Finite Group Databasebeta

← #29671361

Subgroups — #29671361

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458717C211yesmaximal, characteristic, complemented
2#21458173C211yesmaximal, characteristic, complemented
3D4×#11758200C211yesmaximal, characteristic, complemented
4#21457415C211yesmaximal, characteristic
5#21445405C211yesmaximal, characteristic, complemented
6#21445320C211yesmaximal, characteristic, complemented
7#21445290C211yesmaximal, characteristic, complemented
8#21445196C211yesmaximal, characteristic, complemented
9#21445159C211yesmaximal, characteristic, complemented
10#21445105C211yesmaximal, characteristic, complemented
11#21445084C211yesmaximal, characteristic, complemented
12#21444888C211yesmaximal, characteristic, complemented
13#21444712C211yesmaximal, characteristic, complemented
14#21444611C211yesmaximal, characteristic, complemented
15C61×#1629C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17C22×#11758200—21no—
18#16840473C2211yescharacteristic, complemented
19D4×Dic122C2211yescharacteristic
20#16840417C2211yescharacteristic
21#16840169—21no—
22#16840160C2211yescharacteristic, complemented
23#16840158C2211yescharacteristic, complemented
24#16840158C2211yescharacteristic, complemented
25#16840134C2211yescharacteristic
26#16840134C2211yescharacteristic
27#16840094—21no—
28#16840079—21no—
29#16840060—21no—
30#16840051—21no—
31#16840048C2211yescharacteristic, complemented
32#16840043C2211yescharacteristic, complemented
33#16840006—21no—
34#16839990—21no—
35#16839989—21no—
36#16839967C2211yescharacteristic
37#16839967C2211yescharacteristic
38#16839959C2211yescharacteristic
39#16839954C2211yescharacteristic
40#16839913—21no—
41#16839913—21no—
42#16839905—21no—
43#16839900—21no—
44#16839889C2211yescharacteristic, complemented
45#16839871—21no—
46C4×#11758200C2211yescharacteristic
47C4×#11758199C2211yescharacteristic
48#16839840C2211yescharacteristic
49D4×#92418C2211yescharacteristic
50#16839827C2211yescharacteristic, complemented
51#106×D61—21no—
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839494C2211yescharacteristic, complemented
58#16839491C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60D42×C61C2211yescharacteristic
61C61×#492C2211yescharacteristic
62C61×#476C2211yescharacteristic
63C61×#476C2211yescharacteristic
64C61×#331C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C61×D4⋊C8C2211yescharacteristic
67C2×#11758222—21no—
68C22×Dic122—21no—
69#14519803—44no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×#11758200—44no—
78C2×#11758200D411yescharacteristic
79C2×#11758199—21no—
80C2×#11758199D411yescharacteristic
81C2×#11758194—21no—
82C22×#92418—21no—
83C2×C8×D61—21no—
84#14519777C2311yescharacteristic
85#14519777C2311yescharacteristic
86#14519776—21no—
87#14519768—21no—
88#14519759—41no—
89#14519757—21no—
90#14519756—21no—
91#14519756—21no—
92#14519754—41no—
93#14519753—41no—
94#14519748—21no—
95C4×Dic122C2311yescharacteristic
96C4×D244C2311yescharacteristic
97D4×Dic61—21no—
98#14519739—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61—21no—
101#14519711—21no—
102#14519710D411yescharacteristic
103#14519709—21no—
104#14519709—41no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708D411yescharacteristic
110C4⋊#92418C2311yescharacteristic
111C4⋊#92418C2311yescharacteristic
112#14519706—21no—
113#14519702—21no—
114#14519700—21no—
115#14519700—21no—
116#14519700D411yescharacteristic
117#14519699—21no—
118#14519698—21no—
119#14519698—21no—
120#14519698D411yescharacteristic
121#14519698D411yescharacteristic
122#14519697D411yescharacteristic
123C22⋊#92418—21no—
124C22⋊#92418—21no—
125#14519689—21no—
126#14519687—21no—
127C4×#92418C2311yescharacteristic
128C4×#92418C2311yescharacteristic
129C22×D4×C61—21no—
130C22×D4×C61—21no—
131C2×SD16×C61—41no—
132C4⋊Q8×C61C2311yescharacteristic
133#127×C61C2311yescharacteristic
134#121×C61—21no—
135#121×C61—21no—
136#120×C61—41no—
137C4×D4×C61C2311yescharacteristic
138C4×D4×C61C2311yescharacteristic
139#106×C61—21no—
140#106×C61—21no—
141C4⋊C8×C61C2311yescharacteristic
142C4⋊C8×C61C2311yescharacteristic
143#102×C61—21no—
144#102×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—41no—
147C2×Q8×C61—21no—
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—
161C2×C22⋊D61—41no—
162C22×Dic61—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—
171C2×Dic122—42no—
172C2×Dic122—41no—
173C2×Dic122—81no—
174C2×Dic122C2×D411yescharacteristic
175SD16×C61—82no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—41no—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188C22⋊Dic61—41no—
189#11758200—22no—
190#11758200—42no—
191#11758200—42no—
192#11758200—42no—
193#11758200—42no—
194#11758200—42no—
195#11758200—81no—
196#11758200—81no—
197#11758199—42no—
198#11758199SD1614yescomplemented
199#11758198—21no—
200#11758198—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—41no—
206C4⋊Dic61C2×D411yescharacteristic
207C4⋊Dic61C2×D411yescharacteristic
208#11758196—21no—
209#11758196—41no—
210#11758196—41no—
211#11758196—41no—
212C4×Dic61—21no—
213#11758194—42no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—42no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C8×D61—42no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—41no—
225C23×C61—82no—
226C23×C61—82no—
227C23×C61—161no—
228C22×D61—21no—
229Q8×C61—81no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—81no—
238D4×C61—82no—
239D4×C61—82no—
240D4×C61C2×SD1612yes—
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×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C22⋊D61—82no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—82no—
258D244—21no—
259D244C2×SD1612yescomplemented
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—82no—
263C4×D61—161no—
264Dic122—21no—
265Dic122—21no—
266Dic122—41no—
267Dic122—41no—
268Dic122—41no—
269Dic122—41no—
270Dic122—82no—
271C8×C61—22no—
272C8×C61—41no—
273C8×C61—41no—
274#92418—21no—
275#92418—21no—
276#92418—41no—
277#92418—41no—
278#92418C2×SD1612yescomplemented
279#92418AGL(1,ℤ/8)12yescomplemented
280#19890—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—164no—
291C22×C61D4211yescharacteristic
292C2×D61—22no—
293C2×D61—41no—
294C2×D61—41no—
295C4×C61—21no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61C22×SD1611yescharacteristic
300C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C4×C61#57711yescharacteristic
302C4×C61#57711yescharacteristic
303Dic61—41no—
304Dic61—41no—
305Dic61—41no—
306Dic61—81no—
307Dic61—81no—
308#3384—611no—
309#3335—611no—
310D4×SD16—611no—
311#3272—611no—
312#1705—611no—
313#1691—611no—
314#1686—611no—
315#1671—611no—
316#1665—611no—
317#1654—611no—
318#1651—611no—
319#1629D6111yescharacteristic, complemented
320#1622—611no—
321#1590—611no—
322C8×D8—611no—
323C2×C61—42no—
324C2×C61—81no—
325C2×C61—161no—
326C2×C61#327211yescharacteristic
327C2×C61D4×SD1611yescharacteristic
328C2×C61#329111yescharacteristic
329D61—42no—
330C2×AGL(1,ℤ/8)—1221no—
331C22×SD16—1221no—
332#554—611no—
333D4×Q8—611no—
334#547—611no—
335D42C2×D6111yescharacteristic
336#501—1221no—
337#498—611no—
338#498—611no—
339#498—611no—
340#492—611no—
341#492—611no—
342#492C2×D6111yescharacteristic
343#483—1221no—
344#481—1221no—
345#478—1221no—
346#476—1221no—
347#476C2×D6111yescharacteristic
348#476C2×D6111yescharacteristic
349#475—611no—
350#474—611no—
351#468—1221no—
352#465—1221no—
353#465—1221no—
354#461—611no—
355#461—611no—
356#460—611no—
357#459—611no—
358#450—1221no—
359#450—1221no—
360#449—1221no—
361#448—1221no—
362#446—611no—
363#442—1221no—
364C4×SD16—611no—
365C4×D8—611no—
366#436—611no—
367C8×D4—611no—
368C8×D4—611no—
369C2×#106—1221no—
370#335—611no—
371#334—611no—
372#333—611no—
373#332—611no—
374#332—611no—
375#331C2×D6111yescharacteristic
376Q8⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8C2×D6111yescharacteristic
379D4⋊C8C2×D6111yescharacteristic
380C82—611no—
381C61#1989011yescharacteristic, complemented, Sylow
382C2×#41—1221no—
383C22×Q8—1221no—
384C22×D4—21no—
385C22×D4—21no—
386AGL(1,ℤ/8)—2444no—
387C2×SD16—41no—
388C2×SD16—611no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16—1221no—
392C2×SD16—1221no—
393C2×SD16—1221no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—2444no—
397C2×D8—611no—
398C2×D8—1221no—
399C2×M16—1221no—
400C22×C8—1221no—
401C22×C8—1221no—
402C4⋊Q8—611no—
403C4⋊Q8—611no—
404C4⋊Q8—1221no—
405C4⋊Q8C22×D6111yescharacteristic
406#127C22×D6111yescharacteristic
407#125—1221no—
408#123—2441no—
409C22⋊Q8—1221no—
410C22⋊Q8—1221no—
411C22⋊Q8—1221no—
412C22⋊Q8—2441no—
413C22⋊Q8—2441no—
414#121—21no—
415#121—21no—
416#121—1221no—
417#120—41no—
418C4×Q8—611no—
419C4×D4—611no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4C22×D6111yescharacteristic
423C4×D4C22×D6111yescharacteristic
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426#107—611no—
427#107—1221no—
428#106—21no—
429#106—21no—
430#106—611no—
431#106—1221no—
432#106—1221no—
433#106—1221no—
434#106—1221no—
435#106—1221no—
436#106—2441no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—1221no—
440C4⋊C8C22×D6111yescharacteristic
441C4⋊C8C22×D6111yescharacteristic
442#103—611no—
443#103—1221no—
444#103—1221no—
445#103—1221no—
446#102—21no—
447#102—21no—
448#102—611no—
449#102—611no—
450#102—611no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457#97—1221no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8C22×D6111yescharacteristic
461C24—41no—
462#41—4881no—
463#41—2442no—
464C2×Q8—21no—
465C2×Q8—611no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—4881no—
469C2×Q8—2441no—
470C2×Q8—2442no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—81no—
479C2×D4—81no—
480C2×D4—611no—
481C2×D4—2441no—
482C2×D4D4×D6111yescharacteristic
483C2×D4D4×D6111yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—1221no—
487C22×C4—2441no—
488C22×C4—2441no—
489C22×C4—2441no—
490SD16—82no—
491SD16—1222no—
492SD16—4881no—
493SD16—4881no—
494SD16—2442no—
495SD16—2442no—
496SD16—2442no—
497SD16—2442no—
498SD16—2442no—
499D8—614no—
500D8—2442no—
501M16—2442no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—611no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—2442no—
512C2×C8—2442no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—611no—
516C4⋊C4—611no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—2441no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4D4×D6111yescharacteristic
527C4⋊C4D4×D6111yescharacteristic
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—41no—
531C22⋊C4—1221no—
532C22⋊C4—2441no—
533C22⋊C4—2441no—
534C22⋊C4—2441no—
535C42—1221no—
536C42C23×D6111yescharacteristic
537C23—21no—
538C23—21no—
539C23—41no—
540C23—82no—
541C23—82no—
542C23—161no—
543C23—1221no—
544Q8—81no—
545Q8—1221no—
546Q8—1221no—
547Q8—4882no—
548Q8—2441no—
549Q8—2441no—
550Q8—2441no—
551Q8—2441no—
552D4—21no—
553D4—21no—
554D4—21no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—81no—
560D4—82no—
561D4—82no—
562D4—612no—
563D4—1221no—
564D4—4882no—
565D4SD16×D6112yes—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—9761no—
577C2×C4—4882no—
578C2×C4—4882no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4#1451982211yescharacteristic
586C8—22no—
587C8—41no—
588C8—41no—
589C8—612no—
590C8—612no—
591C8—1221no—
592C8—1221no—
593C8—2441no—
594C8—2441no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—164no—
605C22—1222no—
606C22—2441no—
607C22—2441no—
608C22#1684039411yescharacteristic
609C4—21no—
610C4—41no—
611C4—41no—
612C4—81no—
613C4—4881no—
614C4—4881no—
615C4—2441no—
616C4—2441no—
617C4—2441no—
618C4C2×SD16×D6111yescharacteristic
619C4AGL(1,ℤ/8)×D6111yescharacteristic
620C4#1684056811yescharacteristic
621C4#1684057211yescharacteristic
622C2—42no—
623C2—81no—
624C2—161no—
625C2—2442no—
626C2#2145741711yescharacteristic
627C2#2145751311yescharacteristic
628C2#2145761411yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1629C2
socleC22×C61D42
prime radical#21457415C2
hypercenter#1629D61
hypocenterC61#19890
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1629

Complements

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

NH ≅ G/Ntype
1. #21458717C2semidirect
2. #21458173C2semidirect
3. D4×#11758200C2semidirect
5. #21445405C2semidirect
6. #21445320C2semidirect
7. #21445290C2semidirect
8. #21445196C2semidirect
9. #21445159C2semidirect
10. #21445105C2semidirect
11. #21445084C2semidirect
12. #21444888C2semidirect
13. #21444712C2semidirect
14. #21444611C2semidirect
15. C61×#1629C2semidirect
18. #16840473C22semidirect
22. #16840160C22semidirect
23. #16840158C22semidirect
24. #16840158C22semidirect
31. #16840048C22semidirect
32. #16840043C22semidirect
44. #16839889C22semidirect
50. #16839827C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839509C22semidirect
57. #16839494C22semidirect
58. #16839491C22semidirect
59. C8×#92418C22semidirect
198. #11758199SD16semidirect
259. D244C2×SD16semidirect
278. #92418C2×SD16semidirect
279. #92418AGL(1,ℤ/8)semidirect
319. #1629D61semidirect
381. C61#19890semidirect

Maximal subgroups

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

subgroupindex
#214587172
#214581732
D4×#117582002
#214574152
#214454052
#214453202
#214452902
#214451962
#214451592
#214451052
#214450842
#214448882
#214447122
#214446112
C61×#16292
#1989061