Finite Group Databasebeta

← #29677048

Subgroups — #29677048

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458416C211yesmaximal, characteristic, complemented
2#21458118C211yesmaximal, characteristic, complemented
3#21458061C211yesmaximal, characteristic, complemented
4#21445418C211yesmaximal, characteristic, complemented
5#21445368C211yesmaximal, characteristic, complemented
6#21445350C211yesmaximal, characteristic, complemented
7#21445318C211yesmaximal, characteristic, complemented
8#21445281C211yesmaximal, characteristic, complemented
9#21445045C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444970C211yesmaximal, characteristic, complemented
12#21444907C211yesmaximal, characteristic, complemented
13#21444704C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3272C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840417C2211yescharacteristic, complemented
18#16840160C2211yescharacteristic, complemented
19#16840114C2211yescharacteristic, complemented
20#16840114C2211yescharacteristic, complemented
21#16840108C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840097—21no—
24#16840085—21no—
25#16840083—21no—
26#16840079—21no—
27#16840052C2211yescharacteristic
28#16840052C2211yescharacteristic, complemented
29#16840042C2211yescharacteristic
30#16840039C2211yescharacteristic, complemented
31#16840013—21no—
32#16840012—21no—
33#16840006—21no—
34#16839967C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839905—21no—
37#16839896—21no—
38C4×#11758199C2211yescharacteristic
39#16839841C2211yescharacteristic, complemented
40#16839840C2211yescharacteristic
41#16839840C2211yescharacteristic, complemented
42#16839803—21no—
43#16839767—21no—
44#16839761—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51D4⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#547C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#468—21no—
61C61×#461C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#450—21no—
64C61×#449—21no—
65C4×D8×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758200—21no—
69C2×#11758199—21no—
70C2×#11758199D411yescharacteristic
71C2×#11758194—21no—
72C2×#11758194—21no—
73C2×#11758189—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519763—41no—
77#14519759—41no—
78#14519756—21no—
79#14519751—21no—
80#14519748—21no—
81#14519748—21no—
82C4×D244C2311yescharacteristic
83#14519736—21no—
84#14519735—21no—
85C4⋊C4×D61—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710D411yescharacteristic
89#14519709—21no—
90#14519709—21no—
91#14519708—21no—
92#14519708—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—41no—
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—41no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118AGL(1,ℤ/8)×C61—44no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×D8×C61—21no—
123C2×D8×C61D411yescharacteristic
124C2×M16×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126#127×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C2×D4×C61—42no—
155C2×D4×C61—42no—
156C2×D4×C61—41no—
157C2×D4×C61—81no—
158C2×D4×C61—81no—
159C2×D4×C61C2×D411yescharacteristic
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—41no—
166C2×C22⋊D61—41no—
167C2×C22⋊D61—41no—
168#11758222—42no—
169#11758222—81no—
170C2×D244C2×D411yescharacteristic
171C2×C4×D61—21no—
172C2×C4×D61—41no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175SD16×C61—42no—
176SD16×C61—42no—
177SD16×C61—81no—
178D8×C61—22no—
179D8×C61—42no—
180M16×C61—42no—
181C2×C8×C61—21no—
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—41no—
194C22⋊C4×C61—41no—
195C42×C61C2411yescharacteristic
196C22⋊Dic61—41no—
197C22⋊Dic61—41no—
198#11758200—81no—
199#11758199—41no—
200#11758199—81no—
201#11758198—21no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D411yescharacteristic
208#11758196—41no—
209C4×Dic61—41no—
210#11758194—81no—
211#11758194—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218#11758189—42no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—41no—
223C23×C61—82no—
224C23×C61—82no—
225C23×C61—161no—
226C22×D61—21no—
227Q8×C61—41no—
228Q8×C61—41no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—81no—
240D4×C61—82no—
241D4×C61—82no—
242D4×C61—82no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
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—82no—
258C22⋊D61—161no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262D244—21no—
263D244—21no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—82no—
267Dic122—41no—
268Dic122—41no—
269Dic122—81no—
270C8×C61—21no—
271C8×C61—41no—
272C8×C61AGL(1,ℤ/8)12yes—
273#92418—21no—
274#92418—21no—
275#92418—41no—
276#92418—41no—
277#92418—41no—
278#92418—41no—
279#20301—611nomaximal, Sylow
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
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—
297C4×C61—21no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—81no—
302C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C61#57511yescharacteristic
304C4×C61#57611yescharacteristic
305C4×C61#57711yescharacteristic
306Dic61—41no—
307Dic61—81no—
308Dic61—81no—
309#3354—611no—
310#3330—611no—
311#3325—611no—
312#3272D6111yescharacteristic, complemented
313#1708—611no—
314#1699—611no—
315#1697—611no—
316#1691—611no—
317#1685—611no—
318#1646—611no—
319#1639—611no—
320#1635—611no—
321#1625—611no—
322#1589—611no—
323#1589—611no—
324C2×C61—81no—
325C2×C61—81no—
326C2×C61—81no—
327C2×C61—161no—
328C2×C61#327211yescharacteristic
329C2×C61#327311yescharacteristic
330C2×C61#329011yescharacteristic
331D61—81no—
332C2×AGL(1,ℤ/8)—21no—
333#555—611no—
334#547—611no—
335#547C2×D6111yescharacteristic
336D42C2×D6111yescharacteristic
337#498—611no—
338#492C2×D6111yescharacteristic
339#487—611no—
340#487—611no—
341#486—611no—
342#486—611no—
343#484—1221no—
344#482—1221no—
345#482—1221no—
346#481—1221no—
347#476—611no—
348#476—611no—
349#474—611no—
350#474—611no—
351#469—1221no—
352#469—1221no—
353#468—21no—
354#468—1221no—
355#461—611no—
356#461C2×D6111yescharacteristic
357#459—611no—
358#459C2×D6111yescharacteristic
359#450—21no—
360#449—21no—
361#449—1221no—
362#447—1221no—
363C4×D8—611no—
364C4×D8C2×D6111yescharacteristic
365#436—611no—
366#436—611no—
367#436—611no—
368#436C2×D6111yescharacteristic
369#428—1221no—
370#418—1221no—
371#417—1221no—
372#335—611no—
373#335—611no—
374#333—611no—
375#332—611no—
376#332—611no—
377#331—611no—
378D4⋊C8—611no—
379D4⋊C8—611no—
380D4⋊C8—611no—
381D4⋊C8—611no—
382C82—611no—
383C61#2030111yescharacteristic, complemented, Sylow
384C2×#41—21no—
385C2×#41—1221no—
386C22×D4—21no—
387C22×D4—21no—
388AGL(1,ℤ/8)—44no—
389C2×SD16—21no—
390C2×SD16—21no—
391C2×SD16—21no—
392C2×SD16—1221no—
393C2×D8—21no—
394C2×D8—611no—
395C2×D8—1221no—
396C2×D8C22⋊D6111yescharacteristic
397C2×M16—21no—
398C2×M16—1221no—
399C2×M16—1221no—
400C2×M16—1221no—
401C4⋊Q8—611no—
402C4⋊Q8—611no—
403C4⋊Q8C22×D6111yescharacteristic
404#127C22×D6111yescharacteristic
405#124—2441no—
406#123—41no—
407#123—2441no—
408C22⋊Q8—21no—
409C22⋊Q8—1221no—
410#121—21no—
411#121—21no—
412#121—21no—
413#121—1221no—
414#121—1221no—
415#121—1221no—
416#120—41no—
417C4×D4—611no—
418C4×D4C22×D6111yescharacteristic
419C4×D4C22×D6111yescharacteristic
420C4×D4C22×D6111yescharacteristic
421#117—1221no—
422#117—1221no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—1221no—
425#107—611no—
426#107—1221no—
427#107—1221no—
428#107C22⋊D6111yescharacteristic
429#106—1221no—
430#106—1221no—
431#106—1221no—
432#106—1221no—
433C4⋊C8—611no—
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8C22×D6111yescharacteristic
437#103—21no—
438#103—1221no—
439#103—1221no—
440#103—1221no—
441#103—2441no—
442#102—21no—
443#102—611no—
444#102—611no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—2441no—
453#102C22⋊D6111yescharacteristic
454#102C22⋊D6111yescharacteristic
455C22⋊C8—21no—
456C22⋊C8—1221no—
457C22⋊C8—1221no—
458C22⋊C8—1221no—
459C4×C8—611no—
460C4×C8—611no—
461C4×C8C22×D6111yescharacteristic
462C24—41no—
463#41—42no—
464#41—81no—
465#41—4881no—
466#41—2442no—
467C2×Q8—21no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—41no—
478C2×D4—81no—
479C2×D4—81no—
480C2×D4—611no—
481C2×D4—2441no—
482C2×D4—2441no—
483C2×D4D4×D6111yescharacteristic
484C2×D4C2×C22⋊D6111yescharacteristic
485C2×D4C2×C22⋊D6111yescharacteristic
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—41no—
490C22×C4—1221no—
491C22×C4—2441no—
492SD16—42no—
493SD16—42no—
494SD16—81no—
495SD16—4881no—
496D8—22no—
497D8—42no—
498D8—4881no—
499D8—2441no—
500M16—42no—
501M16—4881no—
502M16—4881no—
503M16—2442no—
504C2×C8—21no—
505C2×C8—611no—
506C2×C8—611no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8C2×C22⋊D6111yescharacteristic
512C2×C8C2×C22⋊D6111yescharacteristic
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—41no—
516C4⋊C4—611no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—2441no—
522C4⋊C4D4×D6111yescharacteristic
523C4⋊C4C2×C22⋊D6111yescharacteristic
524C4⋊C4C2×C22⋊D6111yescharacteristic
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—1221no—
531C22⋊C4—2441no—
532C22⋊C4—2441no—
533C22⋊C4—2441no—
534C42—2441no—
535C42C23×D6111yescharacteristic
536C23—21no—
537C23—21no—
538C23—21no—
539C23—41no—
540C23—82no—
541C23—82no—
542C23—161no—
543C23—1221no—
544Q8—41no—
545Q8—41no—
546Q8—4881no—
547Q8—2441no—
548Q8—2441no—
549D4—21no—
550D4—21no—
551D4—21no—
552D4—21no—
553D4—21no—
554D4—21no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—81no—
560D4—82no—
561D4—82no—
562D4—82no—
563D4—1221no—
564D4—1221no—
565D4—9761no—
566D4—4882no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—82no—
578C2×C4—1221no—
579C2×C4—4882no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C22×C22⋊D6111yescharacteristic
586C2×C4#1451982211yescharacteristic
587C8—21no—
588C8—41no—
589C8—1221no—
590C8—1221no—
591C8—2441no—
592C8—2441no—
593C8—2441no—
594C8—2441no—
595C8#1451980312yes—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—164no—
609C22—2441no—
610C22—2441no—
611C22—2441no—
612C22D4×C22⋊D6111yescharacteristic
613C4—21no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—4881no—
619C4—4881no—
620C4—2441no—
621C4C2×#1451980311yescharacteristic
622C4#1684055811yescharacteristic
623C4#1684056611yescharacteristic
624C4#1684057311yescharacteristic
625C2—81no—
626C2—81no—
627C2—81no—
628C2—161no—
629C2—4881no—
630C2#2145741511yescharacteristic
631C2#2145742411yescharacteristic
632C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3272C2
socleC22×C61D42
hypercenter#3272D61
hypocenterC61#20301
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3272

Complements

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

NH ≅ G/Ntype
1. #21458416C2semidirect
2. #21458118C2semidirect
3. #21458061C2semidirect
4. #21445418C2semidirect
5. #21445368C2semidirect
6. #21445350C2semidirect
7. #21445318C2semidirect
8. #21445281C2semidirect
9. #21445045C2semidirect
10. #21444998C2semidirect
11. #21444970C2semidirect
12. #21444907C2semidirect
13. #21444704C2semidirect
14. #21444702C2semidirect
15. C61×#3272C2semidirect
17. #16840417C22semidirect
18. #16840160C22semidirect
19. #16840114C22semidirect
20. #16840114C22semidirect
28. #16840052C22semidirect
30. #16840039C22semidirect
39. #16839841C22semidirect
41. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
53. D4⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
312. #3272D61semidirect
383. C61#20301semidirect

Maximal subgroups

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

subgroupindex
#214584162
#214581182
#214580612
#214454182
#214453682
#214453502
#214453182
#214452812
#214450452
#214449982
#214449702
#214449072
#214447042
#214447022
C61×#32722
#2030161