—611no—
338#483—1221no—
339#481—1221no—
340#478—1221no—
341#476—1221no—
342#476C2×D6111yescharacteristic
343#476C2×D6111yescharacteristic, complemented
344#475—611no—
345#474—611no—
346#468—1221no—
347#465—1221no—
348#465—1221no—
349#461—611no—
350#461—611no—
351#460—611no—
352#459—611no—
353#450—1221no—
354#450—1221no—
355#449—1221no—
356#448—1221no—
357#446—611no—
358#442—1221no—
359C4×SD16—611no—
360C4×D8—611no—
361#436—611no—
362C8×D4—611no—
363C8×D4—611no—
364C2×#103—1221no—
365C2×#102—1221no—
366#335—611no—
367#334—611no—
368#333—611no—
369#332—611no—
370#332C2×D6111yescharacteristic, complemented
371#331—611no—
372Q8⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8C2×D6111yescharacteristic
375D4⋊C8C2×D6111yescharacteristic, complemented
376C82—611no—
377C61#1989111yescharacteristic, complemented, Sylow
378C2×#41—21no—
379C22×Q8—1221no—
380C22×D4—1221no—
381C22×D4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×SD16—611no—
384C2×SD16—1221no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×D8—611no—
390C2×D8—1221no—
391C2×M16—1221no—
392C22×C8—1221no—
393C22×C8—1221no—
394C4⋊Q8—611no—
395C4⋊Q8—1221no—
396C4⋊Q8C22×D6111yescharacteristic
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#125—1221no—
400#123—41no—
401C22⋊Q8—21no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404C22⋊Q8—2441no—
405C22⋊Q8—2441no—
406#121—21no—
407#121—1221no—
408#121—1221no—
409#120—2441no—
410C4×Q8—611no—
411C4×D4—611no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4C22×D6111yescharacteristic
415C4×D4C22×D6111yescharacteristic
416C2×C4⋊C4—21no—
417C2×C4⋊C4—1221no—
418#107—611no—
419#107—1221no—
420#106—21no—
421#106—21no—
422#106—611no—
423#106—1221no—
424#106—1221no—
425#106—1221no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—1221no—
429C4⋊C8C22×D6111yescharacteristic
430C4⋊C8C22×D6111yescharacteristic
431#103—41no—
432#103—611no—
433#103—1221no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#102—21no—
439#102—21no—
440#102—611no—
441#102—611no—
442#102—611no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—2441no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452#97—1221no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8C22×D6111yescharacteristic
456C24—2441no—
457#41—42no—
458#41—81no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—611no—
462C2×Q8—1221no—
463C2×Q8—4881no—
464C2×Q8—2441no—
465C2×Q8—2442no—
466C2×D4—41no—
467C2×D4—611no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—4881no—
472C2×D4—4881no—
473C2×D4—2441no—
474C2×D4—2442no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4D4×D6111yescharacteristic
478C2×D4D4×D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—1221no—
483C22×C4—2441no—
484C22×C4—2441no—
485SD16—4881no—
486SD16—4881no—
487SD16—4881no—
488SD16—2442no—
489SD16—2441no—
490SD16—2442no—
491D8—614no—
492D8—2442no—
493M16—2442no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—4881no—
504C2×C8—4881no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—611no—
511C4⋊C4—611no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4—2441no—
518C4⋊C4D4×D6111yescharacteristic
519C4⋊C4D4×D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—1221no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C42—1221no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—21no—
531C23—1221no—
532C23—9761no—
533C23—4882no—
534C23—4882no—
535C23—2441no—
536Q8—41no—
537Q8—41no—
538Q8—81no—
539Q8—1221no—
540Q8—1221no—
541Q8—4882no—
542Q8—2441no—
543Q8—2441no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—82no—
548D4—612no—
549D4—1221no—
550D4—4881no—
551D4—4882no—
552D4—4882no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4SD16×D6112yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—82no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—9761no—
571C2×C4—4882no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4#1451982211yescharacteristic
578C8—41no—
579C8—41no—
580C8—41no—
581C8—612no—
582C8—1221no—
583C8—1221no—
584C8—1221no—
585C8—2441no—
586C8—2441no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—1222no—
594C22—9764no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—2441no—
599C22—2441no—
600C22#1684039411yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—4881no—
607C4—2441no—
608C4—2441no—
609C4—2441no—
610C4C2×SD16×D6111yescharacteristic
611C4#1684056111yescharacteristic
612C4#1684056411yescharacteristic
613C4#1684057211yescharacteristic
614C2—42no—
615C2—81no—
616C2—9761no—
617C2—2442no—
618C2#2145742311yescharacteristic
619C2#2145751311yescharacteristic
620C2#2145759811yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1635C2
socleC22×C61D42
prime radical#21457419C2
hypercenter#1635D61
hypocenterC61#19891
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1635

Complements

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

NH ≅ G/Ntype
1. #21458717C2semidirect
2. #21457678C2semidirect
3. #21457640C2semidirect
5. #21445422C2semidirect
6. #21445320C2semidirect
7. #21445294C2semidirect
8. #21445245C2semidirect
9. #21445230C2semidirect
10. #21445023C2semidirect
11. #21445000C2semidirect
12. #21444925C2semidirect
13. #21444712C2semidirect
14. #21444611C2semidirect
15. C61×#1635C2semidirect
18. D4×Dic122C22semidirect
21. #16840158C22semidirect
23. #16840114C22semidirect
30. #16840044C22semidirect
43. #16839889C22semidirect
48. D4×#92418C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
57. #16839494C22semidirect
59. C8×#92418C22semidirect
61. C61×#487C22semidirect
63. C61×#476C22semidirect
64. C61×#332C22semidirect
66. C61×D4⋊C8C22semidirect
195. #11758199SD16semidirect
315. #1635D61semidirect
335. #487C2×D61semidirect
343. #476C2×D61semidirect
370. #332C2×D61semidirect
375. D4⋊C8C2×D61semidirect
377. C61#19891semidirect

Maximal subgroups

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

subgroupindex
#214587172
#214576782
#214576402
#214574192
#214454222
#214453202
#214452942
#214452452
#214452302
#214450232
#214450002
#214449252
#214447122
#214446112
C61×#16352
#1989161