Finite Group Databasebeta

← #29670456

Subgroups — #29670456

Order 15616.

614 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1SD16×Dic122C211yesmaximal, characteristic, complemented
2#21458486C211yesmaximal, characteristic, complemented
3#21458107C211yesmaximal, characteristic, complemented
4#21445481C211yesmaximal, characteristic, complemented
5#21445350C211yesmaximal, characteristic, complemented
6#21445279C211yesmaximal, characteristic, complemented
7#21445232C211yesmaximal, characteristic, complemented
8#21445194C211yesmaximal, characteristic, complemented
9#21445019C211yesmaximal, characteristic, complemented
10#21444964C211yesmaximal, characteristic, complemented
11#21444958C211yesmaximal, characteristic, complemented
12#21444921C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3305C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17D4×Dic122C2211yescharacteristic
18#16840160C2211yescharacteristic
19#16840159—21no—
20#16840141C2211yescharacteristic
21#16840114C2211yescharacteristic
22#16840108—21no—
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840103—21no—
26#16840054C2211yescharacteristic
27#16840052—21no—
28#16840052C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840050—21no—
31#16840047C2211yescharacteristic
32#16840045—21no—
33#16840006—21no—
34#16839967C2211yescharacteristic
35#16839959—21no—
36#16839959C2211yescharacteristic
37#16839900—21no—
38Q8×#92418C2211yescharacteristic
39C8×Dic122C2211yescharacteristic
40SD16×Dic61—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839770—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#555C2211yescharacteristic
57C61×#550C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#477C2211yescharacteristic
61C61×#466—21no—
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C61×#416—21no—
67C22×Dic122—21no—
68C2×#11758202—41no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200D411yescharacteristic
72C2×#11758194—21no—
73#14519777—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519776—21no—
78#14519776—41no—
79#14519756—21no—
80#14519754—41no—
81Q8×Dic61—21no—
82C4×Dic122—21no—
83C4×Dic122C2311yescharacteristic
84D4×Dic61—21no—
85#14519710—21no—
86#14519709—21no—
87#14519709—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708D411yescharacteristic
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C2×D8×C61—21no—
118C22×C8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120#127×C61—21no—
121#127×C61C2311yescharacteristic
122#124×C61—41no—
123#121×C61—21no—
124#121×C61—21no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61—21no—
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
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—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×Dic61—41no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—42no—
163C2×Dic122—41no—
164C2×Dic122—41no—
165C2×Dic122—81no—
166C2×Dic122C2×D411yescharacteristic
167SD16×C61—22no—
168D8×C61—81no—
169C2×C8×C61—21no—
170C2×C8×C61—81no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
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—82no—
189#11758200—22no—
190#11758200—42no—
191#11758200—81no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—21no—
202C4×Dic61—41no—
203#11758194—81no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—41no—
213Q8×C61—21no—
214Q8×C61—81no—
215Q8×C61#13712yes—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—82no—
225D4×C61—81no—
226D4×C61—161no—
227D4×C61—161no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—82no—
246Dic122—21no—
247Dic122—41no—
248Dic122—41no—
249Dic122—82no—
250Dic122—81no—
251Dic122—161no—
252Dic122#13512yes—
253C8×C61—21no—
254C8×C61—21no—
255C8×C61—41no—
256#92418—21no—
257#92418—21no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#19828—611nomaximal, Sylow
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61D4211yescharacteristic
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61C2×#13511yescharacteristic
280C4×C61C2×#13711yescharacteristic
281C4×C61#57711yescharacteristic
282C4×C61#57711yescharacteristic
283Dic61—22no—
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287Q8×SD16—611no—
288#3362—611no—
289#3329—611no—
290#3305D6111yescharacteristic
291#1720—611no—
292#1697—611no—
293#1684—611no—
294#1678—611no—
295#1671—611no—
296#1642—611no—
297#1634—611no—
298#1633—611no—
299#1627—611no—
300#1583—611no—
301SD16⋊C8—611no—
302C2×C61—81no—
303C2×C61—81no—
304C2×C61—161no—
305C2×C61#326611yescharacteristic
306C2×C61#328211yescharacteristic
307C2×C61#328811yescharacteristic
308Q82—611no—
309#555C2×D6111yescharacteristic
310#550C2×D6111yescharacteristic
311D4×Q8—611no—
312#498—611no—
313#498—1221no—
314#494—611no—
315#487—611no—
316#486—611no—
317#486—611no—
318#486—1221no—
319#486C2×D6111yescharacteristic
320#485—1221no—
321#482—21no—
322#477—611no—
323#477C2×D6111yescharacteristic
324#476—611no—
325#476—611no—
326#476—1221no—
327#476—1221no—
328#475—611no—
329#475—1221no—
330#468—1221no—
331#466—21no—
332#461—611no—
333#460—611no—
334#460—1221no—
335#459C2×D6111yescharacteristic
336#449—21no—
337#448—1221no—
338C8×Q8—611no—
339C8×Q8—611no—
340C4×SD16—611no—
341C4×SD16—1221no—
342C4×SD16—1221no—
343C4×SD16C2×D6111yescharacteristic
344#436—611no—
345C8×D4C2×D6111yescharacteristic
346#419—1221no—
347#416—21no—
348#335—611no—
349#334—611no—
350#333—611no—
351#333—611no—
352#332—611no—
353#331—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358#322—611no—
359C61#1982811yescharacteristic, complemented, Sylow
360C2×#41—21no—
361C22×Q8—1221no—
362C2×Q16—2441no—
363C2×SD16—611no—
364C2×SD16—1221no—
365C2×SD16—1221no—
366C2×SD16C22⋊D6111yescharacteristic
367C2×D8—21no—
368C2×M16—1221no—
369C22×C8—21no—
370C4⋊Q8—611no—
371C4⋊Q8—611no—
372C4⋊Q8—1221no—
373C4⋊Q8—1221no—
374C4⋊Q8—1221no—
375C4⋊Q8—2441no—
376C4⋊Q8C22×D6111yescharacteristic
377#127—21no—
378#127C22×D6111yescharacteristic
379#124—41no—
380C22⋊Q8—1221no—
381C22⋊Q8—2441no—
382#121—21no—
383#121—21no—
384#121—21no—
385#121—41no—
386C4×Q8—611no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×Q8C22×D6111yescharacteristic
390C4×D4—21no—
391C4×D4—1221no—
392C4×D4C22×D6111yescharacteristic
393C4×D4C22×D6111yescharacteristic
394#117—21no—
395#117—21no—
396#107—21no—
397#107—1221no—
398#106—611no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402#106—1221no—
403#106—1221no—
404#106C22⋊D6111yescharacteristic
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—1221no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411#103—21no—
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103C22⋊D6111yescharacteristic
419#102—21no—
420#102—21no—
421#102—21no—
422#102—611no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—2441no—
429#102C22⋊D6111yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—1221no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—1221no—
435C4×C8—1221no—
436C4×C8C22×D6111yescharacteristic
437#41—42no—
438#41—81no—
439C2×Q8—21no—
440C2×Q8—611no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8—4881no—
444C2×Q8—2441no—
445C2×Q8—2441no—
446C2×Q8—2442no—
447C2×Q8C2×C22⋊D6111yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4D4×D6111yescharacteristic
455C2×D4C2×C22⋊D6111yescharacteristic
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—41no—
459C22×C4—2441no—
460Q16—4882no—
461SD16—22no—
462SD16—1222no—
463SD16—4881no—
464SD16—2442no—
465D8—81no—
466M16—4881no—
467C2×C8—21no—
468C2×C8—81no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8C2×C22⋊D6111yescharacteristic
476C2×C8C2×C22⋊D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—611no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2441no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C4⋊C4D4×D6111yescharacteristic
490C4⋊C4C2×C22⋊D6111yescharacteristic
491C4⋊C4C2×C22⋊D6111yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—2441no—
498C42—21no—
499C42—41no—
500C42—1221no—
501C42—2441no—
502C42C23×D6111yescharacteristic
503C23—21no—
504C23—21no—
505C23—41no—
506Q8—21no—
507Q8—81no—
508Q8—612no—
509Q8—1221no—
510Q8—9761no—
511Q8—4881no—
512Q8—4882no—
513Q8—2441no—
514Q8—2441no—
515Q8#1451980712yes—
516D4—21no—
517D4—21no—
518D4—21no—
519D4—21no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—82no—
525D4—81no—
526D4—161no—
527D4—161no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—82no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—4882no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4#1451982211yescharacteristic
546C8—21no—
547C8—21no—
548C8—41no—
549C8—1221no—
550C8—1221no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22D4×C22⋊D6111yescharacteristic
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—1222no—
572C4—4881no—
573C4—4881no—
574C4—2441no—
575C4#135×D6111yescharacteristic
576C4C2×#1451980711yescharacteristic
577C4#1684056911yescharacteristic
578C4#1684057311yescharacteristic
579C2—81no—
580C2—81no—
581C2—161no—
582C2#2145732511yescharacteristic
583C2#2145752011yescharacteristic
584C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3305C2
socleC22×C61D42
hypercenter#3305D61
hypocenterC61#19828
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3305

Complements

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

NH ≅ G/Ntype
1. SD16×Dic122C2semidirect
2. #21458486C2semidirect
3. #21458107C2semidirect
4. #21445481C2semidirect
5. #21445350C2semidirect
6. #21445279C2semidirect
7. #21445232C2semidirect
8. #21445194C2semidirect
9. #21445019C2semidirect
10. #21444964C2semidirect
11. #21444958C2semidirect
12. #21444921C2semidirect
13. #21444663C2semidirect
14. #21444631C2semidirect
359. C61#19828semidirect

Maximal subgroups

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

subgroupindex
SD16×Dic1222
#214584862
#214581072
#214454812
#214453502
#214452792
#214452322
#214451942
#214450192
#214449642
#214449582
#214449212
#214446632
#214446312
C61×#33052
#1982861