Finite Group Databasebeta

← #29670141

Subgroups — #29670141

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458690C211yesmaximal, characteristic, complemented
2#21457645C211yesmaximal, characteristic, complemented
3#21457439C211yesmaximal, characteristic
4#21445446C211yesmaximal, characteristic, complemented
5#21445316C211yesmaximal, characteristic, complemented
6#21445235C211yesmaximal, characteristic, complemented
7#21445177C211yesmaximal, characteristic, complemented
8#21445157C211yesmaximal, characteristic, complemented
9#21444996C211yesmaximal, characteristic, complemented
10#21444946C211yesmaximal, characteristic, complemented
11#21444915C211yesmaximal, characteristic, complemented
12#21444890C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15C61×#3392C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17#16840472C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840161—21no—
20#16840158C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840134C2211yescharacteristic, complemented
23#16840110C2211yescharacteristic
24#16840109C2211yescharacteristic
25#16840082—21no—
26#16840069—21no—
27#16840059—21no—
28#16840057C2211yescharacteristic, complemented
29#16840052C2211yescharacteristic
30#16840039C2211yescharacteristic
31#16840013—21no—
32#16839994—21no—
33#16839962C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839911—21no—
37#16839907—21no—
38#16839895—21no—
39Q8×#92418C2211yescharacteristic, complemented
40C4×#11758201—21no—
41C4×#11758201C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#102×D61—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#557C2211yescharacteristic, complemented
57C61×#550C2211yescharacteristic
58C61×#492—21no—
59C61×#492C2211yescharacteristic, complemented
60C61×#461—21no—
61C61×#461C2211yescharacteristic
62C61×#460—21no—
63C61×#460C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C22×D244—21no—
68C2×D4×D61—21no—
69#14519802—44no—
70C2×#11758200—41no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201D411yescharacteristic
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758194—21no—
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519769—21no—
82#14519755—41no—
83#14519753—21no—
84#14519752—21no—
85#14519751—21no—
86#14519747—41no—
87C4×D244—21no—
88C4×D244C2311yescharacteristic
89C4⋊C4×D61—21no—
90#14519710—21no—
91#14519710—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708D411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701—21no—
101#14519701D411yescharacteristic
102#14519699—21no—
103#14519699—21no—
104#14519698—21no—
105#14519698—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61D411yescharacteristic
122C4⋊Q8×C61—21no—
123C4⋊Q8×C61C2311yescharacteristic
124#127×C61—21no—
125#127×C61C2311yescharacteristic
126#125×C61—21no—
127#125×C61—41no—
128#121×C61—21no—
129#121×C61—41no—
130C4×Q8×C61—21no—
131C4×Q8×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61C2311yescharacteristic
136#106×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
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×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C2×C22⋊D61—41no—
160D4×D61—42no—
161D4×D61—81no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—42no—
166C2×D244—41no—
167C2×D244—81no—
168C2×D244C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—41no—
171C2×Dic122—21no—
172SD16×C61—41no—
173SD16×C61—42no—
174SD16×C61—81no—
175SD16×C61—81no—
176SD16×C61—81no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C22⋊C4×C61—21no—
191C22⋊C4×C61—41no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—41no—
195C42×C61C2411yescharacteristic
196#11758201—42no—
197#11758201—81no—
198#11758201SD1614yescomplemented
199#11758200—82no—
200#11758199—42no—
201#11758199—42no—
202#11758199—81no—
203#11758198—21no—
204#11758198—41no—
205#11758198—41no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—41no—
210#11758196—41no—
211#11758194—42no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C8×D61—81no—
219C23×C61—21no—
220C23×C61—41no—
221C22×D61—21no—
222C22×D61—41no—
223C22×D61—82no—
224C22×D61—82no—
225C22×D61—161no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—21no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61C2×SD1612yes—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—82no—
239D4×C61—81no—
240D4×C61—161no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
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—82no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—82no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256D244—21no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—81no—
262D244—82no—
263D244C2×SD1612yes—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—161no—
267Dic122—81no—
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418C2×SD1612yes—
277#19807—611nomaximal, Sylow
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61D4211yescharacteristic
285C2×D61—22no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—164no—
292C4×C61—22no—
293C4×C61—21no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61C22×SD1611yescharacteristic
300C4×C61#57511yescharacteristic
301C4×C61#57611yescharacteristic
302C4×C61#57711yescharacteristic
303Dic61—41no—
304Dic61—81no—
305#3392D6111yescharacteristic, complemented
306#3381—611no—
307#3295—611no—
308#3274—611no—
309#1715—611no—
310#1691—611no—
311#1678—611no—
312#1668—611no—
313#1665—611no—
314#1639—611no—
315#1631—611no—
316#1626—611no—
317#1622—611no—
318#1583—611no—
319SD16⋊C8—611no—
320C2×C61—81no—
321C2×C61—161no—
322C2×C61#326811yescharacteristic
323C2×C61D4×SD1611yescharacteristic
324C2×C61#328811yescharacteristic
325D61—42no—
326D61—161no—
327C2×AGL(1,ℤ/8)—1221no—
328#557C2×D6111yescharacteristic, complemented
329#554—611no—
330#550C2×D6111yescharacteristic
331D42—611no—
332#499—1221no—
333#498—611no—
334#493—611no—
335#492—21no—
336#492—611no—
337#492C2×D6111yescharacteristic, complemented
338#486—611no—
339#486—611no—
340#481—1221no—
341#479—1221no—
342#478—1221no—
343#477—611no—
344#476—611no—
345#474—611no—
346#469—1221no—
347#465—1221no—
348#461—21no—
349#461C2×D6111yescharacteristic
350#460—21no—
351#460—611no—
352#460—611no—
353#460C2×D6111yescharacteristic, complemented
354#459—611no—
355#450—1221no—
356#449—1221no—
357#447—1221no—
358C8×Q8—611no—
359C8×Q8C2×D6111yescharacteristic, complemented
360C4×SD16—21no—
361C4×SD16—611no—
362C4×SD16—1221no—
363C4×SD16C2×D6111yescharacteristic
364#436—611no—
365C8×D4—611no—
366C2×#102—1221no—
367#335—611no—
368#334—611no—
369#333—611no—
370#332—611no—
371#331—611no—
372#331—611no—
373Q8⋊C8—611no—
374Q8⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377#322—611no—
378C61#1980711yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C22×D4—1221no—
381C22×D4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×SD16—21no—
384C2×SD16—21no—
385C2×SD16—21no—
386C2×SD16—21no—
387C2×SD16—611no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—2441no—
391C2×SD16C22⋊D6111yescharacteristic
392C2×D8—1221no—
393C2×D8—1221no—
394C2×D8—1221no—
395C2×M16—1221no—
396C22×C8—1221no—
397C4⋊Q8—21no—
398C4⋊Q8—611no—
399C4⋊Q8C22×D6111yescharacteristic
400#127—21no—
401#127—611no—
402#127C22×D6111yescharacteristic
403#125—21no—
404#125—41no—
405#125—1221no—
406C22⋊Q8—1221no—
407C22⋊Q8—2441no—
408#121—21no—
409#121—41no—
410#121—1221no—
411#121—1221no—
412#120—2441no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8C22×D6111yescharacteristic
416C4×Q8C22×D6111yescharacteristic
417C4×D4—21no—
418C4×D4—611no—
419C4×D4—1221no—
420C4×D4C22×D6111yescharacteristic
421C2×C4⋊C4—1221no—
422#107—1221no—
423#107—1221no—
424#106—611no—
425#106—1221no—
426#106—1221no—
427#106—1221no—
428#106C22⋊D6111yescharacteristic
429C4⋊C8—21no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8—611no—
433C4⋊C8—1221no—
434C4⋊C8C22×D6111yescharacteristic
435#103—21no—
436#103—611no—
437#103—1221no—
438#103C22⋊D6111yescharacteristic
439#102—21no—
440#102—611no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—2441no—
449#102C22⋊D6111yescharacteristic
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C4×C8—21no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8—1221no—
456C4×C8C22×D6111yescharacteristic
457C24—2441no—
458#41—42no—
459#41—81no—
460C2×Q8—21no—
461C2×Q8—1221no—
462C2×Q8D4×D6111yescharacteristic
463C2×Q8C2×C22⋊D6111yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—611no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—4881no—
474C2×D4—4881no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4—2441no—
478C2×D4—2442no—
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—1221no—
483C22×C4—2441no—
484SD16—41no—
485SD16—42no—
486SD16—81no—
487SD16—81no—
488SD16—81no—
489SD16—614no—
490SD16—4882no—
491SD16—4881no—
492SD16—2442no—
493D8—4881no—
494D8—2442no—
495D8—2442no—
496M16—2442no—
497C2×C8—21no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—4881no—
505C2×C8C2×C22⋊D6111yescharacteristic
506C2×C8C2×C22⋊D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—611no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2441no—
518C4⋊C4—2441no—
519C4⋊C4D4×D6111yescharacteristic
520C4⋊C4C2×C22⋊D6111yescharacteristic
521C4⋊C4C2×C22⋊D6111yescharacteristic
522C22⋊C4—21no—
523C22⋊C4—41no—
524C22⋊C4—1221no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C42—21no—
528C42—21no—
529C42—41no—
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—41no—
533C23—1221no—
534C23—9761no—
535C23—4882no—
536C23—4882no—
537C23—2441no—
538Q8—21no—
539Q8—21no—
540Q8—21no—
541Q8—41no—
542Q8—41no—
543Q8—4881no—
544Q8SD16×D6112yes—
545D4—21no—
546D4—21no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—82no—
552D4—81no—
553D4—161no—
554D4—612no—
555D4—1221no—
556D4—4881no—
557D4—4882no—
558D4—4882no—
559D4—2441no—
560D4—2441no—
561D4—2441no—
562D4—2441no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—82no—
572C2×C4—1221no—
573C2×C4—9761no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4#1451982211yescharacteristic
580C8—21no—
581C8—21no—
582C8—41no—
583C8—612no—
584C8—1221no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—1222no—
596C22—9764no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—2441no—
601C22—2441no—
602C22D4×C22⋊D6111yescharacteristic
603C4—22no—
604C4—21no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—4881no—
611C4—2441no—
612C4C2×SD16×D6111yescharacteristic
613C4#1684056011yescharacteristic
614C4#1684056511yescharacteristic
615C4#1684056811yescharacteristic
616C2—81no—
617C2—161no—
618C2—9761no—
619C2—2442no—
620C2#2145736011yescharacteristic
621C2SD16×C22⋊D6111yescharacteristic
622C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3392C2
socleC22×C61D42
prime radical#21457439C2
hypercenter#3392D61
hypocenterC61#19807
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3392

Complements

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

NH ≅ G/Ntype
1. #21458690C2semidirect
2. #21457645C2semidirect
4. #21445446C2semidirect
5. #21445316C2semidirect
6. #21445235C2semidirect
7. #21445177C2semidirect
8. #21445157C2semidirect
9. #21444996C2semidirect
10. #21444946C2semidirect
11. #21444915C2semidirect
12. #21444890C2semidirect
13. #21444663C2semidirect
14. #21444629C2semidirect
15. C61×#3392C2semidirect
20. #16840158C22semidirect
22. #16840134C22semidirect
28. #16840057C22semidirect
39. Q8×#92418C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
49. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. #16839484C22semidirect
56. C61×#557C22semidirect
59. C61×#492C22semidirect
63. C61×#460C22semidirect
64. C8×Q8×C61C22semidirect
198. #11758201SD16semidirect
305. #3392D61semidirect
328. #557C2×D61semidirect
337. #492C2×D61semidirect
353. #460C2×D61semidirect
359. C8×Q8C2×D61semidirect
378. C61#19807semidirect

Maximal subgroups

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

subgroupindex
#214586902
#214576452
#214574392
#214454462
#214453162
#214452352
#214451772
#214451572
#214449962
#214449462
#214449152
#214448902
#214446632
#214446292
C61×#33922
#1980761