Finite Group Databasebeta

← #29671398

Subgroups — #29671398

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#21458729C211yesmaximal, characteristic, complemented
2#21458488C211yesmaximal, characteristic, complemented
3#21457435C211yesmaximal, characteristic
4#21445410C211yesmaximal, characteristic, complemented
5#21445340C211yesmaximal, characteristic, complemented
6#21445301C211yesmaximal, characteristic, complemented
7#21445232C211yesmaximal, characteristic, complemented
8#21445228C211yesmaximal, characteristic, complemented
9#21444945C211yesmaximal, characteristic, complemented
10#21444935C211yesmaximal, characteristic, complemented
11#21444896C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14C8×#11758200C211yesmaximal, characteristic, complemented
15C61×#3295C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17Q8×Dic122C2211yescharacteristic, complemented
18#16840433C2211yescharacteristic
19#16840172—21no—
20#16840165C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840136C2211yescharacteristic, complemented
23#16840134C2211yescharacteristic
24#16840109C2211yescharacteristic, complemented
25#16840107—21no—
26#16840066—21no—
27#16840054C2211yescharacteristic, complemented
28#16840052C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840051—21no—
31#16840042—21no—
32#16840013—21no—
33#16839962C2211yescharacteristic, complemented
34#16839960—21no—
35#16839959C2211yescharacteristic
36#16839957C2211yescharacteristic
37#16839905—21no—
38#16839896—21no—
39#16839893C2211yescharacteristic, complemented
40C8×Dic122C2211yescharacteristic, complemented
41#16839862—21no—
42C4×#11758200C2211yescharacteristic
43#16839847—21no—
44#16839840C2211yescharacteristic
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
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—
68#14519803—44no—
69C2×#11758200—21no—
70C2×#11758200—41no—
71C2×#11758200D411yescharacteristic
72C2×#11758201—21no—
73C2×#11758201—41no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758194—21no—
78#14519777—21no—
79#14519777C2311yescharacteristic
80#14519776—21no—
81#14519776—41no—
82#14519774C2311yescharacteristic
83#14519773—21no—
84#14519751—21no—
85#14519750—41no—
86Q8×Dic61—21no—
87C4×Dic122—21no—
88C4×Dic122C2311yescharacteristic
89#14519739—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708D411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697—21no—
111C22⋊#92418—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
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—
163C2×C22⋊D61—41no—
164#11758222—42no—
165#11758222—81no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×C4×D61—41no—
170C2×Dic122—21no—
171C2×Dic122—41no—
172C2×Dic122C2×D411yescharacteristic
173SD16×C61—41no—
174SD16×C61—81no—
175C2×C8×C61—21no—
176C2×C8×C61—81no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193#11758201—42no—
194#11758201—82no—
195#11758200—42no—
196#11758200—82no—
197#11758200SD1614yescomplemented
198#11758199—42no—
199#11758199—42no—
200#11758199—81no—
201#11758198—41no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—21no—
207#11758196—41no—
208C4×Dic61—21no—
209C4×Dic61—41no—
210#11758194—42no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—41no—
220C23×C61—82no—
221C23×C61—82no—
222C23×C61—161no—
223C22×D61—41no—
224Q8×C61—21no—
225Q8×C61—21no—
226Q8×C61—81no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—81no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61C2×SD1612yes—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—161no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C22⋊D61—82no—
252C22⋊D61—161no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256D244—41no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—81no—
261C4×D61—82no—
262Dic122—21no—
263Dic122—81no—
264Dic122—161no—
265Dic122C2×SD1612yes—
266C8×C61—21no—
267C8×C61—21no—
268C8×C61—41no—
269#92418—22no—
270#92418—21no—
271#92418—41no—
272#92418—41no—
273#92418—41no—
274#92418C2×SD1612yes—
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—164no—
286C22×C61D4211yescharacteristic
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—81no—
296C4×C61C22×SD1611yescharacteristic
297C4×C61#57511yescharacteristic
298C4×C61#57711yescharacteristic
299C4×C61#57711yescharacteristic
300Dic61—22no—
301Dic61—41no—
302Dic61—81no—
303#3385—611no—
304#3362—611no—
305#3295D6111yescharacteristic, complemented
306#3274—611no—
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#327511yescharacteristic
322C2×C61D4×SD1611yescharacteristic
323C2×C61#329211yescharacteristic
324D61—161no—
325C2×AGL(1,ℤ/8)—1221no—
326Q82—611no—
327#554C2×D6111yescharacteristic
328#550—611no—
329D42C2×D6111yescharacteristic
330#501—1221no—
331#500—611no—
332#493—611no—
333#492—611no—
334#492—611no—
335#486—611no—
336#486—1221no—
337#486C2×D6111yescharacteristic
338#481—21no—
339#479—1221no—
340#477—611no—
341#476—611no—
342#476—611no—
343#476—1221no—
344#476C2×D6111yescharacteristic
345#474—1221no—
346#469—1221no—
347#465—21no—
348#460—611no—
349#460—611no—
350#460—1221no—
351#460C2×D6111yescharacteristic
352#459—611no—
353#450—21no—
354#449—1221no—
355#447—1221no—
356#446—611no—
357C8×Q8—611no—
358#440—1221no—
359C4×SD16—611no—
360C4×SD16—1221no—
361C4×SD16C2×D6111yescharacteristic
362#436—611no—
363C8×D4C2×D6111yescharacteristic
364C2×#102—21no—
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—1221no—
378C22×D4—21no—
379C22×D4—21no—
380AGL(1,ℤ/8)—2444no—
381C2×SD16—21no—
382C2×SD16—611no—
383C2×SD16—1221no—
384C2×SD16—1221no—
385C2×SD16—2441no—
386C2×SD16—2441no—
387C2×SD16C22⋊D6111yescharacteristic
388C2×D8—1221no—
389C2×D8—1221no—
390C2×D8—1221no—
391C2×M16—1221no—
392C22×C8—21no—
393C4⋊Q8—611no—
394C4⋊Q8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8—2441no—
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#127—1221no—
400#127C22×D6111yescharacteristic
401#125—21no—
402C22⋊Q8—21no—
403C22⋊Q8—41no—
404#121—21no—
405#121—21no—
406#121—1221no—
407#121—2441no—
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—21no—
418#107—1221no—
419#107—1221no—
420#107—1221no—
421#106—21no—
422#106—611no—
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—21no—
438#102—21no—
439#102—611no—
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—4881no—
457#41—2442no—
458C2×Q8—21no—
459C2×Q8—611no—
460C2×Q8—1221no—
461C2×Q8—2441no—
462C2×Q8C2×C22⋊D6111yescharacteristic
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—81no—
471C2×D4—81no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—2441no—
476C2×D4—2441no—
477C2×D4D4×D6111yescharacteristic
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—2441no—
483SD16—41no—
484SD16—81no—
485SD16—614no—
486SD16—4882no—
487SD16—4882no—
488SD16—2442no—
489SD16—2442no—
490D8—4881no—
491D8—2442no—
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×C8C2×C22⋊D6111yescharacteristic
503C2×C8C2×C22⋊D6111yescharacteristic
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—611no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
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—82no—
531C23—82no—
532C23—161no—
533C23—2441no—
534Q8—21no—
535Q8—21no—
536Q8—81no—
537Q8—612no—
538Q8—1221no—
539Q8—9761no—
540Q8—4881no—
541D4—21no—
542D4—21no—
543D4—21no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—81no—
549D4—82no—
550D4—82no—
551D4—9761no—
552D4—4881no—
553D4—4882no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4SD16×D6112yes—
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—41no—
568C2×C4—161no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—4882no—
572C2×C4—2441no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#1451982211yescharacteristic
576C8—21no—
577C8—21no—
578C8—41no—
579C8—612no—
580C8—1222no—
581C8—1221no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—164no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22D4×C22⋊D6111yescharacteristic
599C4—21no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—1222no—
606C4—4881no—
607C4—2441no—
608C4C2×SD16×D6111yescharacteristic
609C4#1684056011yescharacteristic
610C4#1684056811yescharacteristic
611C4#1684057311yescharacteristic
612C2—42no—
613C2—81no—
614C2—161no—
615C2—9761no—
616C2#2145744611yescharacteristic
617C2SD16×C22⋊D6111yescharacteristic
618C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3295C2
socleC22×C61D42
prime radical#21457435C2
hypercenter#3295D61
hypocenterC61#19893
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. #21458729C2semidirect
2. #21458488C2semidirect
4. #21445410C2semidirect
5. #21445340C2semidirect
6. #21445301C2semidirect
7. #21445232C2semidirect
8. #21445228C2semidirect
9. #21444945C2semidirect
10. #21444935C2semidirect
11. #21444896C2semidirect
12. #21444891C2semidirect
13. #21444710C2semidirect
14. C8×#11758200C2semidirect
15. C61×#3295C2semidirect
17. Q8×Dic122C22semidirect
20. #16840165C22semidirect
22. #16840136C22semidirect
24. #16840109C22semidirect
27. #16840054C22semidirect
33. #16839962C22semidirect
39. #16839893C22semidirect
40. C8×Dic122C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839510C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. C8×#92418C22semidirect
197. #11758200SD16semidirect
305. #3295D61semidirect
376. C61#19893semidirect

Maximal subgroups

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

subgroupindex
#214587292
#214584882
#214574352
#214454102
#214453402
#214453012
#214452322
#214452282
#214449452
#214449352
#214448962
#214448912
#214447102
C8×#117582002
C61×#32952
#1989361