Finite Group Databasebeta

← #29665302

Subgroups — #29665302

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456733C211yesmaximal, characteristic
2#21455286C211yesmaximal, characteristic, complemented
3#460×D61C211yesmaximal, characteristic, complemented
4#21454898C211yesmaximal, characteristic, complemented
5#21445264C211yesmaximal, characteristic, complemented
6#21445263C211yesmaximal, characteristic, complemented
7#21445262C211yesmaximal, characteristic, complemented
8#21445261C211yesmaximal, characteristic, complemented
9#21445178C211yesmaximal, characteristic, complemented
10#21445177C211yesmaximal, characteristic, complemented
11#21445176C211yesmaximal, characteristic, complemented
12#21444592C211yesmaximal, characteristic, complemented
13#21444485C211yesmaximal, characteristic, complemented
14#21444465C211yesmaximal, characteristic, complemented
15C61×#1668C211yesmaximal, characteristic, complemented
16C2×#14519806—21no—
17C2×SD16×D61—21no—
18#16840310C2211yescharacteristic
19#127×D61C2211yescharacteristic
20C4×Q8×D61C2211yescharacteristic
21#16840138C2211yescharacteristic
22#16840109C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16839983—21no—
25#16839982—21no—
26#16839973C2211yescharacteristic, complemented
27#16839972C2211yescharacteristic, complemented
28#16839963C2211yescharacteristic
29#16839962C2211yescharacteristic
30#16839961—21no—
31#16839960—21no—
32#16839959C2211yescharacteristic
33#16839959C2211yescharacteristic, complemented
34#16839955—21no—
35#16839955—21no—
36#16839894—21no—
37#16839894—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C8×Dic122—21no—
40C8×Dic122C2211yescharacteristic, complemented
41#16839886—21no—
42#16839886C2211yescharacteristic, complemented
43#16839814—21no—
44#16839814C2211yescharacteristic
45#16839813—21no—
46#16839813C2211yescharacteristic
47#16839784C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839760—21no—
50#102×D61—21no—
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839514C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839507C2211yescharacteristic, complemented
56#16839506C2211yescharacteristic
57#16839506C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#493C2211yescharacteristic
61C61×#460C2211yescharacteristic
62C61×#460C2211yescharacteristic, complemented
63C8×Q8×C61C2211yescharacteristic, complemented
64C61×#335C2211yescharacteristic, complemented
65C61×#331C2211yescharacteristic
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×Q8×D61D411yescharacteristic
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72#14519806—44no—
73C2×Dic244—21no—
74C2×#11758202—21no—
75C2×#11758190—21no—
76C2×#11758200—21no—
77C2×#11758200—21no—
78C2×#11758201—21no—
79SD16×D61—44no—
80C2×#11758199—21no—
81C2×#11758199—21no—
82C2×#11758189—21no—
83C2×C8×D61—21no—
84#14519777C2311yescharacteristic
85#14519774C2311yescharacteristic
86#14519773—21no—
87#14519773—21no—
88#14519773—21no—
89#14519764—21no—
90#14519748—21no—
91#14519748—21no—
92Q8×Dic61D411yescharacteristic
93C4×Dic122—21no—
94C4×Dic122C2311yescharacteristic
95#14519742—21no—
96#14519742D411yescharacteristic
97D4×Dic61—21no—
98D4×Dic61—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61D411yescharacteristic
101C42×D61—21no—
102C42×D61C2311yescharacteristic
103C4⋊#92418—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706D411yescharacteristic
112#14519706D411yescharacteristic
113#14519702—21no—
114#14519700—21no—
115#14519699—21no—
116#14519699—21no—
117#14519699—21no—
118#14519698—21no—
119#14519698—21no—
120#14519698—21no—
121#14519698—21no—
122#14519697—21no—
123#14519689—21no—
124#14519689D411yescharacteristic
125#14519689D411yescharacteristic
126C4×#92418—21no—
127C4×#92418C2311yescharacteristic
128C8×Dic61—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C2×SD16×C61—21no—
132C2×SD16×C61—21no—
133C2×D8×C61—21no—
134C2×D8×C61—21no—
135#127×C61C2311yescharacteristic
136#127×C61C2311yescharacteristic
137C4×Q8×C61C2311yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#102×C61—21no—
143#102×C61—21no—
144C4×C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C22×Dic61—41no—
159Q8×D61—22no—
160Q8×D61D814yescomplemented
161#11758224—42no—
162#11758224—42no—
163D4×D61—42no—
164D4×D61—42no—
165D4×D61—81no—
166D4×D61—81no—
167D4×D61—82no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61#4111yescharacteristic
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C2×D411yescharacteristic
179SD16×C61—42no—
180SD16×C61—42no—
181D8×C61—81no—
182D8×C61—81no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61C2×D411yescharacteristic
187C2×C8×C61C2×D411yescharacteristic
188C4⋊C4×C61—21no—
189C4⋊C4×C61C2×D411yescharacteristic
190C42×C61—21no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193C22⋊Dic61—41no—
194#11758202—42no—
195#11758201—42no—
196#11758200—42no—
197#11758200—42no—
198#11758199—81no—
199#11758199—81no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61#4111yescharacteristic
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218Dic244—42no—
219#11758190—42no—
220#11758189—42no—
221C8×D61—42no—
222C23×C61—41no—
223C23×C61—41no—
224C22×D61—41no—
225C22×D61—82no—
226C22×D61—82no—
227C22×D61—82no—
228C22×D61#12111yescharacteristic
229Q8×C61—21no—
230Q8×C61C2×D812yes—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—81no—
240D4×C61—81no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C22⋊D61—82no—
247C22⋊D61—82no—
248C22⋊D61—82no—
249C22⋊D61—82no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—82no—
254C2×Dic61#12111yescharacteristic
255C2×Dic61#12111yescharacteristic
256C2×Dic61#12111yescharacteristic
257D244—41no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—81no—
262C4×D61—24no—
263C4×D61—21no—
264C4×D61—21no—
265C4×D61—41no—
266C4×D61—81no—
267C4×D61C2×D812yes—
268C4×D61#13512yes—
269Dic122—21no—
270Dic122—24no—
271Dic122—41no—
272Dic122—41no—
273Dic122C2×D812yes—
274C8×C61—21no—
275C8×C61—21no—
276C8×C61—41no—
277C8×C61—41no—
278C8×C61—41no—
279#92418—21no—
280#92418—21no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#19398—611nomaximal, Sylow
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61C2×#12111yescharacteristic
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—164no—
298C2×D61#46612yes—
299C4×C61—22no—
300C4×C61—21no—
301C4×C61—41no—
302C4×C61C22×D811yescharacteristic
303C4×C61C2×#13511yescharacteristic
304C4×C61#57711yescharacteristic
305C4×C61#57711yescharacteristic
306Dic61—21no—
307Dic61—22no—
308Dic61—21no—
309Dic61—41no—
310Dic61—81no—
311Dic61#46612yes—
312#3205—611no—
313#3037—611no—
314C2×#460—611no—
315#2982—611no—
316#1682—611no—
317#1682—611no—
318#1682—611no—
319#1682—611no—
320#1668—611no—
321#1668—611no—
322#1668—611no—
323#1668D6111yescharacteristic, complemented
324#1571—611no—
325#1545—611no—
326#1540—611no—
327C2×C61—161no—
328C2×C61—161no—
329C2×C61#304311yescharacteristic
330C2×C61C2×#46611yescharacteristic
331C2×C61#306811yescharacteristic
332D61—22no—
333D61—161no—
334C2×#137—1221no—
335C22×SD16—1221no—
336#532—611no—
337C2×#127—611no—
338C2×C4×Q8—611no—
339#493—611no—
340#493C2×D6111yescharacteristic
341#486—611no—
342#486—611no—
343#464—1221no—
344#464—1221no—
345#462—611no—
346#462—611no—
347#460—611no—
348#460—611no—
349#460—611no—
350#460—611no—
351#460—1221no—
352#460—1221no—
353#460C2×D6111yescharacteristic
354#460C2×D6111yescharacteristic, complemented
355#459—1221no—
356#459—1221no—
357#447—1221no—
358#447—1221no—
359C8×Q8—611no—
360C8×Q8—611no—
361C8×Q8—611no—
362C8×Q8—1221no—
363C8×Q8—1221no—
364C8×Q8C2×D6111yescharacteristic, complemented
365#431—611no—
366#431—611no—
367#431—1221no—
368#431—1221no—
369#423—611no—
370C2×C4⋊C8—611no—
371#417—1221no—
372C2×#102—1221no—
373#335—611no—
374#335—611no—
375#335—611no—
376#335C2×D6111yescharacteristic, complemented
377#332—611no—
378#332—611no—
379#331—611no—
380#331—611no—
381#331—611no—
382#331—611no—
383#331C2×D6111yescharacteristic
384#331C2×D6111yescharacteristic, complemented
385C61#1939811yescharacteristic, complemented, Sylow
386C2×#41—1221no—
387C22×Q8—611no—
388C22×D4—1221no—
389C22×D4—1221no—
390C22×D4—1221no—
391#137—2444no—
392C2×Q16—1221no—
393C2×Q16—1221no—
394C2×SD16—21no—
395C2×SD16—21no—
396C2×SD16—1221no—
397C2×SD16—1221no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×SD16—2444no—
401C2×D8—21no—
402C2×D8—21no—
403C2×D8—1221no—
404C2×D8—1221no—
405C2×M16—1221no—
406C22×C8—1221no—
407C4⋊Q8—611no—
408#127—611no—
409#127—1221no—
410#127—1221no—
411#127—1221no—
412#127C22×D6111yescharacteristic
413#127C22×D6111yescharacteristic
414#124—1221no—
415#121—1221no—
416#121—1221no—
417C4×Q8—611no—
418C4×Q8—611no—
419C4×Q8—611no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422C4×Q8C22×D6111yescharacteristic
423C4×D4—1221no—
424C4×D4—1221no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C42—611no—
428C2×C42—1221no—
429C4⋊C8—21no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8—611no—
433C4⋊C8—611no—
434C4⋊C8—611no—
435C4⋊C8—1221no—
436C4⋊C8—1221no—
437C4⋊C8—1221no—
438C4⋊C8—1221no—
439C4⋊C8—1221no—
440C4⋊C8C22×D6111yescharacteristic
441C4⋊C8C22×D6111yescharacteristic
442C4⋊C8C22×D6111yescharacteristic
443#103—1221no—
444#103—1221no—
445#102—21no—
446#102—21no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455C22⋊C8—611no—
456C22⋊C8—611no—
457C22⋊C8—1221no—
458C4×C8—21no—
459C4×C8—611no—
460C4×C8—1221no—
461C4×C8—1221no—
462C4×C8—1221no—
463C4×C8—1221no—
464C4×C8C22×D6111yescharacteristic
465C24—2441no—
466#41—2442no—
467#41—2442no—
468C2×Q8—614no—
469C2×Q8—611no—
470C2×Q8—1221no—
471C2×Q8—1222no—
472C2×Q8—1221no—
473C2×Q8D4×D6111yescharacteristic
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—4881no—
484C2×D4—4881no—
485C2×D4—4882no—
486C2×D4—2442no—
487C2×D4—2442no—
488C2×D4—2441no—
489C2×D4—2441no—
490C2×D4—2441no—
491C2×D4—2441no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—2441no—
498Q16—2442no—
499Q16—2442no—
500SD16—42no—
501SD16—42no—
502SD16—2442no—
503SD16—2442no—
504SD16—2442no—
505SD16—2442no—
506D8—81no—
507D8—81no—
508D8—4881no—
509D8—4881no—
510M16—2442no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—2442no—
520C2×C8D4×D6111yescharacteristic
521C2×C8D4×D6111yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—611no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4D4×D6111yescharacteristic
531C22⋊C4—2441no—
532C22⋊C4—2441no—
533C42—21no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42C23×D6111yescharacteristic
541C23—41no—
542C23—41no—
543C23—611no—
544C23—4882no—
545C23—4882no—
546C23—4882no—
547C23—2441no—
548Q8—21no—
549Q8—612no—
550Q8—1224no—
551Q8—1221no—
552Q8—2441no—
553Q8—2441no—
554Q8D8×D6112yes—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—81no—
564D4—81no—
565D4—4881no—
566D4—4882no—
567D4—4882no—
568D4—4882no—
569D4—4882no—
570D4—2441no—
571D4—2441no—
572D4—2441no—
573D4—2441no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—611no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—1224no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—4881no—
587C2×C4—4882no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4C2×D4×D6111yescharacteristic
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4#41×D6111yescharacteristic
593C8—21no—
594C8—21no—
595C8—41no—
596C8—41no—
597C8—41no—
598C8—1221no—
599C8—1221no—
600C8—2441no—
601C8—2441no—
602C8—2441no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—612no—
610C22—1221no—
611C22—1221no—
612C22—9764no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22#121×D6111yescharacteristic
617C4—22no—
618C4—21no—
619C4—41no—
620C4—612no—
621C4—1222no—
622C4—1221no—
623C4—1221no—
624C4—4881no—
625C4—2441no—
626C4C2×D8×D6111yescharacteristic
627C4#135×D6111yescharacteristic
628C4#1684056811yescharacteristic
629C4#1684057211yescharacteristic
630C2—161no—
631C2—161no—
632C2—1222no—
633C2—9761no—
634C2#2145533511yescharacteristic
635C2#466×D6111yescharacteristic
636C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1668C2
socleC22×C61C2×#121
prime radical#21456733C2
hypercenter#1668D61
hypocenterC61#19398
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1668

Complements

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

NH ≅ G/Ntype
2. #21455286C2semidirect
3. #460×D61C2semidirect
4. #21454898C2semidirect
5. #21445264C2semidirect
6. #21445263C2semidirect
7. #21445262C2semidirect
8. #21445261C2semidirect
9. #21445178C2semidirect
10. #21445177C2semidirect
11. #21445176C2semidirect
12. #21444592C2semidirect
13. #21444485C2semidirect
14. #21444465C2semidirect
15. C61×#1668C2semidirect
26. #16839973C22semidirect
27. #16839972C22semidirect
33. #16839959C22semidirect
38. Q8×#92418C22semidirect
40. C8×Dic122C22semidirect
42. #16839886C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839514C22semidirect
54. #16839509C22semidirect
55. #16839507C22semidirect
57. #16839506C22semidirect
62. C61×#460C22semidirect
63. C8×Q8×C61C22semidirect
64. C61×#335C22semidirect
66. C61×#331C22semidirect
160. Q8×D61D8semidirect
323. #1668D61semidirect
354. #460C2×D61semidirect
364. C8×Q8C2×D61semidirect
376. #335C2×D61semidirect
384. #331C2×D61semidirect
385. C61#19398semidirect

Maximal subgroups

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

subgroupindex
#214567332
#214552862
#460×D612
#214548982
#214452642
#214452632
#214452622
#214452612
#214451782
#214451772
#214451762
#214445922
#214444852
#214444652
C61×#16682
#1939861