Finite Group Databasebeta

← #29662448

Subgroups — #29662448

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456394C211yesmaximal, characteristic, complemented
2#21455276C211yesmaximal, characteristic, complemented
3#21455268C211yesmaximal, characteristic, complemented
4#21454775C211yesmaximal, characteristic, complemented
5#21445266C211yesmaximal, characteristic, complemented
6#21445262C211yesmaximal, characteristic, complemented
7#21445259C211yesmaximal, characteristic, complemented
8#21445255C211yesmaximal, characteristic, complemented
9#21445212C211yesmaximal, characteristic, complemented
10#21445207C211yesmaximal, characteristic, complemented
11#21445178C211yesmaximal, characteristic, complemented
12#21445173C211yesmaximal, characteristic, complemented
13C2×#16839514C211yesmaximal, characteristic, complemented
14#21444466C211yesmaximal, characteristic, complemented
15C61×#1540C211yesmaximal, characteristic, complemented
16C2×#14519799—21no—
17C2×#14519797—21no—
18#16840312C2211yescharacteristic
19C4×#11758222C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840138C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16839996—21no—
25#16839996—21no—
26#16839985—21no—
27#16839981—21no—
28#16839978—21no—
29#16839974—21no—
30#16839973C2211yescharacteristic
31#16839969C2211yescharacteristic
32#16839968C2211yescharacteristic
33#16839964C2211yescharacteristic
34#16839963C2211yescharacteristic
35#16839959C2211yescharacteristic
36#16839958C2211yescharacteristic
37#16839954C2211yescharacteristic
38#16839886C2211yescharacteristic
39#16839886C2211yescharacteristic
40C4×#11758199—21no—
41C4×#11758199—21no—
42#16839827C2211yescharacteristic
43#16839827C2211yescharacteristic
44#16839825—21no—
45#16839825—21no—
46#16839813—21no—
47C2×C4⋊#92418C2211yescharacteristic
48#16839759—21no—
49#16839758—21no—
50C2×C4×#92418C2211yescharacteristic
51C4×C8×D61—21no—
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839514C2211yescharacteristic
55#16839514C2211yescharacteristic
56#16839507C2211yescharacteristic
57#16839507C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#532C2211yescharacteristic
61C61×#431C2211yescharacteristic
62C2×C4⋊C8×C61C2211yescharacteristic
63C61×#332C2211yescharacteristic
64C61×#332C2211yescharacteristic
65C61×#331C2211yescharacteristic
66C61×#331C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758222D411yescharacteristic
69#14519799—44no—
70#14519797—44no—
71C2×Dic244—21no—
72C2×#11758202—21no—
73C2×#11758190—21no—
74C2×#11758190—21no—
75C2×#11758200—21no—
76C2×#11758201—21no—
77C2×D488—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758199—21no—
81C22×#92418—21no—
82C22×#92418D411yescharacteristic
83C22×#92418D411yescharacteristic
84C2×C8×D61—21no—
85C2×C8×D61—21no—
86#14519777C2311yescharacteristic
87#14519774C2311yescharacteristic
88#14519765—21no—
89#14519751—21no—
90#14519751—21no—
91C4×Dic122D411yescharacteristic
92C4×Dic122C2311yescharacteristic
93C4×D244—21no—
94C4×D244—21no—
95C4×D244D411yescharacteristic
96C4×D244C2311yescharacteristic
97C4×C22⋊D61—21no—
98C4×C22⋊D61—21no—
99#14519736D411yescharacteristic
100#14519734—21no—
101C42×D61—21no—
102#14519710—21no—
103#14519710—21no—
104C4⋊#92418—21no—
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519706—21no—
109#14519702—21no—
110#14519702—21no—
111#14519701—21no—
112#14519701—21no—
113#14519699—21no—
114#14519699—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519697—21no—
120#14519697—21no—
121#14519689—21no—
122#14519689—21no—
123C4×#92418D411yescharacteristic
124C4×#92418D411yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×#41×C61—21no—
130C22×C8×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133#124×C61—21no—
134#121×C61—21no—
135#121×C61—21no—
136C4×D4×C61—21no—
137C4×D4×C61—21no—
138C2×C42×C61C2311yescharacteristic
139C4⋊C8×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C4⋊C8×C61C2311yescharacteristic
143C4⋊C8×C61C2311yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61—21no—
146#41×C61—42no—
147#41×C61—42no—
148C2×Q8×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C22×C4×C61—41no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4111yescharacteristic
157C2×C22⋊D61—21no—
158C2×C22⋊D61—41no—
159#11758222—22no—
160#11758222—42no—
161#11758222—42no—
162#11758222D814yescomplemented
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—41no—
171C2×Dic122—21no—
172C2×Dic122C2×D411yescharacteristic
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—42no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61#4111yescharacteristic
185C42×C61C2411yescharacteristic
186C22⋊Dic61—21no—
187#11758202—42no—
188#11758201—42no—
189#11758200—42no—
190#11758199—42no—
191#11758199—42no—
192#11758199—42no—
193#11758198—21no—
194#11758198—21no—
195#11758198—41no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C2×D411yescharacteristic
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—21no—
202#11758196—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C2×#92418—22no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—41no—
210C2×#92418—42no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215Dic244—42no—
216D488—42no—
217#11758190—42no—
218#11758190—42no—
219C8×D61—42no—
220C8×D61—42no—
221C23×C61—41no—
222C23×C61#12111yescharacteristic
223C22×D61—21no—
224C22×D61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—81no—
232D4×C61—82no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61#12111yescharacteristic
241C2×C4×C61C2×D812yes—
242C2×C4×C61#13512yes—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C22⋊D61—24no—
247C22⋊D61—82no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—41no—
252D244—21no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—41no—
257D244—81no—
258D244C2×D812yes—
259C4×D61—24no—
260C4×D61—42no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—82no—
264Dic122—21no—
265Dic122—41no—
266Dic122—41no—
267Dic122C2×D812yes—
268C8×C61—41no—
269C8×C61—41no—
270C8×C61—41no—
271C8×C61—41no—
272#92418—21no—
273#92418—21no—
274#92418—21no—
275#92418—41no—
276#92418—41no—
277#92418#13512yescomplemented
278#19125—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61#46612yes—
285C22×C61C2×#12111yescharacteristic
286C2×D61—22no—
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—81no—
296C4×C61#46612yes—
297C4×C61C22×D811yescharacteristic
298C4×C61C2×#13511yescharacteristic
299C4×C61C2×#13511yescharacteristic
300C4×C61#57511yescharacteristic
301Dic61—22no—
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305#3169—611no—
306#3035—611no—
307#3035—611no—
308C8×#41—611no—
309#1682—611no—
310#1682—611no—
311#1681—611no—
312#1681—611no—
313#1674—611no—
314#1673—611no—
315#1668—611no—
316#1667—611no—
317C2×#335—611no—
318#1540—611no—
319#1540D6111yescharacteristic, complemented
320C2×C61—22no—
321C2×C61—161no—
322C2×C61#303511yescharacteristic
323C2×C61C2×#46611yescharacteristic
324C2×C61#306711yescharacteristic
325D61—42no—
326D61—161no—
327C2×#135—1221no—
328C2×#135—1221no—
329#532—611no—
330#532C2×D6111yescharacteristic
331C4×#41—611no—
332#500—611no—
333#493—611no—
334#486—611no—
335#486—611no—
336#466—1221no—
337#466—1221no—
338#464—1221no—
339#464—1221no—
340#463—1221no—
341#463—1221no—
342#462—611no—
343#462—611no—
344#461—611no—
345#461—611no—
346#460—611no—
347#460—611no—
348#459—611no—
349#459—611no—
350C8×Q8—611no—
351C8×Q8—611no—
352C4×D8—1221no—
353C4×D8—1221no—
354C8×D4—611no—
355C8×D4—611no—
356C8×D4—1221no—
357C8×D4—1221no—
358#431—1221no—
359#431C2×D6111yescharacteristic
360C2×C4⋊C8—611no—
361C2×C4⋊C8C2×D6111yescharacteristic
362#416—1221no—
363#416—1221no—
364C2×C4×C8—611no—
365C2×C4×C8—1221no—
366#335—611no—
367#335—611no—
368#335—611no—
369#335—611no—
370#332—611no—
371#332—611no—
372#332C2×D6111yescharacteristic
373#332C2×D6111yescharacteristic
374#331—611no—
375#331—611no—
376#331C2×D6111yescharacteristic
377#331C2×D6111yescharacteristic
378C61#1912511yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C2×#41—611no—
381C2×#41—1221no—
382#135—2444no—
383#135—2444no—
384C2×Q16—1221no—
385C2×Q16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×D8—1221no—
391C2×D8—1221no—
392C2×D8—1221no—
393C2×D8—1221no—
394C22×C8—21no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—1221no—
398C22×C8—1221no—
399C22×C8—1221no—
400C4⋊Q8—611no—
401C4⋊Q8C22×D6111yescharacteristic
402#127—611no—
403#127C22×D6111yescharacteristic
404#124—21no—
405#124—1221no—
406#121—21no—
407#121—21no—
408#121—1221no—
409#121—1221no—
410C4×Q8—611no—
411C4×Q8—611no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—611no—
415C4×D4—611no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1221no—
420#117—611no—
421#117—1221no—
422C2×C42—1221no—
423C2×C42C22×D6111yescharacteristic
424#107—1221no—
425#107—1221no—
426C4⋊C8—21no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8—1221no—
432C4⋊C8C22×D6111yescharacteristic
433C4⋊C8C22×D6111yescharacteristic
434C4⋊C8C22×D6111yescharacteristic
435C4⋊C8C22×D6111yescharacteristic
436#103—1221no—
437#103—1221no—
438#103—1221no—
439#103—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448C22⋊C8—21no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C4×C8—21no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8—1221no—
457C4×C8—1221no—
458#41—42no—
459#41—42no—
460#41—614no—
461#41—1222no—
462#41—2442no—
463#41—2442no—
464C2×Q8—21no—
465C2×Q8—611no—
466C2×Q8—1221no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—611no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—2441no—
477C22×C4—41no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—2441no—
482C22×C4#1175822211yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485Q16—2442no—
486Q16—2442no—
487SD16—2442no—
488SD16—2442no—
489SD16—2442no—
490SD16—2442no—
491D8—2442no—
492D8—2442no—
493D8—2442no—
494D8—2442no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—42no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—1222no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—2442no—
508C2×C8—2442no—
509C2×C8—2441no—
510C2×C8—2442no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—611no—
514C4⋊C4—611no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—1221no—
522C22⋊C4—1221no—
523C22⋊C4—1221no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42#1175822211yescharacteristic
530C42D4×D6111yescharacteristic
531C42D4×D6111yescharacteristic
532C42C23×D6111yescharacteristic
533C23—41no—
534C23—1221no—
535C23—2441no—
536C23#1451975011yescharacteristic
537Q8—41no—
538Q8—41no—
539Q8—612no—
540Q8—1221no—
541Q8—2441no—
542Q8—2441no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—81no—
548D4—82no—
549D4—612no—
550D4—1224no—
551D4—1221no—
552D4—4881no—
553D4—4882no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—82no—
563C2×C4—1221no—
564C2×C4—1224no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—4882no—
568C2×C4—2442no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4#1451975011yescharacteristic
573C2×C4#1451975011yescharacteristic
574C2×C4#1451975011yescharacteristic
575C2×C4D8×D6112yes—
576C2×C4#1451979812yes—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C8—41no—
581C8—41no—
582C8—41no—
583C8—41no—
584C8—612no—
585C8—1221no—
586C8—1221no—
587C8—1221no—
588C8—2441no—
589C8—2441no—
590C22—21no—
591C22—21no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—1222no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—2441no—
600C22—2441no—
601C22#1683999712yes—
602C22C2×#1451975011yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—81no—
607C4—1222no—
608C4—4881no—
609C4—2441no—
610C4—2441no—
611C4#1683999712yes—
612C4C2×D8×D6111yescharacteristic
613C4#135×D6111yescharacteristic
614C4C2×#1451979811yescharacteristic
615C4#1684055911yescharacteristic
616C2—22no—
617C2—161no—
618C2—9761no—
619C2—2442no—
620C2#2145527211yescharacteristic
621C2C2×#1683999711yescharacteristic
622C2#2145553811yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1540C2
socleC22×C61C2×#121
hypercenter#1540D61
hypocenterC61#19125
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
1. #21456394C2semidirect
2. #21455276C2semidirect
3. #21455268C2semidirect
4. #21454775C2semidirect
5. #21445266C2semidirect
6. #21445262C2semidirect
7. #21445259C2semidirect
8. #21445255C2semidirect
9. #21445212C2semidirect
10. #21445207C2semidirect
11. #21445178C2semidirect
12. #21445173C2semidirect
13. C2×#16839514C2semidirect
14. #21444466C2semidirect
15. C61×#1540C2semidirect
162. #11758222D8semidirect
277. #92418#135semidirect
319. #1540D61semidirect
378. C61#19125semidirect

Maximal subgroups

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

subgroupindex
#214563942
#214552762
#214552682
#214547752
#214452662
#214452622
#214452592
#214452552
#214452122
#214452072
#214451782
#214451732
C2×#168395142
#214444662
C61×#15402
#1912561