Finite Group Databasebeta

← #29665177

Subgroups — #29665177

Order 15616.

708 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#21456740C211yesmaximal, characteristic, complemented
2#474×D61C211yesmaximal, characteristic, complemented
3#21455278C211yesmaximal, characteristic
4#21454840C211yesmaximal, characteristic, complemented
5#21445243C211yesmaximal, characteristic, complemented
6#21445242C211yesmaximal, characteristic, complemented
7#21445241C211yesmaximal, characteristic, complemented
8#21445240C211yesmaximal, characteristic, complemented
9#21445159C211yesmaximal, characteristic, complemented
10#21445158C211yesmaximal, characteristic, complemented
11#21445157C211yesmaximal, characteristic, complemented
12#21444588C211yesmaximal, characteristic, complemented
13#21444502C211yesmaximal, characteristic, complemented
14#21444465C211yesmaximal, characteristic, complemented
15C61×#1665C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17#16840310C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840134C2211yescharacteristic, complemented
21#16840113C2211yescharacteristic, complemented
22#16840112C2211yescharacteristic, complemented
23#16840043C2211yescharacteristic
24#16840042C2211yescharacteristic
25#16840041—21no—
26#16840040—21no—
27#16840039C2211yescharacteristic
28#16839976—21no—
29#16839975—21no—
30#16839968C2211yescharacteristic
31#16839967C2211yescharacteristic
32#16839965—21no—
33#16839965—21no—
34#16839954C2211yescharacteristic
35#16839899—21no—
36#16839899—21no—
37C8×D244C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39#16839827C2211yescharacteristic
40C8×C22⋊D61—21no—
41#16839825—21no—
42#16839814C2211yescharacteristic
43#16839813C2211yescharacteristic
44#107×D61—21no—
45#16839784C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839760—21no—
48#102×D61—21no—
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839507C2211yescharacteristic, complemented
58#16839507C2211yescharacteristic, complemented
59#16839506C2211yescharacteristic, complemented
60C61×#492C2211yescharacteristic, complemented
61C61×#474C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69C2×D4×D61D411yescharacteristic
70C22×C4×D61—21no—
71#14519801—44no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758199—21no—
77C2×#11758189—21no—
78C22×#92418—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82#14519777C2311yescharacteristic
83#14519777C2311yescharacteristic
84#14519776—21no—
85#14519776—21no—
86#14519775—21no—
87#14519764—21no—
88#14519748—21no—
89#14519748—21no—
90C4×D244C2311yescharacteristic
91D4×Dic61—21no—
92D4×Dic61—21no—
93D4×Dic61D411yescharacteristic
94#14519739D411yescharacteristic
95C4×C22⋊D61—21no—
96#14519737—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61D411yescharacteristic
100C22⋊C4×D61—21no—
101C42×D61C2311yescharacteristic
102#14519711—21no—
103#14519710—21no—
104#14519710—21no—
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519706—21no—
110#14519706D411yescharacteristic
111#14519706D411yescharacteristic
112#14519702—21no—
113#14519702—21no—
114#14519702—21no—
115#14519701—21no—
116#14519701—21no—
117#14519700—21no—
118#14519699—21no—
119#14519698—21no—
120#14519698—21no—
121#14519697—21no—
122C22⋊#92418—21no—
123#14519689—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689D411yescharacteristic
127#14519689D411yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C2×SD16×C61—21no—
132C2×D8×C61—21no—
133C22×C8×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#107×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—41no—
154C22×Dic61—21no—
155C22×Dic61—41no—
156Q8×D61—81no—
157Q8×D61—81no—
158#11758224—42no—
159#11758224—42no—
160D4×D61—22no—
161D4×D61SD1614yescomplemented
162C2×D244C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—41no—
167C2×C4×D61—81no—
168C2×C4×D61—81no—
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61#4111yescharacteristic
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—41no—
175SD16×C61—81no—
176SD16×C61—81no—
177D8×C61—42no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—81no—
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C42×C61C2411yescharacteristic
189C22⋊Dic61—21no—
190C22⋊Dic61—41no—
191C22⋊Dic61—41no—
192#11758200—42no—
193#11758200—81no—
194#11758200—81no—
195#11758199—42no—
196#11758198—21no—
197#11758198—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61#4111yescharacteristic
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—81no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215#11758190—42no—
216#11758189—42no—
217C8×D61—42no—
218C8×D61—81no—
219C8×D61—81no—
220C23×C61—21no—
221C23×C61—41no—
222C22×D61—21no—
223C22×D61—24no—
224C22×D61—81no—
225C22×D61—81no—
226C22×D61#12111yescharacteristic
227Q8×C61—41no—
228Q8×C61—41no—
229D4×C61—21no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—81no—
235D4×C61C2×SD1612yes—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C22⋊D61—24no—
244C22⋊D61—82no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—81no—
249C2×Dic61—82no—
250C2×Dic61#12111yescharacteristic
251C2×Dic61#12111yescharacteristic
252C2×Dic61#12111yescharacteristic
253D244—21no—
254D244C2×SD1612yes—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—81no—
261C4×D61—81no—
262C4×D61—82no—
263C4×D61C2×SD1612yes—
264C4×D61#13512yes—
265Dic122—41no—
266Dic122—41no—
267Dic122—41no—
268Dic122—41no—
269Dic122—82no—
270C8×C61—21no—
271C8×C61—21no—
272C8×C61—41no—
273C8×C61—41no—
274C8×C61—41no—
275#92418—21no—
276#92418—21no—
277#92418—41no—
278#92418—41no—
279#92418—41no—
280#19385—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61C2×#12111yescharacteristic
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—22no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—84no—
294C2×D61#46512yes—
295C4×C61—21no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61C22×SD1611yescharacteristic
299C4×C61C2×#13511yescharacteristic
300C4×C61#57611yescharacteristic
301C4×C61#57611yescharacteristic
302Dic61—21no—
303Dic61—21no—
304Dic61—41no—
305Dic61—81no—
306Dic61—81no—
307Dic61#46512yes—
308#3206—611no—
309C2×#474—611no—
310#3036—611no—
311#2972—611no—
312#1679—611no—
313#1679—611no—
314#1679—611no—
315#1679—611no—
316#1665—611no—
317#1665—611no—
318#1665—611no—
319#1665D6111yescharacteristic, complemented
320#1570—611no—
321#1549—611no—
322#1540—611no—
323C2×C61—42no—
324C2×C61—161no—
325C2×C61#304111yescharacteristic
326C2×C61C2×#46511yescharacteristic
327C2×C61#306911yescharacteristic
328D61—22no—
329D61—42no—
330C2×AGL(1,ℤ/8)—1221no—
331#532—611no—
332C2×C4⋊Q8—611no—
333C2×C4×D4—611no—
334#492—611no—
335#492C2×D6111yescharacteristic, complemented
336#487—611no—
337#487—611no—
338#474—611no—
339#474—611no—
340#474—611no—
341#474—1221no—
342#474—1221no—
343#474C2×D6111yescharacteristic
344#463—1221no—
345#463—1221no—
346#461—611no—
347#461—611no—
348#461—1221no—
349#461—1221no—
350#459—611no—
351#459C2×D6111yescharacteristic
352#448—1221no—
353#448—1221no—
354C8×D4—611no—
355C8×D4—611no—
356C8×D4—611no—
357C8×D4—1221no—
358C8×D4—1221no—
359C8×D4C2×D6111yescharacteristic
360#431—611no—
361#431—611no—
362C2×#107—1221no—
363#423—611no—
364C2×C4⋊C8—611no—
365#417—1221no—
366C2×#102—1221no—
367C2×C22⋊C8—1221no—
368C2×C22⋊C8—1221no—
369#334—611no—
370#334—611no—
371#334—611no—
372#334C2×D6111yescharacteristic, complemented
373#332—611no—
374#332—611no—
375#332—611no—
376#332—611no—
377#332—611no—
378#332C2×D6111yescharacteristic, complemented
379#331—611no—
380#331C2×D6111yescharacteristic, complemented
381C61#1938511yescharacteristic, complemented, Sylow
382C2×#41—1221no—
383C22×Q8—1221no—
384C22×D4—611no—
385C23×C4—1221no—
386AGL(1,ℤ/8)—2444no—
387C2×SD16—21no—
388C2×SD16—21no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16—1221no—
392C2×SD16—1221no—
393C2×D8—21no—
394C2×D8—1221no—
395C2×M16—1221no—
396C22×C8—21no—
397C22×C8—1221no—
398C22×C8—1221no—
399C22×C8—1221no—
400C22×C8—1221no—
401C4⋊Q8—611no—
402C4⋊Q8—611no—
403C4⋊Q8—1221no—
404C4⋊Q8—1221no—
405C4⋊Q8—1221no—
406C4⋊Q8C22×D6111yescharacteristic
407#127C22×D6111yescharacteristic
408#124—1221no—
409#121—1221no—
410#121—1221no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—611no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4C22×D6111yescharacteristic
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C42—611no—
424#107—21no—
425#107—1221no—
426#107—1221no—
427#107—1221no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8—611no—
433C4⋊C8—1221no—
434C4⋊C8—1221no—
435C4⋊C8C22×D6111yescharacteristic
436C4⋊C8C22×D6111yescharacteristic
437C4⋊C8C22×D6111yescharacteristic
438#103—1221no—
439#103—1221no—
440#103—1221no—
441#103—1221no—
442#103—1221no—
443#103—1221no—
444#102—21no—
445#102—21no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450C22⋊C8—21no—
451C22⋊C8—611no—
452C22⋊C8—611no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457C4×C8—611no—
458C4×C8—1221no—
459C4×C8C22×D6111yescharacteristic
460C24—1221no—
461#41—2442no—
462#41—2442no—
463C2×Q8—21no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—4881no—
467C2×Q8—4881no—
468C2×Q8—2441no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—614no—
473C2×D4—611no—
474C2×D4—1222no—
475C2×D4—1221no—
476C2×D4—2441no—
477C2×D4D4×D6111yescharacteristic
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—4881no—
487C22×C4—4881no—
488C22×C4—2441no—
489C22×C4—2441no—
490SD16—81no—
491SD16—81no—
492SD16—4881no—
493SD16—4881no—
494SD16—2442no—
495SD16—2442no—
496D8—42no—
497D8—2442no—
498M16—2442no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—81no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—4881no—
509C2×C8—4881no—
510C2×C8—4881no—
511C2×C8—2442no—
512C2×C8D4×D6111yescharacteristic
513C2×C8D4×D6111yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—21no—
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⋊C4D4×D6111yescharacteristic
525C22⋊C4—21no—
526C22⋊C4—1221no—
527C22⋊C4—1221no—
528C22⋊C4—1221no—
529C22⋊C4—2441no—
530C22⋊C4—2441no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42C23×D6111yescharacteristic
535C23—21no—
536C23—41no—
537C23—611no—
538C23—1224no—
539C23—1221no—
540C23—4881no—
541C23—4881no—
542Q8—41no—
543Q8—41no—
544Q8—4882no—
545Q8—2441no—
546Q8—2441no—
547Q8—2441no—
548Q8—2441no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—81no—
555D4—612no—
556D4—1224no—
557D4—1221no—
558D4—4882no—
559D4SD16×D6112yes—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—611no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—4881no—
573C2×C4—4881no—
574C2×C4—4882no—
575C2×C4—4882no—
576C2×C4—4881no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C8—21no—
586C8—21no—
587C8—41no—
588C8—41no—
589C8—41no—
590C8—1221no—
591C8—1221no—
592C8—2441no—
593C8—2441no—
594C8—2441no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—612no—
602C22—1221no—
603C22—1221no—
604C22—1222no—
605C22—4884no—
606C22—2441no—
607C22—2441no—
608C22#121×D6111yescharacteristic
609C4—21no—
610C4—41no—
611C4—81no—
612C4—612no—
613C4—1221no—
614C4—1221no—
615C4—4881no—
616C4—4881no—
617C4—2441no—
618C4C2×SD16×D6111yescharacteristic
619C4#135×D6111yescharacteristic
620C4#1684056311yescharacteristic
621C4#1684056611yescharacteristic
622C2—42no—
623C2—161no—
624C2—1222no—
625C2—2442no—
626C2#2145531711yescharacteristic
627C2#465×D6111yescharacteristic
628C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1665C2
socleC22×C61C2×#121
prime radical#21455278C2
hypercenter#1665D61
hypocenterC61#19385
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1665

Complements

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

NH ≅ G/Ntype
1. #21456740C2semidirect
2. #474×D61C2semidirect
4. #21454840C2semidirect
5. #21445243C2semidirect
6. #21445242C2semidirect
7. #21445241C2semidirect
8. #21445240C2semidirect
9. #21445159C2semidirect
10. #21445158C2semidirect
11. #21445157C2semidirect
12. #21444588C2semidirect
13. #21444502C2semidirect
14. #21444465C2semidirect
15. C61×#1665C2semidirect
20. #16840134C22semidirect
21. #16840113C22semidirect
22. #16840112C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839509C22semidirect
56. #16839508C22semidirect
57. #16839507C22semidirect
58. #16839507C22semidirect
59. #16839506C22semidirect
60. C61×#492C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
66. C61×#331C22semidirect
161. D4×D61SD16semidirect
319. #1665D61semidirect
335. #492C2×D61semidirect
372. #334C2×D61semidirect
378. #332C2×D61semidirect
380. #331C2×D61semidirect
381. C61#19385semidirect

Maximal subgroups

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

subgroupindex
#214567402
#474×D612
#214552782
#214548402
#214452432
#214452422
#214452412
#214452402
#214451592
#214451582
#214451572
#214445882
#214445022
#214444652
C61×#16652
#1938561