Finite Group Databasebeta

← #29669751

Subgroups — #29669751

Order 15616.

616 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#21458452C211yesmaximal, characteristic
2#21458151C211yesmaximal, characteristic, complemented
3#21457689C211yesmaximal, characteristic, complemented
4#21445475C211yesmaximal, characteristic, complemented
5#21445396C211yesmaximal, characteristic, complemented
6#21445263C211yesmaximal, characteristic, complemented
7#21445230C211yesmaximal, characteristic, complemented
8#21445220C211yesmaximal, characteristic, complemented
9#21445033C211yesmaximal, characteristic, complemented
10#21444983C211yesmaximal, characteristic, complemented
11#21444945C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444636C211yesmaximal, characteristic, complemented
15C61×#3380C211yesmaximal, characteristic, complemented
16#16840433C2211yescharacteristic
17D4×Dic122C2211yescharacteristic
18#16840136C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16840109C2211yescharacteristic
23#16840107—21no—
24#16840097—21no—
25#16840075—21no—
26#16840054C2211yescharacteristic
27#16840054C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840041—21no—
30#16840010—21no—
31#16840001—21no—
32#16839972C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839955—21no—
36#16839922—21no—
37#16839917—21no—
38C8×Dic122C2211yescharacteristic
39C4×#11758202C2211yescharacteristic
40#16839842—21no—
41#16839840C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839761—21no—
44C2×#14519700—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#554C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#499—21no—
60C61×#478—21no—
61C61×#478—21no—
62C61×#476C2211yescharacteristic
63C61×#476C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×Dic122—21no—
69C2×#11758202—21no—
70C2×#11758202D411yescharacteristic
71C2×#11758201—21no—
72C2×#11758201—41no—
73C2×#11758199—21no—
74C2×#11758194—21no—
75C22×#92418—21no—
76#14519777C2311yescharacteristic
77#14519776—21no—
78#14519774C2311yescharacteristic
79#14519773—21no—
80#14519756—21no—
81#14519754—41no—
82#14519751—21no—
83#14519750—41no—
84C4×Dic122C2311yescharacteristic
85D4×Dic61—21no—
86#14519739—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519701—41no—
96#14519701D411yescharacteristic
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×D8×C61D411yescharacteristic
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#125×C61—21no—
119#125×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61—41no—
123C22⋊Q8×C61—41no—
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61C2311yescharacteristic
128C4×D4×C61C2311yescharacteristic
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131#107×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—41no—
152C22×C4×C61—41no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155C22×Dic61—41no—
156#11758222—42no—
157#11758222—81no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×C4×D61—41no—
162C2×Dic122—21no—
163C2×Dic122—42no—
164C2×Dic122—41no—
165C2×Dic122—81no—
166C2×Dic122C2×D411yescharacteristic
167D8×C61—22no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61—21no—
188C42×C61C2411yescharacteristic
189C22⋊Dic61—41no—
190#11758202—22no—
191#11758202—81no—
192#11758201—81no—
193#11758201—82no—
194#11758199—42no—
195#11758198—41no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—41no—
201C4×Dic61—21no—
202#11758194—81no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—81no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C23×C61—21no—
211C23×C61—21no—
212C22×D61—41no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—81no—
216Q8×C61—81no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61#13712yes—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—161no—
232C2×C4×C61—161no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C22⋊D61—82no—
237C22⋊D61—161no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61—82no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—81no—
247C4×D61—82no—
248Dic122—21no—
249Dic122—41no—
250Dic122—41no—
251Dic122—82no—
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#19781—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61D4211yescharacteristic
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C2×#13511yescharacteristic
281C4×C61C2×#13711yescharacteristic
282C4×C61#57611yescharacteristic
283C4×C61#57811yescharacteristic
284Dic61—22no—
285Dic61—41no—
286Dic61—81no—
287#3380D6111yescharacteristic, complemented
288#3358—611no—
289#3333—611no—
290#3299—611no—
291#1719—611no—
292#1703—611no—
293#1682—611no—
294#1677—611no—
295#1676—611no—
296#1644—611no—
297#1637—611no—
298#1631—611no—
299#1622—611no—
300#1581—611no—
301#1579—611no—
302C2×C61—42no—
303C2×C61—81no—
304C2×C61#327011yescharacteristic
305C2×C61#328011yescharacteristic
306C2×C61#328811yescharacteristic
307D61—161no—
308#554C2×D6111yescharacteristic
309#554C2×D6111yescharacteristic
310#550—611no—
311D4×Q8—611no—
312#500C2×D6111yescharacteristic
313#499—21no—
314#492—611no—
315#487—611no—
316#487—611no—
317#486—611no—
318#486—611no—
319#486—1221no—
320#484—1221no—
321#480—1221no—
322#478—21no—
323#478—21no—
324#477—611no—
325#477—611no—
326#476C2×D6111yescharacteristic
327#476C2×D6111yescharacteristic
328#474—611no—
329#474—1221no—
330#468—1221no—
331#467—1221no—
332#462—611no—
333#460—611no—
334#459—611no—
335#459—1221no—
336#452—1221no—
337#451—1221no—
338C8×Q8—611no—
339C8×Q8C2×D6111yescharacteristic
340C4×Q16—611no—
341C4×D8—21no—
342C4×D8—1221no—
343C4×D8C2×D6111yescharacteristic
344#436—611no—
345C8×D4—611no—
346#417—1221no—
347C2×#103—1221no—
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#1978111yescharacteristic, complemented, Sylow
360C2×#41—1221no—
361C22×Q8—1221no—
362C2×Q16—611no—
363C2×Q16—1221no—
364C2×SD16—1221no—
365C2×SD16—2441no—
366C2×D8—1221no—
367C2×D8C22⋊D6111yescharacteristic
368C2×M16—1221no—
369C22×C8—1221no—
370C4⋊Q8—611no—
371C4⋊Q8—1221no—
372C4⋊Q8C22×D6111yescharacteristic
373C4⋊Q8C22×D6111yescharacteristic
374#127—611no—
375#127—1221no—
376#125—21no—
377#125—21no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—41no—
381C22⋊Q8—41no—
382C22⋊Q8—1221no—
383C22⋊Q8—2441no—
384#121—1221no—
385#121—2441no—
386C4×Q8—611no—
387C4×Q8C22×D6111yescharacteristic
388C4×D4—21no—
389C4×D4—21no—
390C4×D4—1221no—
391C4×D4—1221no—
392C4×D4C22×D6111yescharacteristic
393C4×D4C22×D6111yescharacteristic
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396#107—21no—
397#107—21no—
398#107—611no—
399#107—1221no—
400#107—1221no—
401#107—1221no—
402#107C22⋊D6111yescharacteristic
403#106—21no—
404#106—21no—
405C4⋊C8—21no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411#103—611no—
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—2441no—
419#102—21no—
420#102—21no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102C22⋊D6111yescharacteristic
429#102C22⋊D6111yescharacteristic
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C4×C8—21no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—1221no—
436C4×C8C22×D6111yescharacteristic
437#41—4881no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—611no—
442C2×Q8—1221no—
443C2×Q8—4881no—
444C2×Q8—2441no—
445C2×Q8—2442no—
446C2×Q8D4×D6111yescharacteristic
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4C2×C22⋊D6111yescharacteristic
453C2×D4C2×C22⋊D6111yescharacteristic
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—2441no—
459C22×C4—2441no—
460Q16—1222no—
461Q16—4881no—
462SD16—4882no—
463SD16—4881no—
464D8—22no—
465D8—2442no—
466M16—4881no—
467C2×C8—21no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—4881no—
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—41no—
486C4⋊C4—611no—
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—2441no—
499C22⋊C4—2441no—
500C42—21no—
501C42—1221no—
502C42C23×D6111yescharacteristic
503C23—21no—
504C23—21no—
505C23—2441no—
506Q8—21no—
507Q8—21no—
508Q8—81no—
509Q8—81no—
510Q8—612no—
511Q8—1221no—
512Q8—4882no—
513Q8—2441no—
514Q8—2441no—
515D4—21no—
516D4—21no—
517D4—21no—
518D4—9761no—
519D4—4881no—
520D4—4882no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524D4—2441no—
525D4#1451980812yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—161no—
537C2×C4—161no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—4882no—
541C2×C4—4882no—
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—22no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—4881no—
562C22—4881no—
563C22—4881no—
564C22D4×C22⋊D6111yescharacteristic
565C4—21no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—1222no—
573C4—4881no—
574C4—2441no—
575C4#135×D6111yescharacteristic
576C4C2×#1451980811yescharacteristic
577C4#1684056411yescharacteristic
578C4#1684057711yescharacteristic
579C2—42no—
580C2—81no—
581C2—9761no—
582C2#2145739011yescharacteristic
583C2#2145749811yescharacteristic
584C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3380C2
socleC22×C61D42
prime radical#21458452C2
hypercenter#3380D61
hypocenterC61#19781
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3380

Complements

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

NH ≅ G/Ntype
2. #21458151C2semidirect
3. #21457689C2semidirect
4. #21445475C2semidirect
5. #21445396C2semidirect
6. #21445263C2semidirect
7. #21445230C2semidirect
8. #21445220C2semidirect
9. #21445033C2semidirect
10. #21444983C2semidirect
11. #21444945C2semidirect
12. #21444891C2semidirect
13. #21444649C2semidirect
14. #21444636C2semidirect
15. C61×#3380C2semidirect
287. #3380D61semidirect
359. C61#19781semidirect

Maximal subgroups

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

subgroupindex
#214584522
#214581512
#214576892
#214454752
#214453962
#214452632
#214452302
#214452202
#214450332
#214449832
#214449452
#214448912
#214446492
#214446362
C61×#33802
#1978161