Finite Group Databasebeta

← #29671571

Subgroups — #29671571

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#21458468C211yesmaximal, characteristic, complemented
2#21458450C211yesmaximal, characteristic
3#21458104C211yesmaximal, characteristic, complemented
4#21445430C211yesmaximal, characteristic, complemented
5#21445366C211yesmaximal, characteristic, complemented
6#21445349C211yesmaximal, characteristic, complemented
7#21445279C211yesmaximal, characteristic, complemented
8#21445261C211yesmaximal, characteristic, complemented
9#21445033C211yesmaximal, characteristic, complemented
10#21445019C211yesmaximal, characteristic, complemented
11#21444982C211yesmaximal, characteristic, complemented
12#21444928C211yesmaximal, characteristic, complemented
13#21444702C211yesmaximal, characteristic, complemented
14Q16×#92418C211yesmaximal, characteristic, complemented
15C61×#3333C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17Q8×D244C2211yescharacteristic
18#16840119—21no—
19#16840114C2211yescharacteristic
20#16840113C2211yescharacteristic
21#16840112C2211yescharacteristic
22#16840108—21no—
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840105—21no—
26#16840066—21no—
27#16840049C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840042—21no—
30#16840042C2211yescharacteristic
31#16840019—21no—
32#16839977—21no—
33#16839972C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839954—21no—
36#16839954C2211yescharacteristic
37#16839916—21no—
38Q8×#92418C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C4×#11758200—21no—
41C4×#11758199—21no—
42C4×#11758199C2211yescharacteristic
43#16839841C2211yescharacteristic
44#16839769—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#554C2211yescharacteristic
57C61×#550C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#480—21no—
60C61×#477C2211yescharacteristic
61C61×#468—21no—
62C61×#460C2211yescharacteristic
63C61×#452—21no—
64C4×Q16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#417—21no—
67C2×Q8×D61—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758199—21no—
71C2×#11758199D411yescharacteristic
72C2×#11758189—21no—
73#14519777—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519772—41no—
77#14519765—21no—
78#14519757—21no—
79#14519757—41no—
80#14519753—21no—
81C4×Dic122—21no—
82C4×D244—21no—
83C4×D244C2311yescharacteristic
84#14519735—21no—
85#14519710—21no—
86#14519710D411yescharacteristic
87#14519708—21no—
88#14519708—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519702—21no—
95#14519702—21no—
96#14519701—21no—
97#14519701—41no—
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—41no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110#14519689—21no—
111C4×#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×Q16×C61D411yescharacteristic
117C2×SD16×C61—21no—
118C2×M16×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120#127×C61—21no—
121#127×C61C2311yescharacteristic
122#125×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#103×C61D411yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—41no—
157C22×C4×C61—41no—
158Q8×D61—42no—
159Q8×D61—81no—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—41no—
166Q16×C61—22no—
167SD16×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—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61C2411yescharacteristic
187#11758202—81no—
188#11758200—42no—
189#11758199—41no—
190#11758199—42no—
191#11758198—21no—
192#11758198—41no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—41no—
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—41no—
202C4×Dic61—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—
211C23×C61—41no—
212C22×D61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—81no—
217Q8×C61#13512yes—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—82no—
225D4×C61—81no—
226D4×C61—161no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
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
240C2×Dic61—21no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243D244—21no—
244D244—21no—
245C4×D61—41no—
246C4×D61—41no—
247C4×D61—82no—
248Dic122—41no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—82no—
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×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61D4211yescharacteristic
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61C2×#13511yescharacteristic
281C4×C61#57511yescharacteristic
282C4×C61#57611yescharacteristic
283C4×C61#57811yescharacteristic
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287#3360—611no—
288#3358—611no—
289#3333D6111yescharacteristic, complemented
290#3328—611no—
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—161no—
304C2×C61#327711yescharacteristic
305C2×C61#328011yescharacteristic
306C2×C61#329011yescharacteristic
307D61—81no—
308#556—611no—
309#554C2×D6111yescharacteristic
310#550C2×D6111yescharacteristic
311D4×Q8—611no—
312#488—1221no—
313#487—611no—
314#487—611no—
315#487—611no—
316#486—611no—
317#486—611no—
318#486—1221no—
319#486C2×D6111yescharacteristic
320#485—1221no—
321#480—21no—
322#479—1221no—
323#477C2×D6111yescharacteristic
324#476—611no—
325#475—611no—
326#474—611no—
327#474—1221no—
328#470—1221no—
329#468—21no—
330#463—1221no—
331#462—611no—
332#461—611no—
333#460C2×D6111yescharacteristic
334#459—611no—
335#459—1221no—
336#452—21no—
337#451—1221no—
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—1221no—
347#417—21no—
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—21no—
361C22×Q8—1221no—
362C2×Q16—1221no—
363C2×Q16C22⋊D6111yescharacteristic
364C2×SD16—21no—
365C2×SD16—1221no—
366C2×D8—611no—
367C2×D8—1221no—
368C2×M16—21no—
369C2×M16—1221no—
370C4⋊Q8—611no—
371C4⋊Q8—611no—
372C4⋊Q8—1221no—
373C4⋊Q8C22×D6111yescharacteristic
374#127—21no—
375#127C22×D6111yescharacteristic
376#126—2441no—
377#125—21no—
378#124—1221no—
379C22⋊Q8—21no—
380C22⋊Q8—41no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—2441no—
384#121—21no—
385#121—41no—
386C4×Q8—1221no—
387C4×Q8C22×D6111yescharacteristic
388C4×Q8C22×D6111yescharacteristic
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—611no—
392C4×D4—1221no—
393C4×D4C22×D6111yescharacteristic
394#117—1221no—
395C2×C4⋊C4—21no—
396#107—21no—
397#107—611no—
398#107—1221no—
399#107C22⋊D6111yescharacteristic
400#106—1221no—
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—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—2441no—
417#103C22⋊D6111yescharacteristic
418#103C22⋊D6111yescharacteristic
419#102—21no—
420#102—21no—
421#102—21no—
422#102—611no—
423#102—611no—
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—42no—
438#41—81no—
439C2×Q8—21no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8—2441no—
444C2×Q8—2442no—
445C2×Q8C2×C22⋊D6111yescharacteristic
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—611no—
453C2×D4D4×D6111yescharacteristic
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—1221no—
458C22×C4—2441no—
459Q16—22no—
460Q16—4881no—
461SD16—81no—
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—41no—
481C4⋊C4—41no—
482C4⋊C4—611no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
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—1221no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C42—21no—
500C42—21no—
501C42—2441no—
502C42C23×D6111yescharacteristic
503C23—21no—
504C23—41no—
505C23—1221no—
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—81no—
510Q8—4882no—
511Q8—2441no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515Q8#1451979712yes—
516D4—21no—
517D4—21no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—82no—
523D4—81no—
524D4—161no—
525D4—1221no—
526D4—1221no—
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—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—21no—
548C8—41no—
549C8—1221no—
550C8—1221no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22D4×C22⋊D6111yescharacteristic
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—4881no—
573C4—4881no—
574C4—2441no—
575C4C2×#1451979711yescharacteristic
576C4#1684056111yescharacteristic
577C4#1684056511yescharacteristic
578C4#1684057811yescharacteristic
579C2—81no—
580C2—161no—
581C2—4881no—
582C2#2145747011yescharacteristic
583C2#2145750011yescharacteristic
584C2#2145760111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3333C2
socleC22×C61D42
prime radical#21458450C2
hypercenter#3333D61
hypocenterC61#19906
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3333

Complements

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

NH ≅ G/Ntype
1. #21458468C2semidirect
3. #21458104C2semidirect
4. #21445430C2semidirect
5. #21445366C2semidirect
6. #21445349C2semidirect
7. #21445279C2semidirect
8. #21445261C2semidirect
9. #21445033C2semidirect
10. #21445019C2semidirect
11. #21444982C2semidirect
12. #21444928C2semidirect
13. #21444702C2semidirect
14. Q16×#92418C2semidirect
15. C61×#3333C2semidirect
289. #3333D61semidirect
359. C61#19906semidirect

Maximal subgroups

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

subgroupindex
#214584682
#214584502
#214581042
#214454302
#214453662
#214453492
#214452792
#214452612
#214450332
#214450192
#214449822
#214449282
#214447022
Q16×#924182
C61×#33332
#1990661