Finite Group Databasebeta

← #29677220

Subgroups — #29677220

Order 15616.

624 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
1#21458445C211yesmaximal, characteristic, complemented
2#21458416C211yesmaximal, characteristic
3#21457465C211yesmaximal, characteristic, complemented
4#21445414C211yesmaximal, characteristic, complemented
5#21445340C211yesmaximal, characteristic, complemented
6#21445325C211yesmaximal, characteristic, complemented
7#21445315C211yesmaximal, characteristic, complemented
8#21445306C211yesmaximal, characteristic, complemented
9#21445046C211yesmaximal, characteristic, complemented
10#21445033C211yesmaximal, characteristic, complemented
11#21444984C211yesmaximal, characteristic, complemented
12#21444969C211yesmaximal, characteristic, complemented
13#21444709C211yesmaximal, characteristic, complemented
14#21444704C211yesmaximal, characteristic, complemented
15C61×#3384C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17#16840477C2211yescharacteristic
18#16840417C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840160C2211yescharacteristic
21#16840141C2211yescharacteristic
22#16840114C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840100—21no—
25#16840085—21no—
26#16840079—21no—
27#16840075—21no—
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840049C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16840018—21no—
33#16840012—21no—
34#16840010—21no—
35#16839972C2211yescharacteristic
36#16839964C2211yescharacteristic
37#16839922—21no—
38#16839918—21no—
39C4×#11758201C2211yescharacteristic
40#16839841C2211yescharacteristic
41#16839840C2211yescharacteristic
42#16839840C2211yescharacteristic
43#16839803—21no—
44#16839802—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
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×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758201—21no—
73C2×#11758201D411yescharacteristic
74C2×#11758194—21no—
75C2×#11758194—21no—
76C2×#11758189—21no—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519763—41no—
80#14519759—41no—
81#14519756—21no—
82#14519751—21no—
83#14519748—21no—
84#14519748—21no—
85C4×D244C2311yescharacteristic
86#14519736—21no—
87#14519735—21no—
88C4⋊C4×D61—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519709—21no—
94#14519709—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708D411yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701—21no—
102#14519701—21no—
103#14519701—41no—
104#14519701D411yescharacteristic
105#14519700—21no—
106#14519700—41no—
107#14519698—21no—
108#14519698—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—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×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—41no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166D8×C61—22no—
167C2×C8×C61—21no—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188C22⋊Dic61—41no—
189#11758202—42no—
190#11758202—42no—
191#11758202—81no—
192#11758201—22no—
193#11758201—42no—
194#11758198—21no—
195#11758198—41no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—41no—
202C4×Dic61—41no—
203#11758194—42no—
204#11758194—42no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211#11758189—42no—
212C23×C61—21no—
213C23×C61—21no—
214C22×D61—21no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—81no—
220Q8×C61—82no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—82no—
236C2×C4×C61—161no—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C22⋊D61—82no—
241C22⋊D61—161no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245D244—21no—
246D244—21no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—82no—
250Dic122—41no—
251Dic122—41no—
252Dic122—81no—
253C8×C61—21no—
254C8×C61—41no—
255C8×C61#13712yescomplemented
256#92418—21no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418AGL(1,ℤ/8)12yes—
262#20315—611nomaximal, Sylow
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61D4211yescharacteristic
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C4×C61C2×#13711yescharacteristic
282C4×C61#57711yescharacteristic
283C4×C61#57811yescharacteristic
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287#3384D6111yescharacteristic, complemented
288#3357—611no—
289#3354—611no—
290#3276—611no—
291#1707—611no—
292#1695—611no—
293#1692—611no—
294#1691—611no—
295#1689—611no—
296#1646—611no—
297#1644—611no—
298#1637—611no—
299#1635—611no—
300#1590—611no—
301#1589—611no—
302C2×C61—81no—
303C2×C61—81no—
304C2×C61#327211yescharacteristic
305C2×C61#327611yescharacteristic
306C2×C61#329011yescharacteristic
307D61—81no—
308C2×#137—1221no—
309#555—611no—
310#554C2×D6111yescharacteristic
311D4×Q8C2×D6111yescharacteristic
312#547—611no—
313#501—21no—
314#500—611no—
315#498—611no—
316#498C2×D6111yescharacteristic
317#494—611no—
318#487—611no—
319#487—611no—
320#485—1221no—
321#482—1221no—
322#481—1221no—
323#480—1221no—
324#478—21no—
325#477—611no—
326#476—21no—
327#476—611no—
328#476—611no—
329#476C2×D6111yescharacteristic
330#474—611no—
331#474C2×D6111yescharacteristic
332#470—1221no—
333#469—1221no—
334#468—1221no—
335#462—611no—
336#461—611no—
337#452—1221no—
338#451—1221no—
339#446C2×D6111yescharacteristic
340#442—21no—
341C4×SD16—611no—
342C4×D8C2×D6111yescharacteristic
343#436—611no—
344#436—611no—
345#436—611no—
346#428—1221no—
347#428—1221no—
348#335—611no—
349#334—611no—
350#333—611no—
351#333—611no—
352#332—611no—
353#332—611no—
354Q8⋊C8—611no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358C82—611no—
359C61#2031511yescharacteristic, complemented, Sylow
360C2×#41—1221no—
361C22×Q8—21no—
362#137—2444no—
363C2×Q16—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×SD16—611no—
367C2×SD16—1221no—
368C2×D8C22⋊D6111yescharacteristic
369C2×M16—1221no—
370C2×M16—1221no—
371C2×M16—1221no—
372C4⋊Q8—21no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377#125—21no—
378#124—2441no—
379#123—2441no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—41no—
383C22⋊Q8—41no—
384C22⋊Q8—1221no—
385#121—1221no—
386#121—1221no—
387#121—1221no—
388C4×Q8C22×D6111yescharacteristic
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—611no—
392C4×D4C22×D6111yescharacteristic
393C4×D4C22×D6111yescharacteristic
394#117—1221no—
395#117—1221no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—1221no—
398#107—21no—
399#107—1221no—
400#107—1221no—
401#107—1221no—
402#107—1221no—
403#107C22⋊D6111yescharacteristic
404#106—21no—
405#106—21no—
406#106—21no—
407#106—611no—
408#106—1221no—
409#106—1221no—
410#106—1221no—
411#106—1221no—
412C4⋊C8—21no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8C22×D6111yescharacteristic
417#103—611no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—2441no—
422#103—2441no—
423#102—21no—
424#102—21no—
425#102—611no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102C22⋊D6111yescharacteristic
430#102C22⋊D6111yescharacteristic
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434#97—21no—
435C4×C8—611no—
436C4×C8—611no—
437C4×C8C22×D6111yescharacteristic
438#41—4881no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—81no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8D4×D6111yescharacteristic
448C2×D4—611no—
449C2×D4—2441no—
450C2×D4—2441no—
451C2×D4C2×C22⋊D6111yescharacteristic
452C2×D4C2×C22⋊D6111yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—1221no—
458C22×C4—2441no—
459Q16—4881no—
460Q16—2442no—
461Q16—2442no—
462SD16—1222no—
463SD16—2442no—
464D8—22no—
465M16—2442no—
466M16—2442no—
467M16—2442no—
468C2×C8—21no—
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—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—611no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—2441no—
491C4⋊C4D4×D6111yescharacteristic
492C4⋊C4C2×C22⋊D6111yescharacteristic
493C4⋊C4C2×C22⋊D6111yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—1221no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C42—21no—
503C42—2441no—
504C42C23×D6111yescharacteristic
505C23—21no—
506C23—21no—
507C23—1221no—
508Q8—21no—
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—81no—
513Q8—82no—
514Q8—4881no—
515Q8—2441no—
516Q8—2441no—
517D4—21no—
518D4—21no—
519D4—21no—
520D4—21no—
521D4—1221no—
522D4—1221no—
523D4—9761no—
524D4—4882no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—161no—
537C2×C4—1221no—
538C2×C4—4882no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4#1451982211yescharacteristic
546C8—21no—
547C8—41no—
548C8—612no—
549C8—1221no—
550C8—2441no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C8#1451980812yescomplemented
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22D4×C22⋊D6111yescharacteristic
565C4—21no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—4881no—
573C4—4881no—
574C4—2441no—
575C4AGL(1,ℤ/8)×D6111yescharacteristic
576C4C2×#1451980811yescharacteristic
577C4#1684057211yescharacteristic
578C4#1684057711yescharacteristic
579C2—81no—
580C2—81no—
581C2—4881no—
582C2#2145741311yescharacteristic
583C2#2145746511yescharacteristic
584C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3384C2
socleC22×C61D42
prime radical#21458416C2
hypercenter#3384D61
hypocenterC61#20315
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3384

Complements

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

NH ≅ G/Ntype
1. #21458445C2semidirect
3. #21457465C2semidirect
4. #21445414C2semidirect
5. #21445340C2semidirect
6. #21445325C2semidirect
7. #21445315C2semidirect
8. #21445306C2semidirect
9. #21445046C2semidirect
10. #21445033C2semidirect
11. #21444984C2semidirect
12. #21444969C2semidirect
13. #21444709C2semidirect
14. #21444704C2semidirect
15. C61×#3384C2semidirect
255. C8×C61#137semidirect
287. #3384D61semidirect
359. C61#20315semidirect
554. C8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214584452
#214584162
#214574652
#214454142
#214453402
#214453252
#214453152
#214453062
#214450462
#214450332
#214449842
#214449692
#214447092
#214447042
C61×#33842
#2031561