Finite Group Databasebeta

← #29670635

Subgroups — #29670635

Order 15616.

600 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458524C211yesmaximal, characteristic
2#21458207C211yesmaximal, characteristic, complemented
3#21457752C211yesmaximal, characteristic, complemented
4#21445473C211yesmaximal, characteristic, complemented
5#21445398C211yesmaximal, characteristic, complemented
6#21445258C211yesmaximal, characteristic, complemented
7#21445235C211yesmaximal, characteristic, complemented
8#21445216C211yesmaximal, characteristic, complemented
9#21445024C211yesmaximal, characteristic, complemented
10#21444991C211yesmaximal, characteristic, complemented
11#21444945C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444669C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3360C211yesmaximal, characteristic, complemented
16#16840469C2211yescharacteristic
17#16840433C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840136C2211yescharacteristic
20#16840125—21no—
21#16840114C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840097—21no—
25#16840095—21no—
26#16840086—21no—
27#16840065—21no—
28#16840057C2211yescharacteristic
29#16840054C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16840001—21no—
33#16839993—21no—
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839955—21no—
37#16839922—21no—
38#16839892C2211yescharacteristic
39#16839871—21no—
40C4×#11758202C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43C2×#14519710—21no—
44#16839756—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#556C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#488—21no—
59C61×#487C2211yescharacteristic
60C61×#479—21no—
61C61×#476C2211yescharacteristic
62C61×#463—21no—
63C61×#461C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758202D411yescharacteristic
69C2×#11758201—21no—
70C2×#11758201—41no—
71C2×#11758199—21no—
72C22×#92418—21no—
73C22×#92418—21no—
74#14519777C2311yescharacteristic
75#14519774C2311yescharacteristic
76#14519773—21no—
77#14519768—21no—
78#14519756—21no—
79#14519754—41no—
80#14519751—21no—
81#14519750—41no—
82C4×Dic122C2311yescharacteristic
83D4×Dic61—21no—
84#14519739—21no—
85C2×C4⋊Dic61—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519701D411yescharacteristic
99#14519700—21no—
100#14519700—21no—
101#14519700—41no—
102#14519700D411yescharacteristic
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×SD16×C61—21no—
116C2×D8×C61D411yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—41no—
120#125×C61—21no—
121#124×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131C2×C4⋊C4×C61—21no—
132#107×C61D411yescharacteristic
133#106×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—41no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154C22×Dic61—41no—
155#11758222—42no—
156#11758222—81no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×C4×D61—41no—
161C2×Dic122—21no—
162C2×Dic122C2×D411yescharacteristic
163SD16×C61—42no—
164D8×C61—22no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188#11758202—41no—
189#11758201—81no—
190#11758201—82no—
191#11758199—42no—
192#11758198—41no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199C4×Dic61—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—42no—
205C2×#92418—81no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C23×C61—21no—
209C23×C61—21no—
210C22×D61—41no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—81no—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61#13512yes—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—161no—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C22⋊D61—82no—
235C22⋊D61—161no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—161no—
240D244—41no—
241D244—41no—
242D244—41no—
243D244—41no—
244D244—81no—
245C4×D61—82no—
246Dic122—21no—
247Dic122—21no—
248Dic122—81no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—41no—
252#92418—21no—
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#19840—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61D4211yescharacteristic
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C4×C61—21no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61C2×#13511yescharacteristic
277C4×C61#57511yescharacteristic
278C4×C61#57711yescharacteristic
279C4×C61#57811yescharacteristic
280Dic61—41no—
281Dic61—41no—
282Dic61—81no—
283#3366—611no—
284#3360D6111yescharacteristic, complemented
285#3338—611no—
286#3304—611no—
287#1719—611no—
288#1704—611no—
289#1681—611no—
290#1678—611no—
291#1675—611no—
292#1643—611no—
293#1638—611no—
294#1631—611no—
295#1622—611no—
296#1584—611no—
297#1580—611no—
298C2×C61—42no—
299C2×C61—81no—
300C2×C61#326711yescharacteristic
301C2×C61#328511yescharacteristic
302C2×C61#328811yescharacteristic
303D61—161no—
304#556C2×D6111yescharacteristic
305#554—611no—
306#554C2×D6111yescharacteristic
307#550—611no—
308#500—611no—
309#492—611no—
310#490—1221no—
311#488—21no—
312#487—611no—
313#487C2×D6111yescharacteristic
314#486—611no—
315#486—611no—
316#484—1221no—
317#484—1221no—
318#482—1221no—
319#479—21no—
320#478—1221no—
321#477—611no—
322#477—611no—
323#476—611no—
324#476C2×D6111yescharacteristic
325#475—611no—
326#467—1221no—
327#465—1221no—
328#463—21no—
329#461C2×D6111yescharacteristic
330#460—611no—
331#459—611no—
332#459—1221no—
333#452—1221no—
334#446—611no—
335C8×Q8C2×D6111yescharacteristic
336#442—1221no—
337C4×Q16—611no—
338C4×SD16—21no—
339C4×D8C2×D6111yescharacteristic
340C8×D4—611no—
341C8×D4—611no—
342C2×#107—1221no—
343#416—1221no—
344#335—611no—
345#334—611no—
346#333—611no—
347#333—611no—
348#332—611no—
349#331—611no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
352D4⋊C8—611no—
353D4⋊C8—611no—
354#322—611no—
355C61#1984011yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×Q16—611no—
358C2×SD16—21no—
359C2×SD16—1221no—
360C2×SD16—2441no—
361C2×D8—1221no—
362C2×D8C22⋊D6111yescharacteristic
363C22×C8—1221no—
364C22×C8—1221no—
365C4⋊Q8—611no—
366C4⋊Q8C22×D6111yescharacteristic
367C4⋊Q8C22×D6111yescharacteristic
368#127—611no—
369#127—1221no—
370#126—41no—
371#125—21no—
372#125—1221no—
373#124—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—41no—
377C22⋊Q8—1221no—
378C22⋊Q8—2441no—
379#121—1221no—
380#121—2441no—
381C4×Q8—21no—
382C4×Q8—611no—
383C4×Q8C22×D6111yescharacteristic
384C4×D4—21no—
385C4×D4—1221no—
386C4×D4—1221no—
387C4×D4C22×D6111yescharacteristic
388C4×D4C22×D6111yescharacteristic
389#117—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—1221no—
392#107—611no—
393#107—1221no—
394#107—1221no—
395#107—1221no—
396#107—1221no—
397#107C22⋊D6111yescharacteristic
398#106—21no—
399#106—1221no—
400#106—1221no—
401C4⋊C8—21no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—1221no—
406C4⋊C8C22×D6111yescharacteristic
407#103—21no—
408#103—21no—
409#103—21no—
410#103—611no—
411#103—611no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—2441no—
416#102—21no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102C22⋊D6111yescharacteristic
425#102C22⋊D6111yescharacteristic
426C22⋊C8—1221no—
427C22⋊C8—1221no—
428#97—1221no—
429C4×C8—21no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8C22×D6111yescharacteristic
433#41—4881no—
434#41—2442no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—611no—
438C2×Q8—1221no—
439C2×Q8D4×D6111yescharacteristic
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—2441no—
444C2×D4—2441no—
445C2×D4C2×C22⋊D6111yescharacteristic
446C2×D4C2×C22⋊D6111yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—2441no—
451C22×C4—2441no—
452Q16—2441no—
453SD16—42no—
454SD16—4882no—
455SD16—4881no—
456D8—22no—
457D8—2442no—
458C2×C8—21no—
459C2×C8—611no—
460C2×C8—611no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—4881no—
466C2×C8—2442no—
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—611no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4D4×D6111yescharacteristic
484C4⋊C4C2×C22⋊D6111yescharacteristic
485C4⋊C4C2×C22⋊D6111yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C42—21no—
494C42—41no—
495C42—1221no—
496C42C23×D6111yescharacteristic
497C23—21no—
498C23—21no—
499C23—2441no—
500Q8—21no—
501Q8—21no—
502Q8—41no—
503Q8—41no—
504Q8—81no—
505Q8—1221no—
506Q8—1221no—
507Q8—4881no—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—9761no—
512D4—4881no—
513D4—4882no—
514D4—2441no—
515D4—2441no—
516D4—2441no—
517D4—2441no—
518D4#1451979712yes—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—161no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—9761no—
533C2×C4—4882no—
534C2×C4—2441no—
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C22×C22⋊D6111yescharacteristic
537C2×C4#1451982211yescharacteristic
538C8—21no—
539C8—21no—
540C8—41no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C22—22no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22D4×C22⋊D6111yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—4881no—
565C4—2441no—
566C4—2441no—
567C4C2×#1451979711yescharacteristic
568C4#1684055911yescharacteristic
569C4#1684056911yescharacteristic
570C4#1684057711yescharacteristic
571C2—42no—
572C2—81no—
573C2—9761no—
574C2#2145734411yescharacteristic
575C2#2145755611yescharacteristic
576C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3360C2
socleC22×C61D42
prime radical#21458524C2
hypercenter#3360D61
hypocenterC61#19840
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3360

Complements

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

NH ≅ G/Ntype
2. #21458207C2semidirect
3. #21457752C2semidirect
4. #21445473C2semidirect
5. #21445398C2semidirect
6. #21445258C2semidirect
7. #21445235C2semidirect
8. #21445216C2semidirect
9. #21445024C2semidirect
10. #21444991C2semidirect
11. #21444945C2semidirect
12. #21444891C2semidirect
13. #21444669C2semidirect
14. #21444643C2semidirect
15. C61×#3360C2semidirect
284. #3360D61semidirect
355. C61#19840semidirect

Maximal subgroups

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

subgroupindex
#214585242
#214582072
#214577522
#214454732
#214453982
#214452582
#214452352
#214452162
#214450242
#214449912
#214449452
#214448912
#214446692
#214446432
C61×#33602
#1984061