Finite Group Databasebeta

← #29671368

Subgroups — #29671368

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#21458718C211yesmaximal, characteristic, complemented
2#21457686C211yesmaximal, characteristic, complemented
3#21457415C211yesmaximal, characteristic
4#21445419C211yesmaximal, characteristic, complemented
5#21445320C211yesmaximal, characteristic, complemented
6#21445289C211yesmaximal, characteristic, complemented
7#21445242C211yesmaximal, characteristic, complemented
8#21445225C211yesmaximal, characteristic, complemented
9#21445026C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444969C211yesmaximal, characteristic, complemented
12#21444920C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14C8×#11758199C211yesmaximal, characteristic, complemented
15C61×#3295C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17Q8×D244C2211yescharacteristic, complemented
18#16840417C2211yescharacteristic
19#16840172—21no—
20#16840158C2211yescharacteristic, complemented
21#16840134C2211yescharacteristic
22#16840113C2211yescharacteristic, complemented
23#16840112C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic
25#16840094—21no—
26#16840059—21no—
27#16840052—21no—
28#16840049C2211yescharacteristic, complemented
29#16840047C2211yescharacteristic, complemented
30#16840042C2211yescharacteristic
31#16840006—21no—
32#16839990—21no—
33#16839967C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839913—21no—
37#16839905—21no—
38#16839899—21no—
39#16839893C2211yescharacteristic, complemented
40#16839872—21no—
41C4×#11758199C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic, complemented
44#103×D61—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#481—21no—
60C61×#476C2211yescharacteristic
61C61×#465—21no—
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#102×C61—21no—
67C2×#11758222—21no—
68C2×Q8×D61—21no—
69#14519803—44no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758199—21no—
75C2×#11758199D411yescharacteristic
76C2×#11758194—21no—
77C2×C8×D61—21no—
78#14519777—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519759—41no—
82#14519757—21no—
83#14519757—41no—
84#14519756—21no—
85#14519748—21no—
86C4×D244—21no—
87C4×D244C2311yescharacteristic
88C4⋊C4×D61—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519701—41no—
103#14519700—21no—
104#14519700—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C22×C8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#125×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126#121×C61—21no—
127#121×C61—21no—
128#120×C61—41no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C2×C4⋊C4×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—42no—
154C2×D4×C61—42no—
155C2×D4×C61—81no—
156C2×D4×C61—81no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C2×C22⋊D61—41no—
163Q8×D61—42no—
164Q8×D61—81no—
165#11758222—42no—
166#11758222—81no—
167C2×D244C2×D411yescharacteristic
168C2×C4×D61—21no—
169C2×C4×D61—41no—
170C2×C4×D61—41no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—41no—
174SD16×C61—41no—
175SD16×C61—81no—
176C2×C8×C61—21no—
177C2×C8×C61—81no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—41no—
191C22⋊C4×C61—41no—
192C42×C61—21no—
193C42×C61C2411yescharacteristic
194C22⋊Dic61—41no—
195#11758200—42no—
196#11758200—42no—
197#11758200—81no—
198#11758200—81no—
199#11758199—42no—
200#11758199SD1614yescomplemented
201#11758198—21no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—41no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—41no—
210#11758194—42no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217C8×D61—81no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—41no—
221C23×C61—82no—
222C23×C61—82no—
223C23×C61—161no—
224C22×D61—21no—
225Q8×C61—21no—
226Q8×C61—21no—
227Q8×C61—81no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—81no—
236D4×C61—82no—
237D4×C61—82no—
238D4×C61C2×SD1612yes—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—161no—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C22⋊D61—82no—
253C2×Dic61—21no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256D244—21no—
257D244C2×SD1612yes—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—82no—
261C4×D61—82no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265Dic122—41no—
266Dic122—82no—
267C8×C61—21no—
268C8×C61—21no—
269C8×C61—41no—
270#92418—21no—
271#92418—41no—
272#92418—41no—
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×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—164no—
287C22×C61D4211yescharacteristic
288C2×D61—22no—
289C2×D61—41no—
290C2×D61—41no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61C22×SD1611yescharacteristic
298C4×C61#57511yescharacteristic
299C4×C61#57611yescharacteristic
300C4×C61#57711yescharacteristic
301Dic61—41no—
302Dic61—81no—
303Dic61—81no—
304#3384—611no—
305#3298—611no—
306#3295D6111yescharacteristic, complemented
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—81no—
321C2×C61—161no—
322C2×C61#327311yescharacteristic
323C2×C61D4×SD1611yescharacteristic
324C2×C61#329011yescharacteristic
325D61—42no—
326C2×AGL(1,ℤ/8)—1221no—
327#554C2×D6111yescharacteristic
328D4×Q8—611no—
329#547—611no—
330D42C2×D6111yescharacteristic
331#501—1221no—
332#498—611no—
333#492—611no—
334#487—611no—
335#487—611no—
336#486—611no—
337#486C2×D6111yescharacteristic
338#483—1221no—
339#481—21no—
340#478—1221no—
341#476—611no—
342#476—1221no—
343#476C2×D6111yescharacteristic
344#475—611no—
345#474—611no—
346#468—1221no—
347#465—21no—
348#465—1221no—
349#461—611no—
350#461—611no—
351#460C2×D6111yescharacteristic
352#459—611no—
353#450—21no—
354#450—1221no—
355#449—1221no—
356#448—1221no—
357#446—611no—
358#442—1221no—
359C4×SD16C2×D6111yescharacteristic
360C4×D8—611no—
361#436—611no—
362C8×D4—611no—
363C8×D4C2×D6111yescharacteristic
364C2×#103—1221no—
365C2×#102—21no—
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—1221no—
380C22×D4—21no—
381C22×D4—21no—
382AGL(1,ℤ/8)—2444no—
383C2×SD16—21no—
384C2×SD16—1221no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16C22⋊D6111yescharacteristic
389C2×D8—611no—
390C2×D8—1221no—
391C2×M16—1221no—
392C22×C8—21no—
393C22×C8—1221no—
394C4⋊Q8—611no—
395C4⋊Q8—611no—
396C4⋊Q8—1221no—
397C4⋊Q8C22×D6111yescharacteristic
398#127C22×D6111yescharacteristic
399#125—21no—
400#123—2441no—
401C22⋊Q8—21no—
402C22⋊Q8—41no—
403C22⋊Q8—1221no—
404C22⋊Q8—1221no—
405C22⋊Q8—2441no—
406#121—21no—
407#121—21no—
408#121—1221no—
409#120—41no—
410C4×Q8C22×D6111yescharacteristic
411C4×D4—21no—
412C4×D4—611no—
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—1221no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425#106C22⋊D6111yescharacteristic
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8C22×D6111yescharacteristic
431#103—1221no—
432#103—1221no—
433#103—1221no—
434#103—1221no—
435#103—1221no—
436#103—2441no—
437#103C22⋊D6111yescharacteristic
438#102—21no—
439#102—21no—
440#102—21no—
441#102—21no—
442#102—611no—
443#102—611no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—2441no—
448#102C22⋊D6111yescharacteristic
449C22⋊C8—21no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452#97—1221no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8C22×D6111yescharacteristic
456C24—41no—
457#41—4881no—
458#41—2442no—
459C2×Q8—21no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—4881no—
463C2×Q8—2441no—
464C2×Q8—2442no—
465C2×Q8C2×C22⋊D6111yescharacteristic
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—81no—
474C2×D4—81no—
475C2×D4—611no—
476C2×D4—2441no—
477C2×D4D4×D6111yescharacteristic
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—1221no—
483C22×C4—2441no—
484C22×C4—2441no—
485SD16—41no—
486SD16—81no—
487SD16—4881no—
488SD16—4881no—
489SD16—2442no—
490SD16—2442no—
491D8—614no—
492D8—2442no—
493M16—2442no—
494C2×C8—21no—
495C2×C8—81no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—4881no—
503C2×C8C2×C22⋊D6111yescharacteristic
504C2×C8C2×C22⋊D6111yescharacteristic
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2441no—
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—41no—
532C23—82no—
533C23—82no—
534C23—161no—
535C23—1221no—
536Q8—21no—
537Q8—21no—
538Q8—81no—
539Q8—4882no—
540Q8—2441no—
541Q8—2441no—
542Q8—2441no—
543Q8—2441no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—81no—
552D4—82no—
553D4—82no—
554D4—612no—
555D4—1221no—
556D4—4882no—
557D4SD16×D6112yes—
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—161no—
568C2×C4—1221no—
569C2×C4—4882no—
570C2×C4—4882no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4#1451982211yescharacteristic
578C8—21no—
579C8—21no—
580C8—41no—
581C8—612no—
582C8—1221no—
583C8—2441no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—164no—
597C22—1222no—
598C22—2441no—
599C22—2441no—
600C22D4×C22⋊D6111yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—4881no—
608C4—4881no—
609C4—2441no—
610C4C2×SD16×D6111yescharacteristic
611C4#1684056011yescharacteristic
612C4#1684056511yescharacteristic
613C4#1684056811yescharacteristic
614C2—42no—
615C2—81no—
616C2—161no—
617C2—2442no—
618C2#2145742611yescharacteristic
619C2SD16×C22⋊D6111yescharacteristic
620C2#2145760111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3295C2
socleC22×C61D42
prime radical#21457415C2
hypercenter#3295D61
hypocenterC61#19891
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3295

Complements

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

NH ≅ G/Ntype
1. #21458718C2semidirect
2. #21457686C2semidirect
4. #21445419C2semidirect
5. #21445320C2semidirect
6. #21445289C2semidirect
7. #21445242C2semidirect
8. #21445225C2semidirect
9. #21445026C2semidirect
10. #21444998C2semidirect
11. #21444969C2semidirect
12. #21444920C2semidirect
13. #21444710C2semidirect
14. C8×#11758199C2semidirect
15. C61×#3295C2semidirect
17. Q8×D244C22semidirect
20. #16840158C22semidirect
22. #16840113C22semidirect
23. #16840112C22semidirect
28. #16840049C22semidirect
29. #16840047C22semidirect
39. #16839893C22semidirect
43. C8×D244C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839511C22semidirect
48. #16839509C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
200. #11758199SD16semidirect
306. #3295D61semidirect
377. C61#19891semidirect

Maximal subgroups

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

subgroupindex
#214587182
#214576862
#214574152
#214454192
#214453202
#214452892
#214452422
#214452252
#214450262
#214449982
#214449692
#214449202
#214447102
C8×#117581992
C61×#32952
#1989161