Finite Group Databasebeta

← #29671366

Subgroups — #29671366

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#21457684C211yesmaximal, characteristic, complemented
2#21457637C211yesmaximal, characteristic, complemented
3#21457420C211yesmaximal, characteristic
4#21445421C211yesmaximal, characteristic, complemented
5#21445315C211yesmaximal, characteristic, complemented
6#21445291C211yesmaximal, characteristic, complemented
7#21445240C211yesmaximal, characteristic, complemented
8#21445230C211yesmaximal, characteristic, complemented
9#21445026C211yesmaximal, characteristic, complemented
10#21444997C211yesmaximal, characteristic, complemented
11#21444968C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444709C211yesmaximal, characteristic, complemented
14D8×#92418C211yesmaximal, characteristic, complemented
15C61×#3384C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17#16840416C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840135C2211yescharacteristic
20#16840112C2211yescharacteristic, complemented
21#16840112C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840090—21no—
25#16840080—21no—
26#16840049C2211yescharacteristic
27#16840047C2211yescharacteristic, complemented
28#16840009—21no—
29#16839993—21no—
30#16839993—21no—
31#16839967C2211yescharacteristic
32#16839964C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839914—21no—
36#16839912—21no—
37#16839908—21no—
38#16839901—21no—
39C4×#11758201C2211yescharacteristic
40#16839841C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43C2×#14519701—21no—
44C2×#14519697—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic, complemented
58C61×#501—21no—
59C61×#498C2211yescharacteristic, complemented
60C61×#478—21no—
61C61×#476—21no—
62C61×#476C2211yescharacteristic, complemented
63C61×#474C2211yescharacteristic
64C61×#446C2211yescharacteristic, complemented
65C61×#442—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758224—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519801—44no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201D411yescharacteristic
77C2×#11758199—21no—
78C2×#11758189—21no—
79C22×#92418—21no—
80C22×#92418—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519761—41no—
84#14519757—21no—
85#14519752—21no—
86#14519751—21no—
87#14519751—21no—
88#14519747—41no—
89C4×D244C2311yescharacteristic
90C2×C4⋊Dic61—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708D411yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—41no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×D8×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#125×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—41no—
157C22×C4×C61—41no—
158C2×C22⋊D61—41no—
159C2×C22⋊D61—41no—
160C22×Dic61—41no—
161#11758224—42no—
162#11758224—81no—
163D4×D61—42no—
164D4×D61—81no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—42no—
169C2×D244—41no—
170C2×D244—81no—
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×Dic122—21no—
174D8×C61SD1614yescomplemented
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61C2411yescharacteristic
195#11758201—41no—
196#11758201—81no—
197#11758201—81no—
198#11758200—42no—
199#11758199—42no—
200#11758198—21no—
201#11758198—41no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—41no—
206C4⋊Dic61C2×D411yescharacteristic
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—81no—
212C2×#92418—81no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215#11758190—42no—
216#11758190—81no—
217#11758189—42no—
218C23×C61—21no—
219C23×C61—21no—
220C22×D61—21no—
221C22×D61—41no—
222C22×D61—82no—
223C22×D61—82no—
224C22×D61—161no—
225Q8×C61—21no—
226Q8×C61—21no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—81no—
230Q8×C61—82no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61C2×SD1612yes—
234D4×C61C2×SD1612yes—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—82no—
246C2×C4×C61—161no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C22⋊D61—82no—
251C22⋊D61—82no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—82no—
255D244—21no—
256D244—21no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—81no—
262D244—82no—
263C4×D61—41no—
264C4×D61—41no—
265Dic122—41no—
266Dic122—41no—
267C8×C61—21no—
268C8×C61—41no—
269C8×C61C2×SD1612yes—
270#92418—21no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#19891—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61D4211yescharacteristic
284C2×D61—41no—
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—41no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61—81no—
298C4×C61C22×SD1611yescharacteristic
299C4×C61#57511yescharacteristic
300C4×C61#57611yescharacteristic
301C4×C61#57711yescharacteristic
302Dic61—41no—
303Dic61—81no—
304#3384D6111yescharacteristic, complemented
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#1635—611no—
316#1627—611no—
317#1590—611no—
318C8×D8—611no—
319C2×C61—42no—
320C2×C61—42no—
321C2×C61#327311yescharacteristic
322C2×C61D4×SD1611yescharacteristic
323C2×C61#329011yescharacteristic
324D61—81no—
325D61—161no—
326C2×AGL(1,ℤ/8)—1221no—
327#554C2×D6111yescharacteristic
328D4×Q8C2×D6111yescharacteristic, complemented
329#547—611no—
330D42—611no—
331#501—21no—
332#498C2×D6111yescharacteristic, complemented
333#492—611no—
334#487—611no—
335#487—611no—
336#486—611no—
337#486—611no—
338#483—1221no—
339#481—1221no—
340#478—21no—
341#476—21no—
342#476—611no—
343#476C2×D6111yescharacteristic, complemented
344#475—611no—
345#474C2×D6111yescharacteristic
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#446C2×D6111yescharacteristic, complemented
358#442—21no—
359C4×SD16—611no—
360C4×D8C2×D6111yescharacteristic
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#332—611no—
371#331—611no—
372Q8⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375D4⋊C8—611no—
376C82—611no—
377C61#1989111yescharacteristic, complemented, Sylow
378C2×#41—1221no—
379C22×Q8—21no—
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—1221no—
390C2×D8C22⋊D6111yescharacteristic
391C2×M16—1221no—
392C22×C8—1221no—
393C22×C8—1221no—
394C4⋊Q8—21no—
395C4⋊Q8—611no—
396C4⋊Q8C22×D6111yescharacteristic
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#125—21no—
400#123—2441no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—41no—
404C22⋊Q8—41no—
405C22⋊Q8—1221no—
406#121—1221no—
407#121—1221no—
408#121—1221no—
409#120—2441no—
410C4×Q8C22×D6111yescharacteristic
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—611no—
414C4×D4C22×D6111yescharacteristic
415C4×D4C22×D6111yescharacteristic
416C2×C4⋊C4—21no—
417C2×C4⋊C4—1221no—
418#107—21no—
419#107C22⋊D6111yescharacteristic
420#106—21no—
421#106—21no—
422#106—21no—
423#106—611no—
424#106—1221no—
425#106—1221no—
426C4⋊C8—21no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8C22×D6111yescharacteristic
431#103—611no—
432#103—1221no—
433#103—1221no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#103—2441no—
438#102—21no—
439#102—21no—
440#102—611no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—2441no—
447#102C22⋊D6111yescharacteristic
448#102C22⋊D6111yescharacteristic
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452#97—21no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8C22×D6111yescharacteristic
456C24—2441no—
457#41—4881no—
458#41—2442no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—41no—
462C2×Q8—42no—
463C2×Q8—81no—
464C2×Q8—1221no—
465C2×Q8D4×D6111yescharacteristic
466C2×D4—611no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—4881no—
471C2×D4—4881no—
472C2×D4—2441no—
473C2×D4—2442no—
474C2×D4—2441no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4C2×C22⋊D6111yescharacteristic
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—1221no—
484C22×C4—2441no—
485SD16—4881no—
486SD16—4881no—
487SD16—4881no—
488SD16—2441no—
489SD16—2442no—
490SD16—2442no—
491D8—2442no—
492D8#1175820114yescomplemented
493M16—2442no—
494C2×C8—21no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—4881no—
502C2×C8—4881no—
503C2×C8C2×C22⋊D6111yescharacteristic
504C2×C8C2×C22⋊D6111yescharacteristic
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—611no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—2441no—
517C4⋊C4D4×D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—1221no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C42—21no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—21no—
531C23—1221no—
532C23—9761no—
533C23—4882no—
534C23—4882no—
535C23—2441no—
536Q8—21no—
537Q8—21no—
538Q8—41no—
539Q8—41no—
540Q8—81no—
541Q8—82no—
542Q8—2441no—
543Q8—2441no—
544D4—21no—
545D4—21no—
546D4—1221no—
547D4—1221no—
548D4—4881no—
549D4—4882no—
550D4—4882no—
551D4—4882no—
552D4—2441no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4C2×#1175820112yes—
557D4C2×#1175820112yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—82no—
569C2×C4—161no—
570C2×C4—1221no—
571C2×C4—4882no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4#1451982211yescharacteristic
578C8—21no—
579C8—41no—
580C8—1221no—
581C8—1221no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C8—2441no—
586C8C2×#1175820112yes—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—9764no—
594C22—4881no—
595C22—4881no—
596C22—4881no—
597C22—2441no—
598C22—2441no—
599C22—2441no—
600C22D4×C22⋊D6111yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—4881no—
609C4—2441no—
610C4C22×#1175820111yescharacteristic
611C4#1684055811yescharacteristic
612C4#1684056611yescharacteristic
613C4#1684057011yescharacteristic
614C2—42no—
615C2—42no—
616C2—9761no—
617C2—4881no—
618C2#2145742411yescharacteristic
619C2D4×#1175820111yescharacteristic
620C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3384C2
socleC22×C61D42
prime radical#21457420C2
hypercenter#3384D61
hypocenterC61#19891
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3384

Complements

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

NH ≅ G/Ntype
1. #21457684C2semidirect
2. #21457637C2semidirect
4. #21445421C2semidirect
5. #21445315C2semidirect
6. #21445291C2semidirect
7. #21445240C2semidirect
8. #21445230C2semidirect
9. #21445026C2semidirect
10. #21444997C2semidirect
11. #21444968C2semidirect
12. #21444922C2semidirect
13. #21444709C2semidirect
14. D8×#92418C2semidirect
15. C61×#3384C2semidirect
20. #16840112C22semidirect
21. #16840112C22semidirect
27. #16840047C22semidirect
42. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839509C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
57. D4×Q8×C61C22semidirect
59. C61×#498C22semidirect
62. C61×#476C22semidirect
64. C61×#446C22semidirect
174. D8×C61SD16semidirect
304. #3384D61semidirect
328. D4×Q8C2×D61semidirect
332. #498C2×D61semidirect
343. #476C2×D61semidirect
357. #446C2×D61semidirect
377. C61#19891semidirect
492. D8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214576842
#214576372
#214574202
#214454212
#214453152
#214452912
#214452402
#214452302
#214450262
#214449972
#214449682
#214449222
#214447092
D8×#924182
C61×#33842
#1989161