Finite Group Databasebeta

← #29663476

Subgroups — #29663476

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457150C211yesmaximal, characteristic, complemented
2#21457139C211yesmaximal, characteristic
3#21454794C211yesmaximal, characteristic, complemented
4#21445322C211yesmaximal, characteristic, complemented
5#21445315C211yesmaximal, characteristic, complemented
6#21445288C211yesmaximal, characteristic, complemented
7#21445281C211yesmaximal, characteristic, complemented
8#21445274C211yesmaximal, characteristic, complemented
9#21445267C211yesmaximal, characteristic, complemented
10#21445260C211yesmaximal, characteristic, complemented
11#21445253C211yesmaximal, characteristic, complemented
12C2×#16839515C211yesmaximal, characteristic, complemented
13#21444519C211yesmaximal, characteristic, complemented
14#21444514C211yesmaximal, characteristic, complemented
15C61×#3167C211yesmaximal, characteristic, complemented
16#16840296C2211yescharacteristic
17#16840069—21no—
18#16840062—21no—
19#16840054C2211yescharacteristic
20#16840049C2211yescharacteristic
21#16840044C2211yescharacteristic
22#16840039C2211yescharacteristic
23#16840001—21no—
24#16839996—21no—
25#16839993—21no—
26#16839989—21no—
27#16839984—21no—
28#16839977—21no—
29#16839969C2211yescharacteristic
30#16839964C2211yescharacteristic
31#16839959C2211yescharacteristic
32#16839954C2211yescharacteristic
33#16839892C2211yescharacteristic
34C8×Dic122C2211yescharacteristic
35#16839841C2211yescharacteristic
36#16839833—21no—
37C8×D244C2211yescharacteristic
38C8×C22⋊D61—21no—
39#16839821—21no—
40#16839819C2211yescharacteristic
41C2×C4⋊#92418C2211yescharacteristic
42C2×C4⋊#92418C2211yescharacteristic
43#16839716—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839515C2211yescharacteristic
47#16839515C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#532C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#465—21no—
63C61×#465—21no—
64C4×SD16×C61—21no—
65C4×SD16×C61—21no—
66C2×C4×C8×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758201—21no—
70C2×#11758199—21no—
71C2×#11758189—21no—
72C22×#92418—21no—
73C22×#92418—21no—
74C2×C8×D61—21no—
75#14519771—21no—
76#14519769D411yescharacteristic
77#14519765D411yescharacteristic
78#14519760—21no—
79#14519756D411yescharacteristic
80#14519751D411yescharacteristic
81C4×Dic122C2311yescharacteristic
82C4×D244C2311yescharacteristic
83C4×C22⋊D61—21no—
84#14519734—21no—
85#14519710—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519706—21no—
98#14519706—21no—
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519700—21no—
103#14519698—21no—
104#14519698—21no—
105#14519697—21no—
106#14519697—21no—
107C22⋊#92418—21no—
108#14519689—21no—
109#14519689—21no—
110#14519688—21no—
111#14519687—21no—
112C8×Dic61—21no—
113C2×#41×C61—21no—
114C2×SD16×C61—21no—
115C2×SD16×C61—21no—
116C22×C8×C61D411yescharacteristic
117C22×C8×C61D411yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#125×C61—21no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132#117×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136#103×C61—21no—
137#103×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140C4×C8×C61D411yescharacteristic
141C4×C8×C61D411yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—82no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C22×C4×C61—41no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4111yescharacteristic
156C2×C22⋊D61—21no—
157C2×D244C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×Dic122C2×D411yescharacteristic
160SD16×C61—42no—
161SD16×C61—42no—
162C2×C8×C61—22no—
163C2×C8×C61—41no—
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
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—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C42×C61—41no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61#4111yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61—21no—
185#11758202—81no—
186#11758201—81no—
187#11758200—81no—
188#11758199—81no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193C4⋊Dic61C2×D411yescharacteristic
194C4⋊Dic61C2×D411yescharacteristic
195#11758196—21no—
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199C4×Dic61—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—81no—
207C2×#92418—81no—
208#11758189—42no—
209C8×D61—42no—
210C23×C61—41no—
211C23×C61#12111yescharacteristic
212C22×D61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—81no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—81no—
223D4×C61—82no—
224C2×C4×C61—22no—
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—82no—
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61#12111yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C22⋊D61—42no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243D244—21no—
244D244—21no—
245C4×D61—41no—
246C4×D61—41no—
247Dic122—21no—
248Dic122—21no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—21no—
252C8×C61#13512yes—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#19229—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61#47112yes—
266C22×C61C2×#12111yescharacteristic
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61C2×#13511yescharacteristic
278C4×C61#57511yescharacteristic
279C4×C61#57611yescharacteristic
280C4×C61#57811yescharacteristic
281Dic61—41no—
282Dic61—41no—
283Dic61—41no—
284#3248—611no—
285#3247—611no—
286#3167D6111yescharacteristic, complemented
287#2966—611no—
288#1692—611no—
289#1691—611no—
290#1686—611no—
291#1685—611no—
292#1684—611no—
293#1683—611no—
294#1682—611no—
295#1681—611no—
296C2×#335—611no—
297#1552—611no—
298#1551—611no—
299C2×C61—22no—
300C2×C61—161no—
301C2×C61#303511yescharacteristic
302C2×C61#306111yescharacteristic
303C2×C61C2×#47111yescharacteristic
304D61—81no—
305#533C2×D6111yescharacteristic
306#532C2×D6111yescharacteristic
307#529—611no—
308#498C2×D6111yescharacteristic
309#492C2×D6111yescharacteristic
310#487C2×D6111yescharacteristic
311#486C2×D6111yescharacteristic
312#479—1221no—
313#478—1221no—
314#477—611no—
315#476—611no—
316#475—611no—
317#474—611no—
318#467—1221no—
319#466—1221no—
320#465—21no—
321#465—21no—
322#465—1221no—
323#465—1221no—
324#464—1221no—
325#463—1221no—
326#462—611no—
327#461—611no—
328#460—611no—
329#459—611no—
330#446—611no—
331C8×Q8—611no—
332C4×SD16—21no—
333C4×SD16—21no—
334#436—611no—
335#435—1221no—
336C8×D4—611no—
337C8×D4—1221no—
338#433—1221no—
339#432—611no—
340C2×C4⋊C8—611no—
341C2×C4⋊C8—611no—
342#405—1221no—
343C2×C4×C8C2×D6111yescharacteristic
344#335—611no—
345#335—611no—
346#335—611no—
347#335—611no—
348#333—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353#332—611no—
354#331—611no—
355#331—611no—
356C61#1922911yescharacteristic, complemented, Sylow
357C2×#41—21no—
358C2×Q16—1221no—
359C2×SD16—21no—
360C2×SD16—21no—
361C2×SD16—1221no—
362C2×SD16—1221no—
363C2×D8—1221no—
364C2×M16—1221no—
365C22×C8—1221no—
366C22×C8—1221no—
367C22×C8—1221no—
368C22×C8C22⋊D6111yescharacteristic
369C22×C8C22⋊D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371C4⋊Q8C22×D6111yescharacteristic
372C4⋊Q8C22×D6111yescharacteristic
373#127C22×D6111yescharacteristic
374#126—1221no—
375#125—21no—
376#125—611no—
377#124—21no—
378#124—611no—
379#123—1221no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—611no—
383#121—21no—
384#121—21no—
385#121—611no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—611no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—611no—
392C4×D4—1221no—
393#117—21no—
394#117—1221no—
395C2×C42C22×D6111yescharacteristic
396#107—1221no—
397#107—1221no—
398#106—21no—
399#106—21no—
400#106—1221no—
401#106—1221no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—1221no—
409C4⋊C8—1221no—
410C4⋊C8—1221no—
411C4⋊C8—1221no—
412#103—21no—
413#103—21no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#102—21no—
419#102—21no—
420#102—1221no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427#97—1221no—
428#97—1221no—
429C4×C8—1221no—
430C4×C8C22⋊D6111yescharacteristic
431C4×C8C22⋊D6111yescharacteristic
432C4×C8C22×D6111yescharacteristic
433C4×C8C22×D6111yescharacteristic
434#41—82no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—611no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—611no—
444C2×D4—1221no—
445C22×C4—41no—
446C22×C4—1221no—
447C22×C4D4×D6111yescharacteristic
448C22×C4#1175822411yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450Q16—4881no—
451SD16—42no—
452SD16—42no—
453SD16—4881no—
454SD16—4881no—
455D8—4881no—
456M16—2442no—
457C2×C8—22no—
458C2×C8—41no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—4881no—
466C2×C8—4881no—
467C2×C8—2442no—
468C2×C8C2×C22⋊D6111yescharacteristic
469C2×C8C2×C22⋊D6111yescharacteristic
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—21no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—611no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C22⋊C4—1221no—
493C22⋊C4—1221no—
494C42—41no—
495C42—1221no—
496C42D4×D6111yescharacteristic
497C42#1175822411yescharacteristic
498C42C2×C22⋊D6111yescharacteristic
499C42C23×D6111yescharacteristic
500C23—41no—
501C23—1221no—
502C23#1451974811yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—81no—
508Q8—1221no—
509Q8—1221no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—81no—
515D4—82no—
516D4—1221no—
517D4—1221no—
518D4—2442no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—82no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451974811yescharacteristic
534C2×C4#1451974811yescharacteristic
535C2×C4#1451974811yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C2×C4C2×#1175822411yescharacteristic
539C8—21no—
540C8—21no—
541C8—21no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8#1451979712yes—
549C22—21no—
550C22—21no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—2441no—
555C22—2441no—
556C22—2441no—
557C22#1684002212yes—
558C22C2×#1451974811yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—2441no—
567C4—2441no—
568C4—2441no—
569C4C2×#1451979711yescharacteristic
570C4#1684056011yescharacteristic
571C4#1684056411yescharacteristic
572C4#1684057611yescharacteristic
573C2—22no—
574C2—161no—
575C2—4881no—
576C2#2145526811yescharacteristic
577C2#2145548511yescharacteristic
578C2C2×#1684002211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3167C2
socleC22×C61C2×#121
prime radical#21457139C2
hypercenter#3167D61
hypocenterC61#19229
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21457150C2semidirect
3. #21454794C2semidirect
4. #21445322C2semidirect
5. #21445315C2semidirect
6. #21445288C2semidirect
7. #21445281C2semidirect
8. #21445274C2semidirect
9. #21445267C2semidirect
10. #21445260C2semidirect
11. #21445253C2semidirect
12. C2×#16839515C2semidirect
13. #21444519C2semidirect
14. #21444514C2semidirect
15. C61×#3167C2semidirect
286. #3167D61semidirect
356. C61#19229semidirect

Maximal subgroups

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

subgroupindex
#214571502
#214571392
#214547942
#214453222
#214453152
#214452882
#214452812
#214452742
#214452672
#214452602
#214452532
C2×#168395152
#214445192
#214445142
C61×#31672
#1922961