Finite Group Databasebeta

← #29670335

Subgroups — #29670335

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458126C211yesmaximal, characteristic, complemented
2#21457684C211yesmaximal, characteristic, complemented
3#21457549C211yesmaximal, characteristic, complemented
4#21445469C211yesmaximal, characteristic, complemented
5#21445377C211yesmaximal, characteristic, complemented
6#21445249C211yesmaximal, characteristic, complemented
7#21445240C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic, complemented
10#21445053C211yesmaximal, characteristic, complemented
11#21445019C211yesmaximal, characteristic, complemented
12#21444906C211yesmaximal, characteristic, complemented
13#21444662C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3358C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840417C2211yescharacteristic
18#16840416C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840134C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840099—21no—
25#16840090—21no—
26#16840047C2211yescharacteristic
27#16840042C2211yescharacteristic
28#16840007—21no—
29#16839993—21no—
30#16839989—21no—
31#16839972C2211yescharacteristic
32#16839967C2211yescharacteristic
33#16839964C2211yescharacteristic
34#16839954C2211yescharacteristic
35#16839924—21no—
36#16839905—21no—
37#16839904—21no—
38#16839901—21no—
39C4×#11758201C2211yescharacteristic
40#16839841C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839760—21no—
44C2×#14519701—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839509C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#550C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#486—21no—
59C61×#486C2211yescharacteristic
60C61×#474—21no—
61C61×#474C2211yescharacteristic
62C61×#459—21no—
63C61×#459C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758224—21no—
69#14519797—44no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201D411yescharacteristic
76C2×#11758199—21no—
77C2×#11758189—21no—
78C22×#92418—21no—
79C22×#92418—21no—
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519761—41no—
83#14519759—41no—
84#14519757—21no—
85#14519756—21no—
86#14519751—21no—
87#14519748—21no—
88C4×D244C2311yescharacteristic
89C2×C4⋊Dic61—21no—
90C4⋊C4×D61—21no—
91#14519710—21no—
92#14519710—41no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519702—21no—
99#14519702—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701D411yescharacteristic
105#14519700—21no—
106#14519700—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C22×Q8×C61—21no—
118C2×D8×C61—21no—
119C2×D8×C61D411yescharacteristic
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61—21no—
123#127×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126#121×C61—21no—
127#121×C61—41no—
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C22×C4×C61—41no—
163C2×C22⋊D61—41no—
164C2×C22⋊D61—41no—
165C22×Dic61—41no—
166#11758224—42no—
167#11758224—81no—
168#11758222—42no—
169#11758222—81no—
170C2×D244C2×D411yescharacteristic
171C2×C4×D61—21no—
172C2×C4×D61—41no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175D8×C61—22no—
176D8×C61—42no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194#11758202—42no—
195#11758201—22no—
196#11758201—81no—
197#11758200—42no—
198#11758200—81no—
199#11758200—81no—
200#11758199—42no—
201#11758198—21no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—41no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—41no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—42no—
215C2×#92418—81no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218#11758189—81no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—41no—
222C22×D61—21no—
223Q8×C61—21no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—82no—
227Q8×C61AGL(1,ℤ/8)12yes—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—82no—
236D4×C61—81no—
237D4×C61—161no—
238D4×C61#13512yescomplemented
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—82no—
249C2×C4×C61—82no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—82no—
254C22⋊D61—82no—
255C2×Dic61—21no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—82no—
259D244—21no—
260D244—21no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—82no—
264Dic122—41no—
265Dic122—41no—
266Dic122—41no—
267Dic122—41no—
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#19820—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61D4211yescharacteristic
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—41no—
291C4×C61—22no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61C2×#13511yescharacteristic
298C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C61#57611yescharacteristic
300C4×C61#57711yescharacteristic
301Dic61—41no—
302Dic61—81no—
303Dic61—81no—
304#3358D6111yescharacteristic, complemented
305#3330—611no—
306#3298—611no—
307#3284—611no—
308#1719—611no—
309#1701—611no—
310#1680—611no—
311#1679—611no—
312#1665—611no—
313#1649—611no—
314#1647—611no—
315#1642—611no—
316#1625—611no—
317#1583—611no—
318#1580—611no—
319C2×C61—42no—
320C2×C61—81no—
321C2×C61—161no—
322C2×C61#326411yescharacteristic
323C2×C61#328211yescharacteristic
324C2×C61#328911yescharacteristic
325D61—81no—
326C2×#135—1221no—
327#550C2×D6111yescharacteristic
328D4×Q8C2×D6111yescharacteristic
329#547—611no—
330#547—611no—
331#500—611no—
332#492—611no—
333#487—611no—
334#487—611no—
335#486—21no—
336#486—611no—
337#486C2×D6111yescharacteristic
338#484—1221no—
339#483—1221no—
340#475—611no—
341#474—21no—
342#474—611no—
343#474C2×D6111yescharacteristic
344#468—1221no—
345#465—1221no—
346#465—1221no—
347#462—611no—
348#461—611no—
349#461—611no—
350#459—21no—
351#459—611no—
352#459C2×D6111yescharacteristic
353#452—1221no—
354#449—1221no—
355#449—1221no—
356#448—1221no—
357C8×Q8C2×D6111yescharacteristic
358C4×SD16—611no—
359C4×D8—21no—
360C4×D8C2×D6111yescharacteristic
361#436—611no—
362C8×D4—611no—
363C8×D4—611no—
364#417—1221no—
365C2×#103—1221no—
366#334—611no—
367#334—611no—
368#332—611no—
369#332—611no—
370#332—611no—
371#331—611no—
372Q8⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375D4⋊C8—611no—
376#322—611no—
377C61#1982011yescharacteristic, complemented, Sylow
378C2×#41—21no—
379C2×#41—1221no—
380C2×#41—1221no—
381C22×Q8—21no—
382#135—2444no—
383C2×Q16—1221no—
384C2×SD16—611no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×D8—21no—
390C2×D8—1221no—
391C2×D8C22⋊D6111yescharacteristic
392C2×M16—1221no—
393C22×C8—1221no—
394C22×C8—1221no—
395C4⋊Q8—21no—
396C4⋊Q8—611no—
397C4⋊Q8—611no—
398C4⋊Q8C22×D6111yescharacteristic
399#127—21no—
400#127C22×D6111yescharacteristic
401#123—2441no—
402#123—2441no—
403C22⋊Q8—21no—
404C22⋊Q8—41no—
405C22⋊Q8—1221no—
406C22⋊Q8—1221no—
407#121—21no—
408#121—41no—
409#121—1221no—
410#121—1221no—
411C4×Q8C22×D6111yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—611no—
415C4×D4C22×D6111yescharacteristic
416C4×D4C22×D6111yescharacteristic
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419#107—21no—
420#107—1221no—
421#107—2441no—
422#107C22⋊D6111yescharacteristic
423#106—611no—
424#106—1221no—
425C4⋊C8—21no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8C22×D6111yescharacteristic
430#103—611no—
431#103—1221no—
432#103—1221no—
433#103—1221no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#103—1221no—
439#102—21no—
440#102—21no—
441#102—21no—
442#102—21no—
443#102—611no—
444#102—1221no—
445#102—1221no—
446#102—2441no—
447#102C22⋊D6111yescharacteristic
448#102C22⋊D6111yescharacteristic
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C4×C8—21no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8C22×D6111yescharacteristic
456#41—42no—
457#41—81no—
458#41—4881no—
459#41—4881no—
460#41—2442no—
461#41—2442no—
462C2×Q8—21no—
463C2×Q8—41no—
464C2×Q8—42no—
465C2×Q8—81no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8D4×D6111yescharacteristic
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—611no—
475C2×D4—2441no—
476C2×D4—2441no—
477C2×D4C2×C22⋊D6111yescharacteristic
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—1221no—
484C22×C4—2441no—
485C22×C4—2441no—
486Q16—2442no—
487SD16—1222no—
488SD16—4881no—
489SD16—4881no—
490SD16—4881no—
491SD16—2442no—
492D8—22no—
493D8—42no—
494D8—2442no—
495M16—4881no—
496C2×C8—21no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—4881no—
504C2×C8—2442no—
505C2×C8C2×C22⋊D6111yescharacteristic
506C2×C8C2×C22⋊D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—41no—
511C4⋊C4—611no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—2441no—
517C4⋊C4—2441no—
518C4⋊C4D4×D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C4⋊C4C2×C22⋊D6111yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—1221no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—21no—
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—21no—
532C23—41no—
533C23—1221no—
534Q8—21no—
535Q8—41no—
536Q8—41no—
537Q8—82no—
538Q8—2441no—
539Q8—2441no—
540Q8—2441no—
541Q8—2441no—
542Q8#1451980112yes—
543D4—21no—
544D4—21no—
545D4—21no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—82no—
551D4—81no—
552D4—161no—
553D4—1221no—
554D4—1221no—
555D4—4882no—
556D4—4882no—
557D4#1451979712yescomplemented
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—1221no—
570C2×C4—4882no—
571C2×C4—4882no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C22×C22⋊D6111yescharacteristic
578C2×C4#1451982211yescharacteristic
579C8—21no—
580C8—21no—
581C8—41no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—2441no—
598C22—2441no—
599C22—2441no—
600C22D4×C22⋊D6111yescharacteristic
601C4—22no—
602C4—21no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—4881no—
608C4—4881no—
609C4—2441no—
610C4C2×#1451979711yescharacteristic
611C4C2×#1451980111yescharacteristic
612C4#1684056311yescharacteristic
613C4#1684057011yescharacteristic
614C2—42no—
615C2—81no—
616C2—161no—
617C2—4881no—
618C2#2145729911yescharacteristic
619C2#2145752711yescharacteristic
620C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3358C2
socleC22×C61D42
hypercenter#3358D61
hypocenterC61#19820
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3358

Complements

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

NH ≅ G/Ntype
1. #21458126C2semidirect
2. #21457684C2semidirect
3. #21457549C2semidirect
4. #21445469C2semidirect
5. #21445377C2semidirect
6. #21445249C2semidirect
7. #21445240C2semidirect
8. #21445158C2semidirect
9. #21445068C2semidirect
10. #21445053C2semidirect
11. #21445019C2semidirect
12. #21444906C2semidirect
13. #21444662C2semidirect
14. #21444643C2semidirect
15. C61×#3358C2semidirect
238. D4×C61#135semidirect
304. #3358D61semidirect
377. C61#19820semidirect
557. D4#14519797semidirect

Maximal subgroups

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

subgroupindex
#214581262
#214576842
#214575492
#214454692
#214453772
#214452492
#214452402
#214451582
#214450682
#214450532
#214450192
#214449062
#214446622
#214446432
C61×#33582
#1982061