Finite Group Databasebeta

← #28317348

Subgroups — #28317348

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291666C211yesmaximal, characteristic, complemented
2#20291571C211yesmaximal, characteristic, complemented
3#20290351C211yesmaximal, characteristic, complemented
4#20290346C211yesmaximal, characteristic, complemented
5#20290344C211yesmaximal, characteristic, complemented
6#20290337C211yesmaximal, characteristic, complemented
7#20290334C211yesmaximal, characteristic, complemented
8#20290332C211yesmaximal, characteristic, complemented
9SD16×#88324C211yesmaximal, characteristic, complemented
10#20290295C211yesmaximal, characteristic, complemented
11#20290288C211yesmaximal, characteristic, complemented
12#20290068C211yesmaximal, characteristic, complemented
13C2×D4⋊#88324C211yesmaximal, characteristic, complemented
14#20289920C211yesmaximal, characteristic, complemented
15#20284763C211yesmaximal, characteristic
16#16811674C2211yescharacteristic, complemented
17Q8×#88324C2211yescharacteristic, complemented
18#16811658C2211yescharacteristic
19#16811658C2211yescharacteristic
20#16811657C2211yescharacteristic, complemented
21#16811654—21no—
22#16811654—21no—
23#16811652—21no—
24#16811651C2211yescharacteristic, complemented
25#16811650—21no—
26D4×#88324C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811648—21no—
29#16811644C2211yescharacteristic
30#16811640C2211yescharacteristic
31#16811630—21no—
32C2×C4⋊#88324C2211yescharacteristic
33#16811593—21no—
34C2×C22⋊#88324—21no—
35C2×C4×#88324C2211yescharacteristic
36#16811487C2211yescharacteristic, complemented
37#16811487C2211yescharacteristic, complemented
38#16811484C2211yescharacteristic, complemented
39#16811484C2211yescharacteristic, complemented
40Q8⋊#88324C2211yescharacteristic, complemented
41Q8⋊#88324C2211yescharacteristic, complemented
42#16811471C2211yescharacteristic, complemented
43#16811471C2211yescharacteristic, complemented
44D4⋊#88324C2211yescharacteristic
45D4⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic, complemented
49#88324⋊C8C2211yescharacteristic, complemented
50C8×#88324C2211yescharacteristic, complemented
51C8×#88324C2211yescharacteristic, complemented
52#16811159C2211yescharacteristic
53C2×D4×Dic53C2211yescharacteristic
54#16810956C411yescharacteristic
55#16810942C411yescharacteristic
56#16810925C411yescharacteristic
57#16810925C411yescharacteristic
58#16810840C411yescharacteristic
59#16810840C411yescharacteristic
60#16810757C411yescharacteristic
61SD16×Dic53C2211yescharacteristic
62SD16×Dic53C2211yescharacteristic
63#16810703C2211yescharacteristic
64#16810703C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#450C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719940—21no—
69C2×#11719940—21no—
70C2×D53⋊C8—21no—
71C22×#88324—21no—
72C22×#88324—21no—
73C22×#88324—21no—
74C22×#88324—21no—
75#13133284—21no—
76#13133284D411yescharacteristic
77#13133284D411yescharacteristic
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324—21no—
89C22⋊#88324—21no—
90C22⋊#88324—21no—
91C22⋊#88324D411yescharacteristic
92C22⋊#88324D411yescharacteristic
93#13133272—21no—
94#13133272—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C23×Dic53—21no—
102C2×#11719922C2×C411yescharacteristic
103C2×#11719922C2×C411yescharacteristic
104#13133212—21no—
105#13133208C2×C411yescharacteristic
106#13133206C2×C411yescharacteristic
107#13133201—21no—
108#13133191C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110D4×Dic53—21no—
111D4×Dic53—21no—
112D4×Dic53C2×C411yescharacteristic
113D4×Dic53C2×C411yescharacteristic
114D4×Dic53C2311yescharacteristic
115D4×Dic53C2311yescharacteristic
116#13133179C2311yescharacteristic
117#13133177—21no—
118#13133175—21no—
119C2×C4⋊Dic53C2×C411yescharacteristic
120C2×C22⋊Dic53—21no—
121C2×C4×Dic53C2311yescharacteristic
122#13133150C2×C411yescharacteristic
123#13133150C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133149C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133139C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133138C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C22×D4×C53C2×C411yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139C2×SD16×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#102×C53C2×C411yescharacteristic
142#102×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144C24×C53—21no—
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53—22no—
147C2×D4×C53—21no—
148C2×D4×C53—41no—
149C2×D4×C53C22×C411yescharacteristic
150C2×D4×C53C22×C411yescharacteristic
151C22×C4×C53C22×C411yescharacteristic
152C2×C22⋊D53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—81no—
159C22×Dic53—81no—
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53#4111yescharacteristic
162C2×D212C22×C411yescharacteristic
163C2×C4×D53—21no—
164#11719940—81no—
165#11719940—81no—
166C2×#88324—24no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—81no—
174C2×#88324—81no—
175C2×#88324—81no—
176C2×#88324C2×D411yescharacteristic
177C2×#88324C2×D411yescharacteristic
178C2×#88324C2×D411yescharacteristic
179C2×#88324C2×D411yescharacteristic
180#11719936—81no—
181D53⋊C8—81no—
182SD16×C53—22no—
183SD16×C53—41no—
184C2×C8×C53C22×C411yescharacteristic
185C2×C8×C53C22×C411yescharacteristic
186C4⋊C4×C53—21no—
187C4⋊C4×C53C22×C411yescharacteristic
188C22⋊C4×C53—21no—
189C22⋊Dic53—21no—
190C22⋊Dic53—21no—
191C22⋊Dic53—21no—
192C22⋊Dic53—21no—
193#11719922—22no—
194#11719922—41no—
195#11719921—21no—
196#11719921—21no—
197#11719921—21no—
198#11719921—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53C22×C411yescharacteristic
203C4⋊Dic53C22×C411yescharacteristic
204C4⋊Dic53C22×C411yescharacteristic
205#11719919—21no—
206C4×Dic53—21no—
207C4×Dic53C22×C411yescharacteristic
208C4×Dic53C2×D411yescharacteristic
209C4×Dic53#4111yescharacteristic
210C4×Dic53C2411yescharacteristic
211C2×#88322C22×C411yescharacteristic
212C2×#88322C22×C411yescharacteristic
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—81no—
218C23×C53C4×D411yescharacteristic
219C22×D53—21no—
220Q8×C53—21no—
221Q8×C53—21no—
222D4×C53—21no—
223D4×C53—21no—
224D4×C53—21no—
225D4×C53—42no—
226D4×C53#13112yes—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C23×C411yescharacteristic
233C22⋊D53—42no—
234C2×Dic53—22no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—81no—
245C2×Dic53—81no—
246C2×Dic53—81no—
247C2×Dic53C4×D411yescharacteristic
248C2×Dic53C4×D411yescharacteristic
249C2×Dic53C22×D411yescharacteristic
250C2×Dic53C2×#4111yescharacteristic
251D212—21no—
252D212—21no—
253C4×D53—41no—
254C4×D53—41no—
255#88324—22no—
256#88324—22no—
257#88324—21no—
258#88324—21no—
259#88324—21no—
260#88324—21no—
261#88324—41no—
262#88324—41no—
263#88324—41no—
264#88324—41no—
265C8×C53—21no—
266C8×C53—21no—
267#88322—21no—
268#88322—21no—
269#18014—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—22no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—84no—
279C22×C53#44312yes—
280C22×C53C2×C4×D411yescharacteristic
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—41no—
284C4×C53—21no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53C2×#13111yescharacteristic
288C4×C53#56811yescharacteristic
289Dic53—21no—
290Dic53—21no—
291Dic53—21no—
292Dic53—41no—
293Dic53—41no—
294Dic53—41no—
295Dic53#57611yescharacteristic
296Dic53#57711yescharacteristic
297#3105—531no—
298#2994—531no—
299#2974—531no—
300#1590—531no—
301#1589—531no—
302#1588—531no—
303#1586—531no—
304#1586—531no—
305#1585—531no—
306C8×SD16—531no—
307C8×D8—531no—
308#1573—531no—
309#1494—531no—
310C2×D4⋊C8—531no—
311#1450—531no—
312C2×C53—22no—
313C2×C53—42no—
314C2×C53—81no—
315C2×C53#293311yescharacteristic
316C2×C53#294611yescharacteristic
317C2×C53C2×#44311yescharacteristic
318D53—81no—
319#529—531no—
320C2×C4×D4—531no—
321#484—531no—
322#481—531no—
323#478—531no—
324#478—531no—
325#463—531no—
326#463—531no—
327#450#615711yescharacteristic
328#447—531no—
329#446—531no—
330C8×Q8—531no—
331C4×SD16—531no—
332C4×SD16—531no—
333C4×D8—531no—
334C4×D8—531no—
335#436—531no—
336#436—531no—
337#436—531no—
338#435—1061no—
339#435—1061no—
340#435—1061no—
341C8×D4—531no—
342C8×D4—531no—
343C8×D4—531no—
344C8×D4—1061no—
345#433—1061no—
346#432—531no—
347#431—531no—
348#431—531no—
349#424—1061no—
350C2×C4⋊C8—531no—
351#407—1061no—
352C2×C22⋊C8—1061no—
353C2×C4×C8—531no—
354#335—531no—
355#335—531no—
356#334—531no—
357#334—531no—
358Q8⋊C8—531no—
359Q8⋊C8—531no—
360D4⋊C8—531no—
361D4⋊C8—531no—
362D4⋊C8—531no—
363D4⋊C8—531no—
364D4⋊C8—531no—
365D4⋊C8—531no—
366#322—531no—
367#322—531no—
368C82—531no—
369C82—531no—
370C53#1801411yescharacteristic, complemented, Sylow
371C22×D4C2×#615711yescharacteristic
372C23×C4—1061no—
373C2×SD16C2×#615711yescharacteristic
374C2×SD16C2×#615711yescharacteristic
375C2×D8—531no—
376C2×D8—531no—
377C2×M16—1061no—
378C2×M16—1061no—
379C2×M16—1061no—
380C22×C8—1061no—
381C22×C8—1061no—
382C22×C8—1061no—
383C22×C8—1061no—
384C22×C8—1061no—
385#126—1061no—
386#125—531no—
387#124—531no—
388#123—1061no—
389C22⋊Q8C2×#615711yescharacteristic
390#121—531no—
391C4×Q8—531no—
392C4×D4—531no—
393C4×D4—531no—
394C4×D4—531no—
395C4×D4—531no—
396C4×D4—531no—
397C4×D4—1061no—
398C4×D4—1061no—
399C4×D4—1061no—
400#117—1061no—
401C2×C4⋊C4—531no—
402C2×C22⋊C4—1061no—
403C2×C42—531no—
404#107—531no—
405#107—531no—
406#106—531no—
407#106—531no—
408C4⋊C8—531no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8—531no—
412C4⋊C8—531no—
413C4⋊C8—531no—
414C4⋊C8—531no—
415C4⋊C8—531no—
416C4⋊C8—1061no—
417C4⋊C8—1061no—
418C4⋊C8—1061no—
419#103—531no—
420#103—531no—
421#102—531no—
422#102—531no—
423#102—531no—
424#102—531no—
425#102C2×#615711yescharacteristic
426#102C2×#615711yescharacteristic
427C22⋊C8—531no—
428C22⋊C8—531no—
429C22⋊C8—531no—
430C22⋊C8—1061no—
431C22⋊C8—1061no—
432C22⋊C8—1061no—
433C22⋊C8—1061no—
434C22⋊C8—1061no—
435C22⋊C8—1061no—
436C22⋊C8—1061no—
437C22⋊C8C2×#615711yescharacteristic
438#97—1061no—
439#97—1061no—
440C4×C8—531no—
441C4×C8—531no—
442C4×C8—531no—
443C4×C8—531no—
444C4×C8—531no—
445C4×C8—531no—
446C4×C8—1061no—
447C4×C8—1061no—
448C24—21no—
449C2×Q8C22×#615711yescharacteristic
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4—531no—
454C2×D4—1061no—
455C2×D4C22×#615711yescharacteristic
456C2×D4C22×#615711yescharacteristic
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—4241no—
466C22×C4—4241no—
467C22×C4C22×#615711yescharacteristic
468SD16—22no—
469SD16—41no—
470D8—1062no—
471D8—2121no—
472M16—4241no—
473M16—4241no—
474M16—4241no—
475C2×C8—531no—
476C2×C8—531no—
477C2×C8—531no—
478C2×C8—531no—
479C2×C8—531no—
480C2×C8—531no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1064no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—4241no—
489C2×C8—4241no—
490C2×C8—4241no—
491C2×C8—4241no—
492C2×C8C22×#615711yescharacteristic
493C2×C8C22×#615711yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—531no—
496C4⋊C4—531no—
497C4⋊C4—531no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4C22×#615711yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1061no—
505C22⋊C4—1061no—
506C22⋊C4—1061no—
507C22⋊C4—1061no—
508C22⋊C4—1061no—
509C22⋊C4—1061no—
510C22⋊C4—1061no—
511C22⋊C4—1061no—
512C42—531no—
513C42—531no—
514C42—531no—
515C42—531no—
516C42—1061no—
517C23—21no—
518C23—21no—
519C23—42no—
520C23—42no—
521C23—81no—
522C23—1061no—
523C23D4×#615711yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—42no—
530D4—1061no—
531D4—1061no—
532D4—2122no—
533D4#1313330412yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—531no—
539C2×C4—531no—
540C2×C4—531no—
541C2×C4—531no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1062no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—4241no—
549C2×C4—4241no—
550C2×C4—4241no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4D4×#615711yescharacteristic
558C2×C4C23×#615711yescharacteristic
559C8—21no—
560C8—21no—
561C8—1061no—
562C8—1061no—
563C8—1061no—
564C8—1061no—
565C8—1062no—
566C8—1062no—
567C8—1061no—
568C8—1061no—
569C8—2121no—
570C8—2121no—
571C8—2121no—
572C8—2121no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—84no—
582C22—2121no—
583C22—2121no—
584C22—2121no—
585C22#1681166612yes—
586C22C2×D4×#615711yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—531no—
591C4—531no—
592C4—1061no—
593C4—1061no—
594C4—1061no—
595C4—2121no—
596C4—2121no—
597C4—2121no—
598C4C2×#1313330411yescharacteristic
599C4#1681170111yescharacteristic
600C2—22no—
601C2—42no—
602C2—81no—
603C2—4241no—
604C2#2029144711yescharacteristic
605C2#2029147211yescharacteristic
606C2C2×#1681166611yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#450C4
socleC22×C53C2×C4×D4
prime radical#16810757C4
cosocleC4×Dic53C24
hypercenter#450#6157
hypocenterC53#18014
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20291666C2semidirect
2. #20291571C2semidirect
3. #20290351C2semidirect
4. #20290346C2semidirect
5. #20290344C2semidirect
6. #20290337C2semidirect
7. #20290334C2semidirect
8. #20290332C2semidirect
9. SD16×#88324C2semidirect
10. #20290295C2semidirect
11. #20290288C2semidirect
12. #20290068C2semidirect
13. C2×D4⋊#88324C2semidirect
14. #20289920C2semidirect
16. #16811674C22semidirect
17. Q8×#88324C22semidirect
20. #16811657C22semidirect
24. #16811651C22semidirect
36. #16811487C22semidirect
37. #16811487C22semidirect
38. #16811484C22semidirect
39. #16811484C22semidirect
40. Q8⋊#88324C22semidirect
41. Q8⋊#88324C22semidirect
42. #16811471C22semidirect
43. #16811471C22semidirect
48. #88324⋊C8C22semidirect
49. #88324⋊C8C22semidirect
50. C8×#88324C22semidirect
51. C8×#88324C22semidirect
370. C53#18014semidirect

Maximal subgroups

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

subgroupindex
#202916662
#202915712
#202903512
#202903462
#202903442
#202903372
#202903342
#202903322
SD16×#883242
#202902952
#202902882
#202900682
C2×D4⋊#883242
#202899202
#202847632
#1801453