Finite Group Databasebeta

← #29676854

Subgroups — #29676854

Order 15616.

634 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458535C211yesmaximal, characteristic, complemented
2#21458107C211yesmaximal, characteristic, complemented
3#21458050C211yesmaximal, characteristic, complemented
4#21445430C211yesmaximal, characteristic, complemented
5#21445396C211yesmaximal, characteristic, complemented
6#21445378C211yesmaximal, characteristic, complemented
7#21445366C211yesmaximal, characteristic, complemented
8#21445356C211yesmaximal, characteristic, complemented
9#21445047C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444989C211yesmaximal, characteristic, complemented
12#21444921C211yesmaximal, characteristic, complemented
13#21444707C211yesmaximal, characteristic, complemented
14#21444704C211yesmaximal, characteristic, complemented
15C61×#3325C211yesmaximal, characteristic
16#16840482C2211yescharacteristic
17D4×Dic122C2211yescharacteristic, complemented
18#16840120—21no—
19#16840114C2211yescharacteristic, complemented
20#16840113C2211yescharacteristic, complemented
21#16840112C2211yescharacteristic, complemented
22#16840108C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840103—21no—
25#16840083—21no—
26#16840056—21no—
27#16840054C2211yescharacteristic, complemented
28#16840042C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840006—21no—
31#16840006—21no—
32#16839975—21no—
33#16839967C2211yescharacteristic
34#16839967C2211yescharacteristic, complemented
35#16839959C2211yescharacteristic
36#16839913—21no—
37#16839900—21no—
38#16839892C2211yescharacteristic, complemented
39#16839859—21no—
40C4×#11758200C2211yescharacteristic
41#16839840C2211yescharacteristic
42#16839840C2211yescharacteristic, complemented
43#16839770—21no—
44#16839767—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51#16839492C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#555C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#476C2211yescharacteristic
61C61×#469—21no—
62C61×#459C2211yescharacteristic
63C61×#447—21no—
64C4×D8×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#418—21no—
67C22×Dic122—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200D411yescharacteristic
72C2×#11758194—21no—
73C2×#11758194—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519770—41no—
78#14519764—21no—
79#14519756—21no—
80#14519756—21no—
81#14519754—41no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84D4×Dic61—21no—
85#14519736—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519708D411yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519706—21no—
94#14519702—21no—
95#14519702—21no—
96#14519701—41no—
97#14519700—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—
111C22⋊#92418—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×D4×C61—21no—
116C22×D4×C61—21no—
117C2×D8×C61—21no—
118C2×D8×C61D411yescharacteristic
119C2×M16×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#124×C61—41no—
123#121×C61—21no—
124#121×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#120×C61—41no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#102×C61—21no—
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145C2×Q8×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—41no—
150C2×D4×C61—41no—
151C2×D4×C61—42no—
152C2×D4×C61—42no—
153C2×D4×C61—41no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×Dic61—41no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—42no—
166C2×Dic122—41no—
167C2×Dic122—81no—
168C2×Dic122C2×D411yescharacteristic
169D8×C61—41no—
170D8×C61—81no—
171M16×C61—81no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—41no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189C22⋊Dic61—41no—
190C22⋊Dic61—41no—
191#11758200—41no—
192#11758200—42no—
193#11758200—81no—
194#11758200—81no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—21no—
201#11758196—41no—
202#11758196—41no—
203C4×Dic61—21no—
204C4×Dic61—41no—
205#11758194—81no—
206#11758194—81no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—41no—
217C23×C61—82no—
218C23×C61—82no—
219C23×C61—161no—
220Q8×C61—81no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—81no—
232D4×C61—82no—
233D4×C61—82no—
234D4×C61—161no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—82no—
253Dic122—21no—
254Dic122—21no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259Dic122—82no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—41no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#20287—611nomaximal, Sylow
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—164no—
283C22×C61D4211yescharacteristic
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61#57511yescharacteristic
290C4×C61#57511yescharacteristic
291C4×C61#57611yescharacteristic
292C4×C61#57711yescharacteristic
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61—81no—
297#3367—611no—
298#3329—611no—
299#3325D6111yescharacteristic
300#3324—611no—
301#1711—611no—
302#1703—611no—
303#1701—611no—
304#1699—611no—
305#1698—611no—
306#1646—611no—
307#1639—611no—
308#1638—611no—
309#1627—611no—
310#1590—611no—
311#1589—611no—
312C2×C61—81no—
313C2×C61—81no—
314C2×C61—81no—
315C2×C61—161no—
316C2×C61#327311yescharacteristic
317C2×C61#327511yescharacteristic
318C2×C61#329011yescharacteristic
319#556—611no—
320#555C2×D6111yescharacteristic
321D4×Q8—611no—
322D42C2×D6111yescharacteristic
323#489—1221no—
324#487—611no—
325#487—611no—
326#487—611no—
327#486—611no—
328#486—611no—
329#486C2×D6111yescharacteristic
330#485—1221no—
331#482—21no—
332#482—1221no—
333#477—611no—
334#477—1221no—
335#476C2×D6111yescharacteristic
336#474—611no—
337#474—611no—
338#469—21no—
339#468—1221no—
340#468—1221no—
341#463—1221no—
342#461—611no—
343#461—611no—
344#460—611no—
345#459C2×D6111yescharacteristic
346#450—1221no—
347#448—1221no—
348#447—21no—
349#446—611no—
350#440—1221no—
351C4×SD16—611no—
352C4×D8C2×D6111yescharacteristic
353#436—611no—
354#436—611no—
355#436C2×D6111yescharacteristic
356#419—1221no—
357#418—21no—
358#418—1221no—
359#335—611no—
360#334—611no—
361#333—611no—
362#332—611no—
363#332—611no—
364#331—611no—
365Q8⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369C82—611no—
370C61#2028711yescharacteristic, complemented, Sylow
371C22×Q8—1221no—
372C22×D4—21no—
373C22×D4—21no—
374C2×SD16—611no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—21no—
379C2×D8C22⋊D6111yescharacteristic
380C2×M16—21no—
381C2×M16—1221no—
382C2×M16—1221no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8—1221no—
386C4⋊Q8C22×D6111yescharacteristic
387#127C22×D6111yescharacteristic
388#126—2441no—
389#124—41no—
390#124—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—2441no—
394#121—21no—
395#121—21no—
396#121—21no—
397#121—21no—
398#120—41no—
399C4×Q8—611no—
400C4×Q8—1221no—
401C4×D4—1221no—
402C4×D4C22×D6111yescharacteristic
403C4×D4C22×D6111yescharacteristic
404C4×D4C22×D6111yescharacteristic
405#117—21no—
406#117—21no—
407#117—1221no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107C22⋊D6111yescharacteristic
412#106—21no—
413#106—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8C22×D6111yescharacteristic
419#103—611no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—2441no—
428#102—21no—
429#102—21no—
430#102—21no—
431#102—21no—
432#102—611no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—2441no—
439#102C22⋊D6111yescharacteristic
440#102C22⋊D6111yescharacteristic
441C22⋊C8—21no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444#97—1221no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8C22×D6111yescharacteristic
448C24—41no—
449C2×Q8—21no—
450C2×Q8—611no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—4881no—
454C2×Q8—2441no—
455C2×Q8—2442no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—41no—
464C2×D4—81no—
465C2×D4—81no—
466C2×D4D4×D6111yescharacteristic
467C2×D4C2×C22⋊D6111yescharacteristic
468C2×D4C2×C22⋊D6111yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—2441no—
473SD16—4881no—
474SD16—4881no—
475SD16—2441no—
476SD16—2442no—
477D8—41no—
478D8—81no—
479M16—81no—
480M16—4881no—
481M16—4881no—
482C2×C8—21no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8C2×C22⋊D6111yescharacteristic
490C2×C8C2×C22⋊D6111yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—611no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C42—41no—
514C42—1221no—
515C42—2441no—
516C42C23×D6111yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—41no—
521C23—82no—
522C23—82no—
523C23—161no—
524Q8—81no—
525Q8—1221no—
526Q8—1221no—
527Q8—4882no—
528Q8—2441no—
529Q8—2441no—
530Q8—2441no—
531Q8—2441no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—21no—
536D4—21no—
537D4—21no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—81no—
543D4—82no—
544D4—82no—
545D4—161no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—4882no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4#1451982211yescharacteristic
564C8—21no—
565C8—21no—
566C8—41no—
567C8—1221no—
568C8—1221no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C8—2441no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—164no—
586C22D4×C22⋊D6111yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—4881no—
593C4—4881no—
594C4—2441no—
595C4—2441no—
596C4#1684055811yescharacteristic
597C4#1684056011yescharacteristic
598C4#1684056611yescharacteristic
599C4#1684057311yescharacteristic
600C2—81no—
601C2—81no—
602C2—81no—
603C2—161no—
604C2#2145742411yescharacteristic
605C2#2145744611yescharacteristic
606C2#2145760111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3325C2
socleC22×C61D42
hypercenter#3325D61
hypocenterC61#20287
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3325

Complements

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

NH ≅ G/Ntype
1. #21458535C2semidirect
2. #21458107C2semidirect
3. #21458050C2semidirect
4. #21445430C2semidirect
5. #21445396C2semidirect
6. #21445378C2semidirect
7. #21445366C2semidirect
8. #21445356C2semidirect
9. #21445047C2semidirect
10. #21444998C2semidirect
11. #21444989C2semidirect
12. #21444921C2semidirect
13. #21444707C2semidirect
14. #21444704C2semidirect
17. D4×Dic122C22semidirect
19. #16840114C22semidirect
20. #16840113C22semidirect
21. #16840112C22semidirect
27. #16840054C22semidirect
34. #16839967C22semidirect
38. #16839892C22semidirect
42. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839509C22semidirect
49. #16839507C22semidirect
51. #16839492C22semidirect
54. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
370. C61#20287semidirect

Maximal subgroups

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

subgroupindex
#214585352
#214581072
#214580502
#214454302
#214453962
#214453782
#214453662
#214453562
#214450472
#214449982
#214449892
#214449212
#214447072
#214447042
C61×#33252
#2028761