Finite Group Databasebeta

← #29670121

Subgroups — #29670121

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458682C211yesmaximal, characteristic, complemented
2#21458178C211yesmaximal, characteristic, complemented
3#21457415C211yesmaximal, characteristic
4#21445440C211yesmaximal, characteristic, complemented
5#21445340C211yesmaximal, characteristic, complemented
6#21445225C211yesmaximal, characteristic, complemented
7#21445194C211yesmaximal, characteristic, complemented
8#21445165C211yesmaximal, characteristic, complemented
9#21445026C211yesmaximal, characteristic, complemented
10#21444969C211yesmaximal, characteristic, complemented
11#21444935C211yesmaximal, characteristic, complemented
12#21444890C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3295C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17#16840472C2211yescharacteristic, complemented
18#16840417C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840161—21no—
21#16840160C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic
23#16840134C2211yescharacteristic
24#16840112C2211yescharacteristic, complemented
25#16840091—21no—
26#16840079—21no—
27#16840062—21no—
28#16840059—21no—
29#16840052C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840049C2211yescharacteristic, complemented
32#16840047C2211yescharacteristic, complemented
33#16840006—21no—
34#16839988—21no—
35#16839967C2211yescharacteristic
36#16839954C2211yescharacteristic
37#16839913—21no—
38#16839905—21no—
39Q8×#92418C2211yescharacteristic, complemented
40C4×#11758199—21no—
41C4×#11758199C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic, complemented
44#106×D61—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#481—21no—
60C61×#476C2211yescharacteristic
61C61×#465—21no—
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#102×C61—21no—
67C2×#11758222—21no—
68#14519803—44no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—41no—
73C2×#11758199—21no—
74C2×#11758199D411yescharacteristic
75C2×#11758194—21no—
76C2×C8×D61—21no—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519769—21no—
80#14519759—41no—
81#14519756—21no—
82#14519755—41no—
83#14519753—21no—
84#14519748—21no—
85C4×D244—21no—
86C4×D244C2311yescharacteristic
87C4⋊C4×D61—21no—
88C4⋊C4×D61—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710D411yescharacteristic
92#14519709—21no—
93#14519709—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708—21no—
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—21no—
103#14519701—41no—
104#14519700—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C22×C8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#125×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126#121×C61—21no—
127#121×C61—21no—
128#120×C61—41no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C2×C4⋊C4×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—42no—
154C2×D4×C61—42no—
155C2×D4×C61—81no—
156C2×D4×C61—81no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C2×C22⋊D61—41no—
163#11758222—42no—
164#11758222—81no—
165C2×D244C2×D411yescharacteristic
166C2×C4×D61—21no—
167C2×C4×D61—41no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171SD16×C61—41no—
172SD16×C61—81no—
173C2×C8×C61—21no—
174C2×C8×C61—81no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—41no—
192#11758200—42no—
193#11758200—42no—
194#11758200—81no—
195#11758200—82no—
196#11758199—42no—
197#11758199SD1614yescomplemented
198#11758198—21no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—41no—
206#11758196—41no—
207#11758196—41no—
208#11758194—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C8×D61—42no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—41no—
219C23×C61—82no—
220C23×C61—82no—
221C23×C61—161no—
222C22×D61—21no—
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—81no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—81no—
234D4×C61—82no—
235D4×C61—82no—
236D4×C61C2×SD1612yes—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—161no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C22⋊D61—82no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254D244—21no—
255D244C2×SD1612yes—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—82no—
259C4×D61—161no—
260Dic122—41no—
261Dic122—41no—
262Dic122—81no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—41no—
266#92418—21no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418C2×SD1612yes—
272#19806—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—164no—
283C22×C61D4211yescharacteristic
284C2×D61—22no—
285C2×D61—41no—
286C2×D61—41no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61C22×SD1611yescharacteristic
294C4×C61#57511yescharacteristic
295C4×C61#57711yescharacteristic
296C4×C61#57711yescharacteristic
297Dic61—41no—
298Dic61—81no—
299Dic61—81no—
300#3380—611no—
301#3335—611no—
302#3295D6111yescharacteristic, complemented
303#3272—611no—
304#1714—611no—
305#1695—611no—
306#1677—611no—
307#1671—611no—
308#1666—611no—
309#1643—611no—
310#1635—611no—
311#1629—611no—
312#1622—611no—
313#1583—611no—
314#1580—611no—
315C2×C61—42no—
316C2×C61—81no—
317C2×C61—161no—
318C2×C61#326711yescharacteristic
319C2×C61D4×SD1611yescharacteristic
320C2×C61#328811yescharacteristic
321D61—42no—
322C2×AGL(1,ℤ/8)—1221no—
323#554—611no—
324#554C2×D6111yescharacteristic
325#547—611no—
326D42C2×D6111yescharacteristic
327#500—611no—
328#499—1221no—
329#498—611no—
330#492—611no—
331#492—611no—
332#487—611no—
333#486C2×D6111yescharacteristic
334#483—1221no—
335#481—21no—
336#481—1221no—
337#478—1221no—
338#478—1221no—
339#476—611no—
340#476—611no—
341#476—611no—
342#476C2×D6111yescharacteristic
343#475—611no—
344#468—1221no—
345#465—21no—
346#465—1221no—
347#461—611no—
348#460C2×D6111yescharacteristic
349#459—611no—
350#450—21no—
351#450—1221no—
352#449—1221no—
353C8×Q8—611no—
354C4×SD16C2×D6111yescharacteristic
355C4×D8—611no—
356C4×D8—1221no—
357#436—611no—
358C8×D4—611no—
359C8×D4C2×D6111yescharacteristic
360C2×#106—1221no—
361C2×#102—21no—
362#334—611no—
363#334—611no—
364#333—611no—
365#333—611no—
366#332—611no—
367#331—611no—
368Q8⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372#322—611no—
373C61#1980611yescharacteristic, complemented, Sylow
374C2×#41—1221no—
375C22×D4—21no—
376C22×D4—21no—
377AGL(1,ℤ/8)—2444no—
378C2×SD16—21no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16—1221no—
382C2×SD16—2441no—
383C2×SD16C22⋊D6111yescharacteristic
384C2×D8—611no—
385C2×D8—1221no—
386C2×M16—1221no—
387C22×C8—21no—
388C22×C8—1221no—
389C4⋊Q8—611no—
390C4⋊Q8—611no—
391C4⋊Q8C22×D6111yescharacteristic
392#127C22×D6111yescharacteristic
393#125—21no—
394#125—1221no—
395#123—2441no—
396C22⋊Q8—21no—
397C22⋊Q8—41no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8—2441no—
401#121—21no—
402#121—21no—
403#121—1221no—
404#120—41no—
405C4×Q8C22×D6111yescharacteristic
406C4×D4—21no—
407C4×D4—611no—
408C4×D4—1221no—
409C4×D4C22×D6111yescharacteristic
410C4×D4C22×D6111yescharacteristic
411C2×C4⋊C4—21no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414#107—611no—
415#107—1221no—
416#107—1221no—
417#106—21no—
418#106—1221no—
419#106—1221no—
420#106—1221no—
421#106—1221no—
422#106—1221no—
423#106—1221no—
424#106C22⋊D6111yescharacteristic
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—1221no—
429C4⋊C8C22×D6111yescharacteristic
430#103—1221no—
431#103—1221no—
432#103—2441no—
433#103C22⋊D6111yescharacteristic
434#102—21no—
435#102—21no—
436#102—21no—
437#102—21no—
438#102—611no—
439#102—611no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102C22⋊D6111yescharacteristic
445C22⋊C8—21no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—1221no—
451C4×C8C22×D6111yescharacteristic
452C24—41no—
453#41—4881no—
454#41—2442no—
455C2×Q8—21no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8C2×C22⋊D6111yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—81no—
467C2×D4—81no—
468C2×D4—611no—
469C2×D4—2441no—
470C2×D4D4×D6111yescharacteristic
471C2×D4C2×C22⋊D6111yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—41no—
475C22×C4—1221no—
476C22×C4—2441no—
477C22×C4—2441no—
478SD16—41no—
479SD16—81no—
480SD16—4881no—
481SD16—4882no—
482SD16—2442no—
483SD16—2442no—
484D8—614no—
485D8—2442no—
486M16—2442no—
487C2×C8—21no—
488C2×C8—81no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—2442no—
496C2×C8C2×C22⋊D6111yescharacteristic
497C2×C8C2×C22⋊D6111yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4D4×D6111yescharacteristic
512C4⋊C4C2×C22⋊D6111yescharacteristic
513C4⋊C4C2×C22⋊D6111yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—1221no—
519C22⋊C4—2441no—
520C22⋊C4—2441no—
521C42—21no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—21no—
525C23—41no—
526C23—82no—
527C23—82no—
528C23—161no—
529C23—1221no—
530Q8—21no—
531Q8—21no—
532Q8—81no—
533Q8—4881no—
534Q8—2441no—
535Q8—2441no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—81no—
544D4—82no—
545D4—82no—
546D4—612no—
547D4—1221no—
548D4—4882no—
549D4SD16×D6112yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—161no—
560C2×C4—1221no—
561C2×C4—9761no—
562C2×C4—4882no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4#1451982211yescharacteristic
570C8—21no—
571C8—21no—
572C8—41no—
573C8—612no—
574C8—1221no—
575C8—2441no—
576C8—2441no—
577C8—2441no—
578C8—2441no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—164no—
589C22—1222no—
590C22—2441no—
591C22—2441no—
592C22D4×C22⋊D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—81no—
599C4—4881no—
600C4—4881no—
601C4—2441no—
602C4C2×SD16×D6111yescharacteristic
603C4#1684056011yescharacteristic
604C4#1684056811yescharacteristic
605C4#1684057311yescharacteristic
606C2—42no—
607C2—81no—
608C2—161no—
609C2—2442no—
610C2#2145734011yescharacteristic
611C2SD16×C22⋊D6111yescharacteristic
612C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3295C2
socleC22×C61D42
prime radical#21457415C2
hypercenter#3295D61
hypocenterC61#19806
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3295

Complements

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

NH ≅ G/Ntype
1. #21458682C2semidirect
2. #21458178C2semidirect
4. #21445440C2semidirect
5. #21445340C2semidirect
6. #21445225C2semidirect
7. #21445194C2semidirect
8. #21445165C2semidirect
9. #21445026C2semidirect
10. #21444969C2semidirect
11. #21444935C2semidirect
12. #21444890C2semidirect
13. #21444663C2semidirect
14. #21444641C2semidirect
15. C61×#3295C2semidirect
17. #16840472C22semidirect
19. #16840165C22semidirect
21. #16840160C22semidirect
24. #16840112C22semidirect
31. #16840049C22semidirect
32. #16840047C22semidirect
39. Q8×#92418C22semidirect
43. C8×D244C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839510C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. #16839484C22semidirect
197. #11758199SD16semidirect
302. #3295D61semidirect
373. C61#19806semidirect

Maximal subgroups

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

subgroupindex
#214586822
#214581782
#214574152
#214454402
#214453402
#214452252
#214451942
#214451652
#214450262
#214449692
#214449352
#214448902
#214446632
#214446412
C61×#32952
#1980661