Finite Group Databasebeta

← #29670395

Subgroups — #29670395

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458170C211yesmaximal, characteristic, complemented
2#21458126C211yesmaximal, characteristic
3#21457730C211yesmaximal, characteristic, complemented
4#21445466C211yesmaximal, characteristic, complemented
5#21445391C211yesmaximal, characteristic, complemented
6#21445249C211yesmaximal, characteristic, complemented
7#21445244C211yesmaximal, characteristic, complemented
8#21445196C211yesmaximal, characteristic, complemented
9#21445075C211yesmaximal, characteristic, complemented
10#21445052C211yesmaximal, characteristic, complemented
11#21445033C211yesmaximal, characteristic, complemented
12#21444969C211yesmaximal, characteristic, complemented
13#21444662C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3360C211yesmaximal, characteristic, complemented
16#16840417C2211yescharacteristic
17#16840416C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840114C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840103—21no—
24#16840099—21no—
25#16840090—21no—
26#16840080—21no—
27#16840049C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840039C2211yescharacteristic
31#16840007—21no—
32#16840001—21no—
33#16839993—21no—
34#16839972C2211yescharacteristic
35#16839967C2211yescharacteristic
36#16839922—21no—
37#16839904—21no—
38C4×#11758201C2211yescharacteristic
39#16839841C2211yescharacteristic
40D4×#92418C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42C2×#14519708—21no—
43#16839760—21no—
44#16839756—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
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×#11758224—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758201D411yescharacteristic
72C2×#11758189—21no—
73C22×#92418—21no—
74C22×#92418—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519761—41no—
78#14519759—41no—
79#14519757—21no—
80#14519756—21no—
81#14519751—21no—
82#14519748—21no—
83C4×D244C2311yescharacteristic
84C2×C4⋊Dic61—21no—
85C4⋊C4×D61—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—41no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708D411yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519701—21no—
100#14519701—41no—
101#14519701D411yescharacteristic
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519698—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—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#11758224—42no—
156#11758224—81no—
157#11758222—42no—
158#11758222—81no—
159C2×D244C2×D411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—41no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164SD16×C61—42no—
165D8×C61—22no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61—41no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758202—81no—
190#11758201—41no—
191#11758200—81no—
192#11758198—21no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—41no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—42no—
206C2×#92418—81no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758189—81no—
210C23×C61—21no—
211C23×C61—21no—
212C22×D61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—81no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61#13512yes—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—161no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C22⋊D61—82no—
237C22⋊D61—82no—
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—82no—
242D244—21no—
243D244—21no—
244C4×D61—41no—
245C4×D61—41no—
246C4×D61—82no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—41no—
251C8×C61—21no—
252C8×C61—21no—
253C8×C61—41no—
254#92418—21no—
255#92418—21no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#19824—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61D4211yescharacteristic
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C4×C61—21no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61C2×#13511yescharacteristic
279C4×C61#57511yescharacteristic
280C4×C61#57711yescharacteristic
281C4×C61#57711yescharacteristic
282Dic61—41no—
283Dic61—81no—
284Dic61—81no—
285#3360D6111yescharacteristic, complemented
286#3335—611no—
287#3330—611no—
288#3302—611no—
289#1718—611no—
290#1703—611no—
291#1680—611no—
292#1679—611no—
293#1671—611no—
294#1650—611no—
295#1647—611no—
296#1644—611no—
297#1635—611no—
298#1583—611no—
299#1580—611no—
300C2×C61—42no—
301C2×C61—81no—
302C2×C61#326711yescharacteristic
303C2×C61#328211yescharacteristic
304C2×C61#328911yescharacteristic
305D61—81no—
306#556C2×D6111yescharacteristic
307#554C2×D6111yescharacteristic
308#547—611no—
309#547—611no—
310#500—611no—
311#498—611no—
312#488—21no—
313#487—611no—
314#487—611no—
315#487—611no—
316#487C2×D6111yescharacteristic
317#485—1221no—
318#484—1221no—
319#483—1221no—
320#481—1221no—
321#479—21no—
322#476—611no—
323#476C2×D6111yescharacteristic
324#475—611no—
325#474—611no—
326#474—611no—
327#468—1221no—
328#467—1221no—
329#465—1221no—
330#463—21no—
331#462—611no—
332#461—611no—
333#461C2×D6111yescharacteristic
334#452—1221no—
335#449—1221no—
336C8×Q8C2×D6111yescharacteristic
337C4×SD16—21no—
338C4×SD16—611no—
339C4×D8C2×D6111yescharacteristic
340#436—611no—
341C8×D4—611no—
342C8×D4—611no—
343C2×#106—1221no—
344#417—1221no—
345#416—1221no—
346#334—611no—
347#334—611no—
348#333—611no—
349#332—611no—
350#332—611no—
351#332—611no—
352Q8⋊C8—611no—
353D4⋊C8—611no—
354D4⋊C8—611no—
355D4⋊C8—611no—
356#322—611no—
357C61#1982411yescharacteristic, complemented, Sylow
358C2×#41—1221no—
359C2×#41—1221no—
360C2×Q16—1221no—
361C2×SD16—21no—
362C2×SD16—611no—
363C2×SD16—1221no—
364C2×D8C22⋊D6111yescharacteristic
365C2×M16—1221no—
366C22×C8—1221no—
367C22×C8—1221no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370C4⋊Q8C22×D6111yescharacteristic
371C4⋊Q8C22×D6111yescharacteristic
372#126—41no—
373#125—21no—
374#124—21no—
375#123—2441no—
376#123—2441no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—41no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382#121—1221no—
383#121—1221no—
384C4×Q8—21no—
385C4×Q8C22×D6111yescharacteristic
386C4×D4—21no—
387C4×D4—611no—
388C4×D4C22×D6111yescharacteristic
389C4×D4C22×D6111yescharacteristic
390#117—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394#107—1221no—
395#107—1221no—
396#107—2441no—
397#107C22⋊D6111yescharacteristic
398#106—21no—
399#106—611no—
400#106—1221no—
401#106—1221no—
402#106—1221no—
403#106—1221no—
404C4⋊C8—21no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8C22×D6111yescharacteristic
409#103—21no—
410#103—21no—
411#103—21no—
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#103—2441no—
421#102—21no—
422#102—611no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102C22⋊D6111yescharacteristic
427#102C22⋊D6111yescharacteristic
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C4×C8—21no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8C22×D6111yescharacteristic
435#41—4881no—
436#41—4881no—
437#41—2442no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8D4×D6111yescharacteristic
444C2×D4—611no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4C2×C22⋊D6111yescharacteristic
448C2×D4C2×C22⋊D6111yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—1221no—
453C22×C4—2441no—
454C22×C4—2441no—
455Q16—4881no—
456SD16—42no—
457SD16—4881no—
458SD16—2441no—
459D8—22no—
460M16—4881no—
461C2×C8—21no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—4881no—
469C2×C8—2442no—
470C2×C8C2×C22⋊D6111yescharacteristic
471C2×C8C2×C22⋊D6111yescharacteristic
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—611no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—2441no—
486C4⋊C4—2441no—
487C4⋊C4D4×D6111yescharacteristic
488C4⋊C4C2×C22⋊D6111yescharacteristic
489C4⋊C4C2×C22⋊D6111yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—1221no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C42—21no—
499C42—41no—
500C42C23×D6111yescharacteristic
501C23—21no—
502C23—21no—
503C23—1221no—
504Q8—21no—
505Q8—21no—
506Q8—41no—
507Q8—41no—
508Q8—81no—
509Q8—2441no—
510Q8—2441no—
511Q8—2441no—
512Q8—2441no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—1221no—
517D4—1221no—
518D4—4882no—
519D4—4882no—
520D4#1451979712yes—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—161no—
532C2×C4—1221no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4C2×D4×D6111yescharacteristic
540C2×C4C22×C22⋊D6111yescharacteristic
541C2×C4#1451982211yescharacteristic
542C8—21no—
543C8—21no—
544C8—41no—
545C8—1221no—
546C8—1221no—
547C8—2441no—
548C8—2441no—
549C8—2441no—
550C8—2441no—
551C22—22no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—2441no—
558C22—2441no—
559C22—2441no—
560C22D4×C22⋊D6111yescharacteristic
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—4881no—
569C4—4881no—
570C4—2441no—
571C4C2×#1451979711yescharacteristic
572C4#1684055911yescharacteristic
573C4#1684056911yescharacteristic
574C4#1684057011yescharacteristic
575C2—42no—
576C2—81no—
577C2—4881no—
578C2#2145734411yescharacteristic
579C2#2145752711yescharacteristic
580C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3360C2
socleC22×C61D42
prime radical#21458126C2
hypercenter#3360D61
hypocenterC61#19824
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3360

Complements

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

NH ≅ G/Ntype
1. #21458170C2semidirect
3. #21457730C2semidirect
4. #21445466C2semidirect
5. #21445391C2semidirect
6. #21445249C2semidirect
7. #21445244C2semidirect
8. #21445196C2semidirect
9. #21445075C2semidirect
10. #21445052C2semidirect
11. #21445033C2semidirect
12. #21444969C2semidirect
13. #21444662C2semidirect
14. #21444643C2semidirect
15. C61×#3360C2semidirect
285. #3360D61semidirect
357. C61#19824semidirect

Maximal subgroups

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

subgroupindex
#214581702
#214581262
#214577302
#214454662
#214453912
#214452492
#214452442
#214451962
#214450752
#214450522
#214450332
#214449692
#214446622
#214446432
C61×#33602
#1982461