Finite Group Databasebeta

← #29670316

Subgroups — #29670316

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458454C211yesmaximal, characteristic, complemented
2#21458061C211yesmaximal, characteristic, complemented
3#21457771C211yesmaximal, characteristic, complemented
4#21445456C211yesmaximal, characteristic, complemented
5#21445382C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445228C211yesmaximal, characteristic, complemented
8#21445157C211yesmaximal, characteristic, complemented
9#21445045C211yesmaximal, characteristic, complemented
10#21444984C211yesmaximal, characteristic, complemented
11#21444882C211yesmaximal, characteristic, complemented
12#21444873C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15D4×SD16×C61C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840434C2211yescharacteristic, complemented
18#16840138C2211yescharacteristic
19#16840114C2211yescharacteristic, complemented
20#16840110—21no—
21#16840110C2211yescharacteristic, complemented
22#16840109C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic
24#16840088—21no—
25#16840083—21no—
26#16840057C2211yescharacteristic, complemented
27#16840052C2211yescharacteristic
28#16840039—21no—
29#16840039C2211yescharacteristic, complemented
30#16840013—21no—
31#16839997—21no—
32#16839957—21no—
33#16839957C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839906—21no—
37#16839896—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C4×#11758199—21no—
40C4×#11758199C2211yescharacteristic
41#16839840C2211yescharacteristic
42C8×D244C2211yescharacteristic, complemented
43#16839767—21no—
44#16839757—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C22×SD16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#461C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C61×#448—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758226—21no—
68C2×#11758199—21no—
69C2×#11758199—21no—
70C2×#11758199—21no—
71C2×#11758199—41no—
72C2×#11758199D411yescharacteristic
73C2×#11758194—21no—
74C2×C8×D61—21no—
75#14519777C2311yescharacteristic
76#14519774—21no—
77#14519774C2311yescharacteristic
78#14519763—41no—
79#14519752—21no—
80#14519752—41no—
81#14519751—21no—
82#14519748—21no—
83C4×D244—21no—
84C4×D244C2311yescharacteristic
85#14519736—21no—
86#14519735—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710D411yescharacteristic
90#14519708—21no—
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519700—41no—
97#14519699—21no—
98#14519699—21no—
99#14519698—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698D411yescharacteristic
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—44no—
123C2×SD16×C61D411yescharacteristic
124C22×C8×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—41no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—42no—
157C2×D4×C61—42no—
158C2×D4×C61—81no—
159C2×D4×C61—81no—
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—41no—
165C2×C22⋊D61—41no—
166#11758226—42no—
167#11758226—81no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—41no—
172C2×D244—41no—
173C2×D244C2×D411yescharacteristic
174C2×C4×D61—21no—
175C2×C4×D61—41no—
176C2×Dic122—21no—
177SD16×C61—22no—
178SD16×C61—42no—
179SD16×C61—42no—
180SD16×C61—42no—
181SD16×C61—81no—
182C2×C8×C61—21no—
183C2×C8×C61—42no—
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—41no—
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—41no—
196C22⋊C4×C61—41no—
197C42×C61—21no—
198C42×C61C2411yescharacteristic
199C22⋊Dic61—41no—
200#11758199—22no—
201#11758199—42no—
202#11758199—81no—
203#11758199—81no—
204#11758199—82no—
205#11758198—21no—
206#11758198—41no—
207#11758198—41no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61C2×D411yescharacteristic
211C4×Dic61—41no—
212#11758194—81no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C8×D61—81no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—41no—
223C23×C61—82no—
224C23×C61—82no—
225C23×C61—161no—
226C22×D61—21no—
227C22×D61—41no—
228Q8×C61—21no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—82no—
232Q8×C61AGL(1,ℤ/8)12yes—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—81no—
241D4×C61—82no—
242D4×C61—82no—
243D4×C61#13512yes—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—82no—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C22⋊D61—161no—
258C2×Dic61—21no—
259C2×Dic61—41no—
260D244—21no—
261D244—21no—
262D244—41no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—82no—
267D244—81no—
268D244—161no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—82no—
272Dic122—81no—
273C8×C61—21no—
274C8×C61—21no—
275C8×C61—41no—
276#92418—21no—
277#92418—21no—
278#92418—41no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#19819—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—164no—
293C22×C61D4211yescharacteristic
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C4×C61—22no—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—81no—
306C4×C61C2×#13511yescharacteristic
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C61#57611yescharacteristic
309C4×C61#57711yescharacteristic
310Dic61—41no—
311Dic61—81no—
312#3358—611no—
313#3325—611no—
314#3305—611no—
315D4×SD16D6111yescharacteristic, complemented
316#1717—611no—
317#1701—611no—
318#1678—611no—
319#1677—611no—
320#1665—611no—
321#1646—611no—
322#1637—611no—
323#1621—611no—
324#1619—611no—
325#1583—611no—
326#1580—611no—
327C2×C61—42no—
328C2×C61—81no—
329C2×C61—161no—
330C2×C61#326411yescharacteristic
331C2×C61#328211yescharacteristic
332C2×C61#328811yescharacteristic
333D61—81no—
334D61—161no—
335C22×SD16—21no—
336#555—611no—
337#550—611no—
338D4×Q8C2×D6111yescharacteristic
339D42C2×D6111yescharacteristic
340#493—611no—
341#492C2×D6111yescharacteristic
342#487—611no—
343#486—611no—
344#486—611no—
345#486—611no—
346#486—1221no—
347#482—1221no—
348#482—1221no—
349#477—611no—
350#476—611no—
351#474—611no—
352#474—1221no—
353#469—1221no—
354#466—1221no—
355#465—21no—
356#461C2×D6111yescharacteristic
357#460C2×D6111yescharacteristic
358#459—611no—
359#459—611no—
360#459—611no—
361#459—1221no—
362#450—21no—
363#449—1221no—
364#448—21no—
365#447—1221no—
366C8×Q8—611no—
367C4×SD16C2×D6111yescharacteristic
368C4×D8—611no—
369C4×D8—1221no—
370#436—611no—
371C8×D4—611no—
372C8×D4C2×D6111yescharacteristic
373#418—1221no—
374#416—1221no—
375#334—611no—
376#334—611no—
377#333—611no—
378#332—611no—
379#331—611no—
380#331—611no—
381Q8⋊C8—611no—
382D4⋊C8—611no—
383D4⋊C8—611no—
384D4⋊C8—611no—
385#322—611no—
386C61#1981911yescharacteristic, complemented, Sylow
387C2×#41—1221no—
388C22×Q8—21no—
389C22×D4—21no—
390C22×D4—21no—
391C2×SD16—21no—
392C2×SD16—21no—
393C2×SD16—21no—
394C2×SD16—21no—
395C2×SD16—44no—
396C2×SD16C22⋊D6111yescharacteristic
397C2×D8—611no—
398C2×D8—1221no—
399C2×D8—1221no—
400C2×D8—1221no—
401C2×D8—2441no—
402C2×M16—1221no—
403C22×C8—21no—
404C22×C8—1221no—
405C4⋊Q8—21no—
406C4⋊Q8—611no—
407C4⋊Q8C22×D6111yescharacteristic
408#127—611no—
409#127—1221no—
410#127C22×D6111yescharacteristic
411#124—2441no—
412C22⋊Q8—21no—
413C22⋊Q8—41no—
414#121—21no—
415#121—21no—
416#121—1221no—
417#121—1221no—
418#121—1221no—
419#121—2441no—
420#120—41no—
421C4×Q8C22×D6111yescharacteristic
422C4×D4—21no—
423C4×D4—611no—
424C4×D4—1221no—
425C4×D4C22×D6111yescharacteristic
426C4×D4C22×D6111yescharacteristic
427#117—1221no—
428#117—1221no—
429#107—611no—
430#107—1221no—
431#107—1221no—
432#106—1221no—
433#106C22⋊D6111yescharacteristic
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—1221no—
438C4⋊C8C22×D6111yescharacteristic
439#103—21no—
440#103—1221no—
441#103—2441no—
442#103C22⋊D6111yescharacteristic
443#102—21no—
444#102—611no—
445#102—611no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102C22⋊D6111yescharacteristic
458C22⋊C8—21no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—1221no—
464C4×C8C22×D6111yescharacteristic
465C24—41no—
466#41—4881no—
467#41—2442no—
468C2×Q8—21no—
469C2×Q8—41no—
470C2×Q8—42no—
471C2×Q8—81no—
472C2×Q8—1221no—
473C2×Q8C2×C22⋊D6111yescharacteristic
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—81no—
482C2×D4—81no—
483C2×D4—611no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—2441no—
488C2×D4—2441no—
489C2×D4—2441no—
490C2×D4D4×D6111yescharacteristic
491C2×D4C2×C22⋊D6111yescharacteristic
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—41no—
495C22×C4—1221no—
496C22×C4—2441no—
497SD16—22no—
498SD16—42no—
499SD16—42no—
500SD16—42no—
501SD16—81no—
502D8—1222no—
503D8—4881no—
504D8—4881no—
505D8—4882no—
506D8—2442no—
507M16—4881no—
508C2×C8—21no—
509C2×C8—42no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—4881no—
517C2×C8C2×C22⋊D6111yescharacteristic
518C2×C8C2×C22⋊D6111yescharacteristic
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—41no—
523C4⋊C4—611no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4D4×D6111yescharacteristic
527C4⋊C4C2×C22⋊D6111yescharacteristic
528C4⋊C4C2×C22⋊D6111yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—1221no—
534C22⋊C4—2441no—
535C22⋊C4—2441no—
536C22⋊C4—2441no—
537C42—21no—
538C42—2441no—
539C42C23×D6111yescharacteristic
540C23—21no—
541C23—21no—
542C23—41no—
543C23—82no—
544C23—82no—
545C23—161no—
546C23—1221no—
547C23—2441no—
548Q8—21no—
549Q8—41no—
550Q8—41no—
551Q8—82no—
552Q8—4881no—
553Q8#1451980312yes—
554D4—21no—
555D4—21no—
556D4—21no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—81no—
562D4—82no—
563D4—82no—
564D4—1221no—
565D4—1221no—
566D4—9761no—
567D4—9761no—
568D4—4881no—
569D4—4882no—
570D4—2441no—
571D4—2441no—
572D4—2441no—
573D4—2441no—
574D4#1451979812yes—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—82no—
585C2×C4—1221no—
586C2×C4—4882no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4C2×D4×D6111yescharacteristic
591C2×C4C22×C22⋊D6111yescharacteristic
592C2×C4#1451982211yescharacteristic
593C8—21no—
594C8—21no—
595C8—41no—
596C8—1221no—
597C8—1221no—
598C8—2441no—
599C8—2441no—
600C8—2441no—
601C8—2441no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—164no—
612C22—4881no—
613C22—4881no—
614C22—4881no—
615C22—2441no—
616C22—2441no—
617C22—2441no—
618C22D4×C22⋊D6111yescharacteristic
619C4—22no—
620C4—21no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—4881no—
626C4—2441no—
627C4C2×#1451979811yescharacteristic
628C4C2×#1451980311yescharacteristic
629C4#1684056511yescharacteristic
630C4#1684056811yescharacteristic
631C2—42no—
632C2—81no—
633C2—161no—
634C2—9761no—
635C2—4881no—
636C2#2145729511yescharacteristic
637C2#2145752611yescharacteristic
638C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×SD16×C61C2
socleC22×C61D42
hypercenterD4×SD16D61
hypocenterC61#19819
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×SD16

Complements

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

NH ≅ G/Ntype
1. #21458454C2semidirect
2. #21458061C2semidirect
3. #21457771C2semidirect
4. #21445456C2semidirect
5. #21445382C2semidirect
6. #21445233C2semidirect
7. #21445228C2semidirect
8. #21445157C2semidirect
9. #21445045C2semidirect
10. #21444984C2semidirect
11. #21444882C2semidirect
12. #21444873C2semidirect
13. #21444663C2semidirect
14. #21444641C2semidirect
15. D4×SD16×C61C2semidirect
17. #16840434C22semidirect
19. #16840114C22semidirect
21. #16840110C22semidirect
22. #16840109C22semidirect
26. #16840057C22semidirect
29. #16840039C22semidirect
38. Q8×#92418C22semidirect
42. C8×D244C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. #16839484C22semidirect
315. D4×SD16D61semidirect
386. C61#19819semidirect

Maximal subgroups

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

subgroupindex
#214584542
#214580612
#214577712
#214454562
#214453822
#214452332
#214452282
#214451572
#214450452
#214449842
#214448822
#214448732
#214446632
#214446412
D4×SD16×C612
#1981961