Finite Group Databasebeta

← #29671577

Subgroups — #29671577

Order 15616.

608 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#21458464C211yesmaximal, characteristic, complemented
2#21458454C211yesmaximal, characteristic
3#21458159C211yesmaximal, characteristic, complemented
4#21445432C211yesmaximal, characteristic, complemented
5#21445363C211yesmaximal, characteristic, complemented
6#21445352C211yesmaximal, characteristic, complemented
7#21445275C211yesmaximal, characteristic, complemented
8#21445265C211yesmaximal, characteristic, complemented
9#21445031C211yesmaximal, characteristic, complemented
10#21445018C211yesmaximal, characteristic, complemented
11#21444984C211yesmaximal, characteristic, complemented
12#21444929C211yesmaximal, characteristic, complemented
13#21444702C211yesmaximal, characteristic, complemented
14Q16×#92418C211yesmaximal, characteristic, complemented
15C61×#3328C211yesmaximal, characteristic, complemented
16#16840472C2211yescharacteristic
17#16840434C2211yescharacteristic
18#16840116—21no—
19#16840114C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840110—21no—
22#16840110C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840109C2211yescharacteristic
25#16840077—21no—
26#16840069—21no—
27#16840057C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840039—21no—
30#16840039C2211yescharacteristic
31#16840011—21no—
32#16839974—21no—
33#16839964C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957—21no—
36#16839957C2211yescharacteristic
37#16839923—21no—
38Q8×#92418C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C4×#11758201—21no—
41C4×#11758199—21no—
42C4×#11758199C2211yescharacteristic
43#16839841C2211yescharacteristic
44#16839762—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#556C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#485—21no—
60C61×#475C2211yescharacteristic
61C61×#470—21no—
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C4×Q16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#419—21no—
67C2×#11758226—21no—
68C2×#11758201—21no—
69C2×#11758201—21no—
70C2×#11758199—21no—
71C2×#11758199D411yescharacteristic
72C2×#11758189—21no—
73#14519777C2311yescharacteristic
74#14519774—21no—
75#14519774C2311yescharacteristic
76#14519769—21no—
77#14519755—41no—
78#14519753—21no—
79#14519752—21no—
80#14519752—41no—
81C4×D244—21no—
82C4×D244—21no—
83C4×D244C2311yescharacteristic
84C4⋊C4×D61—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710D411yescharacteristic
88#14519708—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519701—21no—
95#14519701—21no—
96#14519701—21no—
97#14519701—21no—
98#14519700—41no—
99#14519699—21no—
100#14519699—21no—
101#14519698—21no—
102#14519698D411yescharacteristic
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—41no—
109#14519697D411yescharacteristic
110#14519689—21no—
111C4×#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×Q16×C61—21no—
117C2×Q16×C61D411yescharacteristic
118C2×M16×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—42no—
148C2×Q8×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154#11758226—42no—
155#11758226—81no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—41no—
160C2×D244—41no—
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166Q16×C61—22no—
167Q16×C61—81no—
168M16×C61—81no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
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—41no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—41no—
190C42×C61C2411yescharacteristic
191#11758201—42no—
192#11758201—81no—
193#11758199—41no—
194#11758199—42no—
195#11758198—21no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—41no—
202#11758196—41no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758189—81no—
210C23×C61—21no—
211C22×D61—21no—
212C22×D61—41no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—82no—
221Q8×C61#13512yes—
222D4×C61—21no—
223D4×C61—21no—
224C2×C4×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—82no—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C2×Dic61—21no—
238C2×Dic61—41no—
239D244—21no—
240D244—21no—
241D244—41no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—82no—
246D244—81no—
247D244—161no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—82no—
251C4×D61—161no—
252Dic122—81no—
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#19906—611nomaximal, Sylow
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61D4211yescharacteristic
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61C2×#13511yescharacteristic
282C4×C61#57511yescharacteristic
283C4×C61#57611yescharacteristic
284C4×C61#57811yescharacteristic
285Dic61—41no—
286Dic61—81no—
287#3360—611no—
288#3358—611no—
289#3333—611no—
290#3328D6111yescharacteristic, complemented
291#1711—611no—
292#1699—611no—
293#1697—611no—
294#1684—611no—
295#1682—611no—
296#1644—611no—
297#1642—611no—
298#1637—611no—
299#1628—611no—
300#1589—611no—
301C8×Q16—611no—
302C2×C61—81no—
303C2×C61#327711yescharacteristic
304C2×C61#328011yescharacteristic
305C2×C61#329011yescharacteristic
306D61—81no—
307D61—161no—
308#556C2×D6111yescharacteristic
309#554—611no—
310#550—611no—
311D4×Q8C2×D6111yescharacteristic
312#488—1221no—
313#487—611no—
314#487—611no—
315#487C2×D6111yescharacteristic
316#486—611no—
317#486—611no—
318#486—611no—
319#486—1221no—
320#485—21no—
321#480—1221no—
322#479—1221no—
323#477—611no—
324#476—611no—
325#475C2×D6111yescharacteristic
326#474—611no—
327#474—1221no—
328#470—21no—
329#468—1221no—
330#463—1221no—
331#462C2×D6111yescharacteristic
332#461—611no—
333#460—611no—
334#459—611no—
335#459—1221no—
336#452—1221no—
337#451—21no—
338C8×Q8—611no—
339C8×Q8—611no—
340C4×Q16C2×D6111yescharacteristic
341C4×SD16—1221no—
342C4×D8—611no—
343C4×D8—1221no—
344#436—611no—
345#436C2×D6111yescharacteristic
346#419—21no—
347#417—1221no—
348#335—611no—
349#335—611no—
350#333—611no—
351#332—611no—
352#332—611no—
353#331—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358C82—611no—
359C61#1990611yescharacteristic, complemented, Sylow
360C2×#41—1221no—
361C22×Q8—21no—
362C2×Q16—21no—
363C2×Q16C22⋊D6111yescharacteristic
364C2×SD16—1221no—
365C2×SD16—1221no—
366C2×D8—611no—
367C2×D8—1221no—
368C2×M16—21no—
369C2×M16—1221no—
370C4⋊Q8—21no—
371C4⋊Q8—611no—
372C4⋊Q8C22×D6111yescharacteristic
373C4⋊Q8C22×D6111yescharacteristic
374#127—611no—
375#127—1221no—
376#126—41no—
377#125—1221no—
378#124—21no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—41no—
382C22⋊Q8—1221no—
383C22⋊Q8—2441no—
384#121—1221no—
385#121—2441no—
386C4×Q8—21no—
387C4×Q8C22×D6111yescharacteristic
388C4×Q8C22×D6111yescharacteristic
389C4×D4—21no—
390C4×D4—611no—
391C4×D4—1221no—
392C4×D4—1221no—
393C4×D4C22×D6111yescharacteristic
394#117—21no—
395C2×C4⋊C4—1221no—
396#107—611no—
397#107—1221no—
398#107—1221no—
399#107C22⋊D6111yescharacteristic
400#106—21no—
401#106—1221no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8C22×D6111yescharacteristic
408#103—21no—
409#103—21no—
410#103—21no—
411#103—21no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—2441no—
417#103C22⋊D6111yescharacteristic
418#103C22⋊D6111yescharacteristic
419#102—611no—
420#102—611no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—2441no—
430C22⋊C8—21no—
431C22⋊C8—1221no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—1221no—
435C4×C8—1221no—
436C4×C8C22×D6111yescharacteristic
437#41—4881no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—41no—
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—1221no—
445C2×Q8C2×C22⋊D6111yescharacteristic
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×D4—611no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4D4×D6111yescharacteristic
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—1221no—
457C22×C4—2441no—
458C22×C4—2441no—
459Q16—22no—
460Q16—81no—
461SD16—4881no—
462SD16—2442no—
463D8—2441no—
464D8—2442no—
465M16—81no—
466M16—4881no—
467C2×C8—21no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8C2×C22⋊D6111yescharacteristic
475C2×C8C2×C22⋊D6111yescharacteristic
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C4⋊C4D4×D6111yescharacteristic
490C4⋊C4C2×C22⋊D6111yescharacteristic
491C4⋊C4C2×C22⋊D6111yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—1221no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C42—21no—
500C42—21no—
501C42—41no—
502C42C23×D6111yescharacteristic
503C23—21no—
504C23—1221no—
505C23—2441no—
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—82no—
514Q8—4881no—
515Q8#1451979712yes—
516D4—21no—
517D4—21no—
518D4—1221no—
519D4—1221no—
520D4—9761no—
521D4—4881no—
522D4—4882no—
523D4—2441no—
524D4—2441no—
525D4—2441no—
526D4—2441no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—1221no—
538C2×C4—9761no—
539C2×C4—4882no—
540C2×C4—2441no—
541C2×C4—2441no—
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—41no—
556C22—41no—
557C22—41no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22D4×C22⋊D6111yescharacteristic
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—4881no—
574C4—2441no—
575C4C2×#1451979711yescharacteristic
576C4#1684056111yescharacteristic
577C4#1684056511yescharacteristic
578C4#1684057811yescharacteristic
579C2—81no—
580C2—9761no—
581C2—4881no—
582C2#2145747011yescharacteristic
583C2#2145750011yescharacteristic
584C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3328C2
socleC22×C61D42
prime radical#21458454C2
hypercenter#3328D61
hypocenterC61#19906
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3328

Complements

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

NH ≅ G/Ntype
1. #21458464C2semidirect
3. #21458159C2semidirect
4. #21445432C2semidirect
5. #21445363C2semidirect
6. #21445352C2semidirect
7. #21445275C2semidirect
8. #21445265C2semidirect
9. #21445031C2semidirect
10. #21445018C2semidirect
11. #21444984C2semidirect
12. #21444929C2semidirect
13. #21444702C2semidirect
14. Q16×#92418C2semidirect
15. C61×#3328C2semidirect
290. #3328D61semidirect
359. C61#19906semidirect

Maximal subgroups

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

subgroupindex
#214584642
#214584542
#214581592
#214454322
#214453632
#214453522
#214452752
#214452652
#214450312
#214450182
#214449842
#214449292
#214447022
Q16×#924182
C61×#33282
#1990661