Finite Group Databasebeta

← #28317570

Subgroups — #28317570

Order 13568.

720 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291552C211yesmaximal, characteristic, complemented
2#20291455C211yesmaximal, characteristic, complemented
3#20290359C211yesmaximal, characteristic, complemented
4#20290349C211yesmaximal, characteristic, complemented
5#20290334C211yesmaximal, characteristic, complemented
6#20290333C211yesmaximal, characteristic, complemented
7#20290321C211yesmaximal, characteristic, complemented
8#20290314C211yesmaximal, characteristic, complemented
9#20290312C211yesmaximal, characteristic, complemented
10#20290295C211yesmaximal, characteristic, complemented
11#20290279C211yesmaximal, characteristic, complemented
12#20290056C211yesmaximal, characteristic, complemented
13#20290050C211yesmaximal, characteristic, complemented
14#20289938C211yesmaximal, characteristic, complemented
15#20283528C211yesmaximal, characteristic
16C2×D53⋊M16—21no—
17#16811674C2211yescharacteristic, complemented
18#16811674C2211yescharacteristic, complemented
19#16811657C2211yescharacteristic
20#16811657C2211yescharacteristic
21#16811654—21no—
22#16811654—21no—
23#16811653—21no—
24#16811653—21no—
25#16811651C2211yescharacteristic
26#16811651C2211yescharacteristic
27D4×#88324C2211yescharacteristic, complemented
28D4×#88324C2211yescharacteristic, complemented
29#16811644C2211yescharacteristic
30#16811643—21no—
31#16811626C2211yescharacteristic
32#16811595—21no—
33C4×#11719940C2211yescharacteristic
34C4×#11719940C2211yescharacteristic
35C2×#13133272—21no—
36#16811487C2211yescharacteristic, complemented
37#16811486C2211yescharacteristic, complemented
38#16811485C2211yescharacteristic, complemented
39#16811484C2211yescharacteristic, complemented
40Q8⋊#88324C2211yescharacteristic, complemented
41Q8⋊#88324C2211yescharacteristic, complemented
42#16811471C2211yescharacteristic
43#16811471C2211yescharacteristic
44#16811471C2211yescharacteristic
45#16811471C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic, complemented
47D4⋊#88324C2211yescharacteristic, complemented
48#88324⋊C8C2211yescharacteristic, complemented
49#88324⋊C8C2211yescharacteristic, complemented
50#16811464C2211yescharacteristic, complemented
51C8×#88324C2211yescharacteristic, complemented
52C2×#13133242C411yescharacteristic
53#41×Dic53C2211yescharacteristic
54C2×#13133179C2211yescharacteristic
55#16810732C411yescharacteristic
56#16810732C411yescharacteristic
57#16810726C411yescharacteristic
58#16810726C411yescharacteristic
59#16810708C2211yescharacteristic
60#16810708C2211yescharacteristic
61#16810703C2211yescharacteristic
62#16810703C2211yescharacteristic
63#16810663C411yescharacteristic
64#16810629C411yescharacteristic
65M16×Dic53C2211yescharacteristic
66C53×#417C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719936—21no—
69D53⋊M16—44no—
70C2×#11719940—21no—
71C2×#11719940—21no—
72C2×#11719940D411yescharacteristic
73C2×#11719940D411yescharacteristic
74C2×D53⋊C8—21no—
75C2×D53⋊C8—21no—
76C22×#88324—21no—
77C22×#88324—21no—
78#13133284—21no—
79#13133284—21no—
80#13133284—21no—
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C4⋊#88324C2311yescharacteristic
85#13133274—21no—
86#13133274—21no—
87#13133274—21no—
88#13133274—21no—
89C22⋊#88324—21no—
90C22⋊#88324—21no—
91C22⋊#88324—21no—
92#13133272—21no—
93#13133272—21no—
94#13133272—21no—
95#13133272—21no—
96#13133272D411yescharacteristic
97#13133272D411yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C4×#88324C2311yescharacteristic
102C22×D212C2×C411yescharacteristic
103C22×C4×D53—21no—
104#13133242—24no—
105C2×#11719924C2×C411yescharacteristic
106C2×#11719924C2×C411yescharacteristic
107C2×#11719922C2×C411yescharacteristic
108C2×#11719922C2×C411yescharacteristic
109C2×#11719917C2×C411yescharacteristic
110Q8×Dic53C2×C411yescharacteristic
111Q8×Dic53C2311yescharacteristic
112D4×Dic53—21no—
113D4×Dic53—21no—
114D4×Dic53C2×C411yescharacteristic
115D4×Dic53C2311yescharacteristic
116#13133179—21no—
117#13133179—21no—
118#13133179C2×C411yescharacteristic
119#13133179C2×C411yescharacteristic
120#13133179C2311yescharacteristic
121#13133179C2311yescharacteristic
122#13133176—21no—
123#13133176C2×C411yescharacteristic
124C2×#11719921—21no—
125C2×C4×Dic53—21no—
126C2×C4×Dic53C2311yescharacteristic
127#13133150C2×C411yescharacteristic
128#13133150C2×C411yescharacteristic
129#13133148C2×C411yescharacteristic
130#13133148C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133139C2×C411yescharacteristic
133#13133139C2×C411yescharacteristic
134#13133139C2×C411yescharacteristic
135#13133127C2×C411yescharacteristic
136#13133127C2×C411yescharacteristic
137C8×Dic53C2311yescharacteristic
138C8×Dic53C2311yescharacteristic
139C2×#41×C53C2×C411yescharacteristic
140C2×M16×C53C2×C411yescharacteristic
141C2×C4⋊C4×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#103×C53C2×C411yescharacteristic
144#102×C53C2×C411yescharacteristic
145#102×C53C2×C411yescharacteristic
146C23×D53—21no—
147#41×C53—22no—
148#41×C53—41no—
149C2×Q8×C53C22×C411yescharacteristic
150C2×D4×C53—21no—
151C2×D4×C53C22×C411yescharacteristic
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53C22×C411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53#4111yescharacteristic
159C2×D212—22no—
160C2×D212—21no—
161C2×D212—41no—
162C2×D212C22×C411yescharacteristic
163C2×D212C22×C411yescharacteristic
164C2×C4×D53—21no—
165C2×C4×D53—21no—
166C2×C4×D53—21no—
167C2×C4×D53—21no—
168C2×C4×D53—42no—
169C2×C4×D53—42no—
170#11719940—22no—
171#11719940—41no—
172#11719940—42no—
173#11719940—42no—
174C2×#88324—21no—
175C2×#88324—21no—
176C2×#88324—21no—
177C2×#88324—21no—
178C2×#88324—21no—
179C2×#88324—21no—
180C2×#88324—42no—
181C2×#88324—42no—
182C2×#88324C2×D411yescharacteristic
183C2×#88324C2×D411yescharacteristic
184C2×#88324C2×D411yescharacteristic
185C2×#88324C2×D411yescharacteristic
186#11719936—42no—
187#11719936—42no—
188D53⋊C8—42no—
189D53⋊C8—42no—
190M16×C53—41no—
191C2×C8×C53C22×C411yescharacteristic
192C2×C8×C53C22×C411yescharacteristic
193C4⋊C4×C53—21no—
194C4⋊C4×C53C22×C411yescharacteristic
195C4⋊C4×C53C22×C411yescharacteristic
196C22⋊Dic53—21no—
197C22⋊Dic53—21no—
198C22⋊Dic53—21no—
199#11719924—22no—
200#11719924—22no—
201#11719922—22no—
202#11719922M1614yescomplemented
203#11719921—21no—
204#11719921—21no—
205#11719921—21no—
206#11719921—21no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53C22×C411yescharacteristic
210C4⋊Dic53C22×C411yescharacteristic
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—21no—
214C4×Dic53C22×C411yescharacteristic
215C4×Dic53C2×D411yescharacteristic
216C4×Dic53#4111yescharacteristic
217C4×Dic53C2411yescharacteristic
218#11719917—22no—
219C2×#88322C22×C411yescharacteristic
220C2×#88322C22×C411yescharacteristic
221C23×C53—21no—
222C23×C53C4×D411yescharacteristic
223C22×D53—21no—
224C22×D53—21no—
225C22×D53—42no—
226C22×D53—42no—
227C22×D53—81no—
228Q8×C53—21no—
229Q8×C53—21no—
230D4×C53—21no—
231D4×C53—42no—
232D4×C53C2×M1612yescomplemented
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53C4×D411yescharacteristic
241C2×C4×C53C23×C411yescharacteristic
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—81no—
250C2×Dic53C4×D411yescharacteristic
251C2×Dic53C4×D411yescharacteristic
252C2×Dic53C22×D411yescharacteristic
253C2×Dic53C2×#4111yescharacteristic
254D212—21no—
255D212—21no—
256D212—21no—
257D212—42no—
258D212C2×M1612yes—
259C4×D53—42no—
260C4×D53—42no—
261C4×D53—41no—
262C4×D53—41no—
263C4×D53—41no—
264C4×D53—41no—
265#88324—22no—
266#88324—21no—
267#88324—21no—
268#88324—21no—
269#88324—41no—
270#88324—41no—
271#88324—41no—
272#88324—41no—
273#88324—41no—
274#88324AGL(1,ℤ/8)12yescomplemented
275C8×C53—21no—
276C8×C53—21no—
277#88322—21no—
278#88322C2×M1612yescomplemented
279#18046—531nomaximal, Sylow
280C22×C53—21no—
281C22×C53—22no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53C2×C4×D411yescharacteristic
287C2×D53—22no—
288C2×D53—41no—
289C2×D53—41no—
290C2×D53—41no—
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—84no—
294C4×C53—21no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53C22×M1611yescharacteristic
299C4×C53#56811yescharacteristic
300Dic53—21no—
301Dic53—21no—
302Dic53—21no—
303Dic53—41no—
304Dic53—41no—
305Dic53C2×AGL(1,ℤ/8)11yescharacteristic
306Dic53#57511yescharacteristic
307#2970—531no—
308C4×AGL(1,ℤ/8)—531no—
309D4×M16—531no—
310#1592—531no—
311#1590—531no—
312#1586—531no—
313#1585—531no—
314#1582—531no—
315#1581—531no—
316#1580—531no—
317C8×D8—531no—
318#1569—531no—
319#1490—531no—
320#1489—531no—
321#1455—531no—
322C2×C53—41no—
323C2×C53—42no—
324C2×C53D4×M1611yescharacteristic
325C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
326C2×C53#295611yescharacteristic
327D53—42no—
328D53—81no—
329C2×AGL(1,ℤ/8)—531no—
330C22×M16—1061no—
331C4×#41—531no—
332C2×C4×D4—531no—
333#446—531no—
334#446—531no—
335#442—531no—
336#442—531no—
337#440—531no—
338#440—531no—
339C4×SD16—531no—
340C4×SD16—531no—
341C4×D8—531no—
342C4×D8—531no—
343#436—531no—
344#436—531no—
345#435—1061no—
346#435—1061no—
347#435—1061no—
348#435—1061no—
349C8×D4—531no—
350C8×D4—531no—
351C8×D4—531no—
352C8×D4—531no—
353#432—531no—
354#432—1061no—
355#428—531no—
356#423—531no—
357#418—531no—
358#417#615711yescharacteristic
359#407—1061no—
360C4×M16—531no—
361C4×M16—531no—
362C4×M16—531no—
363C2×#97—1061no—
364#335—531no—
365#335—531no—
366#334—531no—
367#334—531no—
368Q8⋊C8—531no—
369Q8⋊C8—531no—
370D4⋊C8—531no—
371D4⋊C8—531no—
372D4⋊C8—531no—
373D4⋊C8—531no—
374D4⋊C8—531no—
375D4⋊C8—531no—
376#322—531no—
377#322—531no—
378#322—531no—
379C82—531no—
380C53#1804611yescharacteristic, complemented, Sylow
381C2×#41C2×#615711yescharacteristic
382C22×D4—531no—
383C23×C4—1061no—
384AGL(1,ℤ/8)—1064no—
385C2×SD16—531no—
386C2×SD16—531no—
387C2×D8—531no—
388C2×D8—531no—
389C2×M16—531no—
390C2×M16—531no—
391C2×M16—531no—
392C2×M16—1061no—
393C2×M16—1061no—
394C2×M16—1061no—
395C2×M16—1061no—
396C2×M16—2124no—
397C2×M16C2×#615711yescharacteristic
398C22×C8—1061no—
399C22×C8—1061no—
400C22×C8—1061no—
401C22×C8—1061no—
402C4×Q8—531no—
403C4×Q8—531no—
404C4×D4—531no—
405C4×D4—531no—
406C4×D4—531no—
407C4×D4—531no—
408C4×D4—531no—
409C4×D4—531no—
410C4×D4—1061no—
411C4×D4—1061no—
412C4×D4—1061no—
413C4×D4—1061no—
414#117—531no—
415#117—1061no—
416C2×C4⋊C4C2×#615711yescharacteristic
417C2×C22⋊C4—1061no—
418C2×C42—531no—
419C2×C42—1061no—
420#107—531no—
421#107—531no—
422#106—531no—
423#106—531no—
424C4⋊C8—531no—
425C4⋊C8—531no—
426C4⋊C8—531no—
427C4⋊C8—531no—
428C4⋊C8—1061no—
429C4⋊C8—1061no—
430C4⋊C8—1061no—
431#103C2×#615711yescharacteristic
432#103C2×#615711yescharacteristic
433#102—531no—
434#102—531no—
435#102—531no—
436#102—531no—
437#102C2×#615711yescharacteristic
438#102C2×#615711yescharacteristic
439C22⋊C8—1061no—
440C22⋊C8—1061no—
441C22⋊C8—1061no—
442C22⋊C8—1061no—
443C22⋊C8—1061no—
444C22⋊C8—1061no—
445C22⋊C8—1061no—
446#97—531no—
447#97—531no—
448#97—531no—
449#97—531no—
450#97—1061no—
451#97—1061no—
452#97—1061no—
453#97—1061no—
454C4×C8—531no—
455C4×C8—531no—
456C4×C8—531no—
457C4×C8—531no—
458C4×C8—531no—
459C4×C8—531no—
460C24—1061no—
461#41—22no—
462#41—41no—
463C2×Q8C22×#615711yescharacteristic
464C2×D4—21no—
465C2×D4—531no—
466C2×D4—531no—
467C2×D4—1062no—
468C2×D4—1061no—
469C2×D4—2121no—
470C2×D4C22×#615711yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—2122no—
482C22×C4—2122no—
483C22×C4C22×#615711yescharacteristic
484SD16—1062no—
485SD16—1062no—
486D8—534no—
487D8—1062no—
488M16—41no—
489M16—1062no—
490M16—1062no—
491M16—2122no—
492M16—2121no—
493M16—2122no—
494M16—2122no—
495M16—2122no—
496C2×C8—531no—
497C2×C8—531no—
498C2×C8—531no—
499C2×C8—531no—
500C2×C8—531no—
501C2×C8—531no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—2122no—
509C2×C8—2122no—
510C2×C8—2122no—
511C2×C8—2122no—
512C2×C8C22×#615711yescharacteristic
513C2×C8C22×#615711yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—531no—
516C4⋊C4—531no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4C22×#615711yescharacteristic
520C4⋊C4C22×#615711yescharacteristic
521C22⋊C4—1061no—
522C22⋊C4—1061no—
523C22⋊C4—1061no—
524C22⋊C4—1061no—
525C22⋊C4—1061no—
526C22⋊C4—1061no—
527C22⋊C4—1061no—
528C42—531no—
529C42—531no—
530C42—531no—
531C42—531no—
532C42—1061no—
533C42—1061no—
534C42—1061no—
535C23—21no—
536C23—1061no—
537C23—1061no—
538C23—4241no—
539C23—2122no—
540C23—2122no—
541C23D4×#615711yescharacteristic
542Q8—21no—
543Q8—21no—
544D4—21no—
545D4—42no—
546D4—532no—
547D4—1061no—
548D4—1061no—
549D4—1061no—
550D4—2122no—
551D4D53⋊M1612yescomplemented
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—531no—
560C2×C4—531no—
561C2×C4—531no—
562C2×C4—531no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—4241no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4D4×#615711yescharacteristic
578C2×C4C23×#615711yescharacteristic
579C8—21no—
580C8—21no—
581C8—532no—
582C8—532no—
583C8—1061no—
584C8—1061no—
585C8—1061no—
586C8—1062no—
587C8—1061no—
588C8—2121no—
589C8—2121no—
590C8—2121no—
591C8—2121no—
592C8—2121no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—41no—
598C22—41no—
599C22—1062no—
600C22—4244no—
601C22—2121no—
602C22—2121no—
603C22—2121no—
604C22—2121no—
605C22—2121no—
606C22C2×D4×#615711yescharacteristic
607C4—21no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—531no—
612C4—531no—
613C4—1061no—
614C4—1061no—
615C4—1061no—
616C4—2121no—
617C4—2121no—
618C4C2×D53⋊M1611yescharacteristic
619C4#1681170111yescharacteristic
620C2—41no—
621C2—42no—
622C2—4241no—
623C2—2122no—
624C2#2029145311yescharacteristic
625C2AGL(1,ℤ/8)×#615711yescharacteristic
626C2#2029149011yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#417C4
socleC22×C53C2×C4×D4
prime radicalC2×#13133242C4
cosocleC4×Dic53C24
hypercenter#417#6157
hypocenterC53#18046
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #20291552C2semidirect
2. #20291455C2semidirect
3. #20290359C2semidirect
4. #20290349C2semidirect
5. #20290334C2semidirect
6. #20290333C2semidirect
7. #20290321C2semidirect
8. #20290314C2semidirect
9. #20290312C2semidirect
10. #20290295C2semidirect
11. #20290279C2semidirect
12. #20290056C2semidirect
13. #20290050C2semidirect
14. #20289938C2semidirect
17. #16811674C22semidirect
18. #16811674C22semidirect
27. D4×#88324C22semidirect
28. D4×#88324C22semidirect
36. #16811487C22semidirect
37. #16811486C22semidirect
38. #16811485C22semidirect
39. #16811484C22semidirect
40. Q8⋊#88324C22semidirect
41. Q8⋊#88324C22semidirect
46. D4⋊#88324C22semidirect
47. D4⋊#88324C22semidirect
48. #88324⋊C8C22semidirect
49. #88324⋊C8C22semidirect
50. #16811464C22semidirect
51. C8×#88324C22semidirect
202. #11719922M16semidirect
232. D4×C53C2×M16semidirect
274. #88324AGL(1,ℤ/8)semidirect
278. #88322C2×M16semidirect
380. C53#18046semidirect
551. D4D53⋊M16semidirect

Maximal subgroups

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

subgroupindex
#202915522
#202914552
#202903592
#202903492
#202903342
#202903332
#202903212
#202903142
#202903122
#202902952
#202902792
#202900562
#202900502
#202899382
#202835282
#1804653