Finite Group Databasebeta

← #29671402

Subgroups — #29671402

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458727C211yesmaximal, characteristic, complemented
2#21458486C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic, complemented
4#21445408C211yesmaximal, characteristic, complemented
5#21445342C211yesmaximal, characteristic, complemented
6#21445300C211yesmaximal, characteristic, complemented
7#21445233C211yesmaximal, characteristic, complemented
8#21445226C211yesmaximal, characteristic, complemented
9#21444946C211yesmaximal, characteristic, complemented
10#21444935C211yesmaximal, characteristic, complemented
11#21444896C211yesmaximal, characteristic, complemented
12#21444889C211yesmaximal, characteristic, complemented
13#21444707C211yesmaximal, characteristic, complemented
14SD16×#92418C211yesmaximal, characteristic, complemented
15C61×#3274C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic, complemented
17#16840469C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic, complemented
20#16840134C2211yescharacteristic
21#16840134C2211yescharacteristic, complemented
22#16840108—21no—
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic, complemented
25#16840078—21no—
26#16840071—21no—
27#16840057C2211yescharacteristic, complemented
28#16840052—21no—
29#16840052C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840040—21no—
33#16839989—21no—
34#16839959—21no—
35#16839959C2211yescharacteristic
36#16839959C2211yescharacteristic, complemented
37#16839913—21no—
38#16839892C2211yescharacteristic, complemented
39Q8×#92418C2211yescharacteristic, complemented
40#16839863—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519698—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839506C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#550C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#469—21no—
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C61×#447—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—21no—
69C2×#11758200—41no—
70C2×#11758200—41no—
71C2×#11758200D411yescharacteristic
72C22×#92418—21no—
73#14519777—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519776—41no—
78#14519768—21no—
79#14519756—21no—
80#14519754—41no—
81Q8×Dic61—21no—
82C4×Dic122—21no—
83C4×Dic122C2311yescharacteristic
84D4×Dic61—21no—
85C2×C4⋊Dic61—21no—
86#14519711—21no—
87#14519711—21no—
88#14519710—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519706—21no—
98#14519701—21no—
99#14519700D411yescharacteristic
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—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×C61D411yescharacteristic
121C2×D8×C61—21no—
122C2×D8×C61—21no—
123C2×D8×C61—21no—
124C2×M16×C61—21no—
125#127×C61—21no—
126#127×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#121×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—41no—
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×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61—42no—
160C2×D4×C61—42no—
161C2×D4×C61—81no—
162C2×D4×C61—81no—
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×D411yescharacteristic
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—41no—
168C22×Dic61—41no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—41no—
172C2×Dic122C2×D411yescharacteristic
173SD16×C61—42no—
174SD16×C61SD1614yescomplemented
175D8×C61—42no—
176D8×C61—42no—
177D8×C61—81no—
178M16×C61—42no—
179C2×C8×C61—21no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C42×C61—21no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193#11758200—41no—
194#11758200—42no—
195#11758200—82no—
196#11758200—82no—
197#11758200—81no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—21no—
206#11758196—41no—
207#11758196—41no—
208C4×Dic61—21no—
209C4×Dic61—41no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—81no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—82no—
222C23×C61—82no—
223C23×C61—161no—
224Q8×C61—21no—
225Q8×C61C2×SD1612yes—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—82no—
238D4×C61—81no—
239D4×C61—81no—
240D4×C61—82no—
241D4×C61—82no—
242D4×C61—161no—
243D4×C61C2×SD1612yes—
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—82no—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—161no—
261Dic122—21no—
262Dic122—21no—
263Dic122—81no—
264Dic122—81no—
265Dic122—161no—
266C8×C61—21no—
267C8×C61—41no—
268C8×C61C2×SD1612yes—
269#92418—22no—
270#92418—21no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#19893—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—164no—
289C22×C61D4211yescharacteristic
290C4×C61—22no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61C22×SD1611yescharacteristic
296C4×C61#57511yescharacteristic
297C4×C61#57711yescharacteristic
298C4×C61#57711yescharacteristic
299Dic61—41no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303#3385—611no—
304#3362—611no—
305#3295—611no—
306#3274D6111yescharacteristic
307#1706—611no—
308#1695—611no—
309#1688—611no—
310#1678—611no—
311#1677—611no—
312#1631—611no—
313#1629—611no—
314#1623—611no—
315#1622—611no—
316#1590—611no—
317C8×SD16—611no—
318C2×C61—42no—
319C2×C61—81no—
320C2×C61—161no—
321C2×C61—161no—
322C2×C61#327511yescharacteristic
323C2×C61D4×SD1611yescharacteristic
324C2×C61#329211yescharacteristic
325C2×AGL(1,ℤ/8)—21no—
326Q82—611no—
327#554—611no—
328#550C2×D6111yescharacteristic
329D42C2×D6111yescharacteristic
330#501—1221no—
331#500—611no—
332#493C2×D6111yescharacteristic
333#492—611no—
334#492—611no—
335#486—611no—
336#486—611no—
337#486—1221no—
338#481—1221no—
339#479—1221no—
340#477—611no—
341#476—611no—
342#476—611no—
343#476—611no—
344#476—1221no—
345#474—1221no—
346#469—21no—
347#465—1221no—
348#460—611no—
349#460—611no—
350#460—1221no—
351#460C2×D6111yescharacteristic
352#459C2×D6111yescharacteristic
353#450—1221no—
354#449—21no—
355#447—21no—
356#446—611no—
357C8×Q8—611no—
358#440—1221no—
359C4×SD16—611no—
360C4×SD16—1221no—
361C4×SD16C2×D6111yescharacteristic
362#436C2×D6111yescharacteristic
363C8×D4—611no—
364C2×#102—1221no—
365#334—611no—
366#334—611no—
367#333—611no—
368#333—611no—
369#331—611no—
370#331—611no—
371Q8⋊C8—611no—
372Q8⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375C82—611no—
376C61#1989311yescharacteristic, complemented, Sylow
377C2×#41—21no—
378C22×D4—21no—
379C22×D4—21no—
380AGL(1,ℤ/8)—44no—
381C2×SD16—21no—
382C2×SD16—611no—
383C2×SD16—1221no—
384C2×SD16—1221no—
385C2×SD16—2441no—
386C2×SD16—2441no—
387C2×SD16C22⋊D6111yescharacteristic
388C2×D8—21no—
389C2×D8—21no—
390C2×D8—21no—
391C2×M16—21no—
392C22×C8—1221no—
393C4⋊Q8—611no—
394C4⋊Q8—611no—
395C4⋊Q8—1221no—
396C4⋊Q8—1221no—
397C4⋊Q8—2441no—
398#127—21no—
399#127C22×D6111yescharacteristic
400#127C22×D6111yescharacteristic
401#125—1221no—
402C22⋊Q8—1221no—
403C22⋊Q8—2441no—
404#121—21no—
405#121—21no—
406#121—21no—
407#121—41no—
408#120—41no—
409C4×Q8—611no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412C4×Q8C22×D6111yescharacteristic
413C4×D4—21no—
414C4×D4—1221no—
415C4×D4C22×D6111yescharacteristic
416C4×D4C22×D6111yescharacteristic
417C2×C4⋊C4—1221no—
418#107—1221no—
419#107—1221no—
420#107—1221no—
421#106—611no—
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⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432#103—611no—
433#103—1221no—
434#103C22⋊D6111yescharacteristic
435#102—21no—
436#102—21no—
437#102—611no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102C22⋊D6111yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—1221no—
450#97—1221no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—1221no—
454C4×C8C22×D6111yescharacteristic
455C24—41no—
456#41—42no—
457#41—81no—
458C2×Q8—611no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—2441no—
462C2×Q8C2×C22⋊D6111yescharacteristic
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—81no—
476C2×D4—81no—
477C2×D4D4×D6111yescharacteristic
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—2441no—
483SD16—42no—
484SD16—4882no—
485SD16—4882no—
486SD16—4881no—
487SD16—2441no—
488SD16—2442no—
489SD16#1175820014yescomplemented
490D8—42no—
491D8—42no—
492D8—81no—
493M16—42no—
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×C8C2×C22⋊D6111yescharacteristic
503C2×C8C2×C22⋊D6111yescharacteristic
504C4⋊C4—21no—
505C4⋊C4—611no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—2441no—
512C4⋊C4—2441no—
513C4⋊C4—2441no—
514C4⋊C4—2441no—
515C4⋊C4D4×D6111yescharacteristic
516C4⋊C4C2×C22⋊D6111yescharacteristic
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—2441no—
523C42—21no—
524C42—1221no—
525C42—2441no—
526C42C23×D6111yescharacteristic
527C23—21no—
528C23—21no—
529C23—41no—
530C23—41no—
531C23—82no—
532C23—82no—
533C23—161no—
534Q8—21no—
535Q8—1221no—
536Q8—1221no—
537Q8—9761no—
538Q8—4881no—
539Q8—4881no—
540Q8C2×#1175820012yes—
541D4—21no—
542D4—21no—
543D4—21no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—82no—
553D4—81no—
554D4—81no—
555D4—82no—
556D4—82no—
557D4—161no—
558D4C2×#1175820012yes—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—82no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—9761no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#1451982211yescharacteristic
576C8—21no—
577C8—41no—
578C8—1222no—
579C8—1221no—
580C8—1221no—
581C8—2441no—
582C8—2441no—
583C8—2441no—
584C8C2×#1175820012yes—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—164no—
598C22D4×C22⋊D6111yescharacteristic
599C4—22no—
600C4—21no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—4881no—
605C4—4881no—
606C4—2441no—
607C4—2441no—
608C4C22×#1175820011yescharacteristic
609C4#1684055811yescharacteristic
610C4#1684057211yescharacteristic
611C4#1684057311yescharacteristic
612C2—42no—
613C2—81no—
614C2—161no—
615C2—161no—
616C2#2145745011yescharacteristic
617C2D4×#1175820011yescharacteristic
618C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3274C2
socleC22×C61D42
hypercenter#3274D61
hypocenterC61#19893
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3274

Complements

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

NH ≅ G/Ntype
1. #21458727C2semidirect
2. #21458486C2semidirect
3. #21457643C2semidirect
4. #21445408C2semidirect
5. #21445342C2semidirect
6. #21445300C2semidirect
7. #21445233C2semidirect
8. #21445226C2semidirect
9. #21444946C2semidirect
10. #21444935C2semidirect
11. #21444896C2semidirect
12. #21444889C2semidirect
13. #21444707C2semidirect
14. SD16×#92418C2semidirect
16. Q8×Dic122C22semidirect
19. #16840167C22semidirect
21. #16840134C22semidirect
24. #16840108C22semidirect
27. #16840057C22semidirect
36. #16839959C22semidirect
38. #16839892C22semidirect
39. Q8×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839510C22semidirect
50. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. C8×#92418C22semidirect
174. SD16×C61SD16semidirect
376. C61#19893semidirect
489. SD16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214587272
#214584862
#214576432
#214454082
#214453422
#214453002
#214452332
#214452262
#214449462
#214449352
#214448962
#214448892
#214447072
SD16×#924182
C61×#32742
#1989361