Finite Group Databasebeta

← #29670980

Subgroups — #29670980

Order 15616.

594 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#21458803C211yesmaximal, characteristic, complemented
2#21458405C211yesmaximal, characteristic, complemented
3#21457741C211yesmaximal, characteristic
4#21445493C211yesmaximal, characteristic, complemented
5#21445315C211yesmaximal, characteristic, complemented
6#21445279C211yesmaximal, characteristic, complemented
7#21445208C211yesmaximal, characteristic, complemented
8#21445182C211yesmaximal, characteristic, complemented
9#21445075C211yesmaximal, characteristic, complemented
10#21445053C211yesmaximal, characteristic, complemented
11#21445047C211yesmaximal, characteristic, complemented
12#21444990C211yesmaximal, characteristic, complemented
13#21444648C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3340C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17#16840417C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840144—21no—
21#16840141C2211yescharacteristic
22#16840114C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840103—21no—
25#16840096—21no—
26#16840084—21no—
27#16840049C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840005—21no—
31#16839989—21no—
32#16839984—21no—
33#16839981—21no—
34#16839967C2211yescharacteristic
35#16839967C2211yescharacteristic
36#16839964C2211yescharacteristic
37#16839905—21no—
38Q8×#92418C2211yescharacteristic
39C4×#11758202—21no—
40C4×#11758199C2211yescharacteristic
41#16839841C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839801—21no—
44#16839756—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#556C2211yescharacteristic
57C61×#555C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#485—21no—
60C61×#482—21no—
61C61×#477C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C61×#465—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C61×#427—21no—
67C2×#11758222—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758199D411yescharacteristic
74C2×#11758189—21no—
75C22×#92418—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519772—41no—
79#14519765—21no—
80#14519759—41no—
81#14519756—21no—
82#14519753—21no—
83#14519748—21no—
84C4×Dic122—21no—
85C4×D244C2311yescharacteristic
86#14519735—21no—
87C4⋊C4×D61—21no—
88#14519711—21no—
89#14519711—21no—
90#14519710—21no—
91#14519710—41no—
92#14519710D411yescharacteristic
93#14519708—21no—
94#14519708—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519701—41no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×SD16×C61D411yescharacteristic
116C22×C8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—41no—
120#124×C61—21no—
121#124×C61—41no—
122C22⋊Q8×C61—21no—
123#121×C61—21no—
124#121×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61C2311yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131#117×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—41no—
148C2×D4×C61C2×D411yescharacteristic
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C2×C22⋊D61—41no—
153#11758222—42no—
154#11758222—81no—
155C2×D244C2×D411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—41no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160SD16×C61—22no—
161C2×C8×C61—21no—
162C2×C8×C61—42no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C42×C61—21no—
181C42×C61—41no—
182C42×C61—41no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185#11758202—42no—
186#11758202—81no—
187#11758201—81no—
188#11758200—81no—
189#11758200—81no—
190#11758199—41no—
191#11758198—21no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—41no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—81no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758189—42no—
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—21no—
222D4×C61—161no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C22⋊D61—82no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240D244—21no—
241D244—21no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—82no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249C8×C61—21no—
250C8×C61—41no—
251C8×C61#13512yes—
252#92418—21no—
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#19863—611nomaximal, Sylow
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61D4211yescharacteristic
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
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#57611yescharacteristic
279C4×C61#57811yescharacteristic
280Dic61—41no—
281Dic61—81no—
282Dic61—81no—
283#3393—611no—
284#3353—611no—
285#3340D6111yescharacteristic, complemented
286#3303—611no—
287#1723—611no—
288#1691—611no—
289#1684—611no—
290#1673—611no—
291#1669—611no—
292#1650—611no—
293#1647—611no—
294#1646—611no—
295#1638—611no—
296#1581—611no—
297SD16⋊C8—611no—
298C2×C61—81no—
299C2×C61—81no—
300C2×C61#326811yescharacteristic
301C2×C61#328311yescharacteristic
302C2×C61#328911yescharacteristic
303D61—81no—
304#556—611no—
305#556C2×D6111yescharacteristic
306#555C2×D6111yescharacteristic
307#547—611no—
308#500—611no—
309#500—611no—
310#498C2×D6111yescharacteristic
311#495—1221no—
312#494—611no—
313#487—611no—
314#487—611no—
315#485—21no—
316#485—1221no—
317#484—1221no—
318#482—21no—
319#482—1221no—
320#477C2×D6111yescharacteristic
321#476—611no—
322#475—611no—
323#474—611no—
324#474C2×D6111yescharacteristic
325#468—1221no—
326#465—21no—
327#465—1221no—
328#464—1221no—
329#464—1221no—
330#461—611no—
331#461—611no—
332#461—611no—
333#449—1221no—
334C8×Q8—611no—
335C4×Q16—1221no—
336C4×SD16C2×D6111yescharacteristic
337C4×D8—611no—
338#436—611no—
339C8×D4—611no—
340C8×D4C2×D6111yescharacteristic
341#428—1221no—
342#427—21no—
343#416—1221no—
344#335—611no—
345#335—611no—
346#333—611no—
347#332—611no—
348#332—611no—
349#332—611no—
350Q8⋊C8—611no—
351D4⋊C8—611no—
352D4⋊C8—611no—
353D4⋊C8—611no—
354#322—611no—
355C61#1986311yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×Q16—1221no—
358C2×Q16—1221no—
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×SD16C22⋊D6111yescharacteristic
363C2×D8—611no—
364C2×M16—1221no—
365C22×C8—21no—
366C22×C8—1221no—
367C4⋊Q8—611no—
368C4⋊Q8—611no—
369C4⋊Q8C22×D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371#126—41no—
372#126—2441no—
373#124—21no—
374#124—41no—
375#124—1221no—
376#123—2441no—
377C22⋊Q8—21no—
378C22⋊Q8—1221no—
379C22⋊Q8—1221no—
380#121—21no—
381#121—21no—
382#121—1221no—
383C4×Q8—21no—
384C4×Q8—1221no—
385C4×Q8C22×D6111yescharacteristic
386C4×D4—611no—
387C4×D4C22×D6111yescharacteristic
388C4×D4C22×D6111yescharacteristic
389#117—21no—
390#117—21no—
391#117—21no—
392#117—1221no—
393C2×C4⋊C4—1221no—
394#107—21no—
395#107—21no—
396#107—611no—
397#107—1221no—
398#107—1221no—
399#107—1221no—
400#107—2441no—
401#106—21no—
402#106—21no—
403#106—1221no—
404#106—1221no—
405#106C22⋊D6111yescharacteristic
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411#103—21no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—2441no—
419#103C22⋊D6111yescharacteristic
420#102—21no—
421#102—611no—
422#102—611no—
423#102—1221no—
424#102—1221no—
425#102C22⋊D6111yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—1221no—
428C22⋊C8—1221no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—1221no—
432C4×C8C22×D6111yescharacteristic
433#41—4881no—
434#41—2442no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×D4—41no—
441C2×D4—611no—
442C2×D4—2441no—
443C2×D4D4×D6111yescharacteristic
444C2×D4C2×C22⋊D6111yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1221no—
448C22×C4—2441no—
449Q16—4881no—
450Q16—2442no—
451SD16—22no—
452SD16—4881no—
453SD16—4881no—
454SD16—4881no—
455D8—2441no—
456M16—2442no—
457C2×C8—21no—
458C2×C8—42no—
459C2×C8—611no—
460C2×C8—611no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—4881no—
466C2×C8C2×C22⋊D6111yescharacteristic
467C2×C8C2×C22⋊D6111yescharacteristic
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—611no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4D4×D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C4⋊C4C2×C22⋊D6111yescharacteristic
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—1221no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C42—21no—
495C42—41no—
496C42—41no—
497C42—2441no—
498C42C23×D6111yescharacteristic
499C23—21no—
500C23—21no—
501C23—1221no—
502Q8—21no—
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—81no—
507Q8—2441no—
508Q8—2441no—
509Q8—2441no—
510Q8—2441no—
511D4—21no—
512D4—21no—
513D4—21no—
514D4—21no—
515D4—161no—
516D4—1221no—
517D4—1221no—
518D4—4882no—
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—1221no—
530C2×C4—4882no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C22×C22⋊D6111yescharacteristic
537C2×C4#1451982211yescharacteristic
538C8—21no—
539C8—41no—
540C8—1221no—
541C8—1221no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8#1451979712yes—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—2441no—
554C22—2441no—
555C22—2441no—
556C22D4×C22⋊D6111yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—4881no—
565C4—4881no—
566C4—2441no—
567C4C2×#1451979711yescharacteristic
568C4#1684055911yescharacteristic
569C4#1684056411yescharacteristic
570C4#1684057711yescharacteristic
571C2—81no—
572C2—81no—
573C2—4881no—
574C2#2145735911yescharacteristic
575C2#2145753711yescharacteristic
576C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3340C2
socleC22×C61D42
prime radical#21457741C2
hypercenter#3340D61
hypocenterC61#19863
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3340

Complements

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

NH ≅ G/Ntype
1. #21458803C2semidirect
2. #21458405C2semidirect
4. #21445493C2semidirect
5. #21445315C2semidirect
6. #21445279C2semidirect
7. #21445208C2semidirect
8. #21445182C2semidirect
9. #21445075C2semidirect
10. #21445053C2semidirect
11. #21445047C2semidirect
12. #21444990C2semidirect
13. #21444648C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3340C2semidirect
285. #3340D61semidirect
355. C61#19863semidirect

Maximal subgroups

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

subgroupindex
#214588032
#214584052
#214577412
#214454932
#214453152
#214452792
#214452082
#214451822
#214450752
#214450532
#214450472
#214449902
#214446482
SD16⋊#924182
C61×#33402
#1986361