Finite Group Databasebeta

← #29677245

Subgroups — #29677245

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458837C211yesmaximal, characteristic
2#21458434C211yesmaximal, characteristic, complemented
3#21457481C211yesmaximal, characteristic, complemented
4#21445407C211yesmaximal, characteristic, complemented
5#21445322C211yesmaximal, characteristic, complemented
6#21445320C211yesmaximal, characteristic, complemented
7#21445291C211yesmaximal, characteristic, complemented
8#21445286C211yesmaximal, characteristic, complemented
9#21445088C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21444945C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444714C211yesmaximal, characteristic, complemented
14#21444704C211yesmaximal, characteristic, complemented
15C61×#3384C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17#16840482C2211yescharacteristic
18#16840433C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840150—21no—
22#16840136C2211yescharacteristic
23#16840136C2211yescharacteristic
24#16840135C2211yescharacteristic
25#16840100—21no—
26#16840092—21no—
27#16840054C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840018—21no—
30#16840010—21no—
31#16839976—21no—
32#16839969C2211yescharacteristic
33#16839967C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839961—21no—
36#16839957C2211yescharacteristic
37#16839922—21no—
38#16839901—21no—
39#16839892C2211yescharacteristic
40#16839861—21no—
41C4×#11758202C2211yescharacteristic
42#16839840C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839802—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#498C2211yescharacteristic
60C61×#478—21no—
61C61×#476—21no—
62C61×#476C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#442—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519808—44no—
69C2×#11758202—21no—
70C2×#11758202D411yescharacteristic
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—41no—
78C2×#11758199—21no—
79C2×#11758194—21no—
80C2×#11758194—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519773—21no—
84#14519770—41no—
85#14519764—21no—
86#14519756—21no—
87#14519751—21no—
88#14519750—41no—
89C4×Dic122C2311yescharacteristic
90#14519742—21no—
91#14519739—21no—
92#14519736—21no—
93#14519710—21no—
94#14519710D411yescharacteristic
95#14519708—21no—
96#14519708—21no—
97#14519708—21no—
98#14519708—41no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519706—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—21no—
108#14519700D411yescharacteristic
109#14519697—21no—
110#14519697—21no—
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519687—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×D8×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#125×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C22×C4×C61—41no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—41no—
159#11758222—42no—
160#11758222—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122C2×D411yescharacteristic
167D8×C61—22no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189C22⋊Dic61—41no—
190#11758202—22no—
191#11758202—42no—
192#11758201—42no—
193#11758201—42no—
194#11758201—81no—
195#11758201—82no—
196#11758200—81no—
197#11758199—81no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—21no—
203#11758196—41no—
204C4×Dic61—21no—
205C4×Dic61—41no—
206#11758194—42no—
207#11758194—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214#11758190—42no—
215#11758190—81no—
216C23×C61—21no—
217C23×C61—21no—
218C22×D61—41no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—81no—
224Q8×C61—82no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—82no—
240C2×C4×C61—161no—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C22⋊D61—82no—
245C22⋊D61—161no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—81no—
254C4×D61—82no—
255Dic122—21no—
256Dic122—21no—
257Dic122—41no—
258Dic122—41no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61#13712yescomplemented
262#92418—22no—
263#92418—21no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418AGL(1,ℤ/8)12yes—
268#20317—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61D4211yescharacteristic
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C61C2×#13711yescharacteristic
288C4×C61#57711yescharacteristic
289C4×C61#57711yescharacteristic
290Dic61—41no—
291Dic61—41no—
292Dic61—81no—
293#3397—611no—
294#3384D6111yescharacteristic, complemented
295#3356—611no—
296#3278—611no—
297#1705—611no—
298#1692—611no—
299#1691—611no—
300#1686—611no—
301#1685—611no—
302#1652—611no—
303#1651—611no—
304#1631—611no—
305#1622—611no—
306#1591—611no—
307#1589—611no—
308C2×C61—81no—
309C2×C61—81no—
310C2×C61#327211yescharacteristic
311C2×C61#327811yescharacteristic
312C2×C61#329111yescharacteristic
313D61—161no—
314C2×#137—1221no—
315#556—611no—
316#554C2×D6111yescharacteristic
317#550—611no—
318D4×Q8C2×D6111yescharacteristic
319#501—21no—
320#498—611no—
321#498—611no—
322#498C2×D6111yescharacteristic
323#496—1221no—
324#492—611no—
325#492—611no—
326#492—611no—
327#485—1221no—
328#483—1221no—
329#478—21no—
330#477—611no—
331#476—21no—
332#476C2×D6111yescharacteristic
333#475—611no—
334#474C2×D6111yescharacteristic
335#470—1221no—
336#468—1221no—
337#463—1221no—
338#462—611no—
339#461—611no—
340#460—611no—
341#460—1221no—
342#459—611no—
343#452—1221no—
344#448—1221no—
345#446—611no—
346#446C2×D6111yescharacteristic
347#442—21no—
348#440—1221no—
349C4×Q16—611no—
350C4×D8C2×D6111yescharacteristic
351#436—611no—
352#436—611no—
353#428—1221no—
354#335—611no—
355#335—611no—
356#333—611no—
357#332—611no—
358#332—611no—
359#331—611no—
360Q8⋊C8—611no—
361Q8⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364C82—611no—
365C61#2031711yescharacteristic, complemented, Sylow
366C2×#41—1221no—
367C22×Q8—21no—
368#137—2444no—
369C2×Q16—611no—
370C2×Q16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—2441no—
378C2×D8—1221no—
379C2×D8C22⋊D6111yescharacteristic
380C2×M16—1221no—
381C2×M16—1221no—
382C4⋊Q8—21no—
383C4⋊Q8—611no—
384C4⋊Q8C22×D6111yescharacteristic
385C4⋊Q8C22×D6111yescharacteristic
386#127—611no—
387#127—1221no—
388#126—2441no—
389#125—21no—
390#124—1221no—
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C22⋊Q8—41no—
394C22⋊Q8—41no—
395C22⋊Q8—1221no—
396#121—1221no—
397#121—2441no—
398C4×Q8—611no—
399C4×Q8—1221no—
400C4×Q8C22×D6111yescharacteristic
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—1221no—
404C4×D4C22×D6111yescharacteristic
405C4×D4C22×D6111yescharacteristic
406#117—1221no—
407C2×C4⋊C4—21no—
408#107—21no—
409#107—611no—
410#107—1221no—
411#107C22⋊D6111yescharacteristic
412#106—21no—
413#106—21no—
414#106—21no—
415#106—1221no—
416#106—1221no—
417#106—1221no—
418#106—2441no—
419C4⋊C8—21no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—1221no—
424C4⋊C8C22×D6111yescharacteristic
425#103—611no—
426#103—611no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#103—1221no—
431#102—21no—
432#102—21no—
433#102—1221no—
434#102—1221no—
435#102C22⋊D6111yescharacteristic
436#102C22⋊D6111yescharacteristic
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439#97—21no—
440#97—1221no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8C22×D6111yescharacteristic
444#41—4881no—
445#41—2442no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—41no—
449C2×Q8—42no—
450C2×Q8—81no—
451C2×Q8—611no—
452C2×Q8—1221no—
453C2×Q8D4×D6111yescharacteristic
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—2441no—
458C2×D4—2441no—
459C2×D4C2×C22⋊D6111yescharacteristic
460C2×D4C2×C22⋊D6111yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—2441no—
466Q16—1222no—
467Q16—2442no—
468SD16—4881no—
469SD16—4882no—
470SD16—4881no—
471SD16—4881no—
472SD16—2442no—
473SD16—2442no—
474SD16—2442no—
475D8—22no—
476D8—4881no—
477M16—2442no—
478M16—2442no—
479C2×C8—21no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8C2×C22⋊D6111yescharacteristic
487C2×C8C2×C22⋊D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—611no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—2441no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C42—21no—
512C42—1221no—
513C42—2441no—
514C42C23×D6111yescharacteristic
515C23—21no—
516C23—21no—
517C23—2441no—
518Q8—21no—
519Q8—21no—
520Q8—41no—
521Q8—41no—
522Q8—81no—
523Q8—82no—
524Q8—1221no—
525Q8—1221no—
526Q8—2441no—
527Q8—2441no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—9761no—
533D4—4881no—
534D4—4882no—
535D4—2441no—
536D4—2441no—
537D4—2441no—
538D4—2441no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—82no—
550C2×C4—161no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—4882no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—21no—
559C8—41no—
560C8—612no—
561C8—1222no—
562C8—1221no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8#1451980812yescomplemented
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22D4×C22⋊D6111yescharacteristic
577C4—21no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—4881no—
585C4—2441no—
586C4—2441no—
587C4AGL(1,ℤ/8)×D6111yescharacteristic
588C4C2×#1451980811yescharacteristic
589C4#1684057011yescharacteristic
590C4#1684057211yescharacteristic
591C2—81no—
592C2—81no—
593C2—9761no—
594C2#2145741311yescharacteristic
595C2#2145748111yescharacteristic
596C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3384C2
socleC22×C61D42
prime radical#21458837C2
hypercenter#3384D61
hypocenterC61#20317
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3384

Complements

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

NH ≅ G/Ntype
2. #21458434C2semidirect
3. #21457481C2semidirect
4. #21445407C2semidirect
5. #21445322C2semidirect
6. #21445320C2semidirect
7. #21445291C2semidirect
8. #21445286C2semidirect
9. #21445088C2semidirect
10. #21445082C2semidirect
11. #21444945C2semidirect
12. #21444891C2semidirect
13. #21444714C2semidirect
14. #21444704C2semidirect
15. C61×#3384C2semidirect
261. C8×C61#137semidirect
294. #3384D61semidirect
365. C61#20317semidirect
566. C8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214588372
#214584342
#214574812
#214454072
#214453222
#214453202
#214452912
#214452862
#214450882
#214450822
#214449452
#214448912
#214447142
#214447042
C61×#33842
#2031761