Finite Group Databasebeta

← #27354227

Subgroups — #27354227

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284721C211yesmaximal, characteristic
2#20283806C211yesmaximal, characteristic, complemented
3#20283736C211yesmaximal, characteristic
4#20273575C211yesmaximal, characteristic, complemented
5#20273569C211yesmaximal, characteristic, complemented
6#20273552C211yesmaximal, characteristic, complemented
7#20273550C211yesmaximal, characteristic, complemented
8#20273527C211yesmaximal, characteristic, complemented
9#20273525C211yesmaximal, characteristic, complemented
10C8×#11719913C211yesmaximal, characteristic, complemented
11#20273492C211yesmaximal, characteristic, complemented
12#20273156C211yesmaximal, characteristic
13#20273134C211yesmaximal, characteristic, complemented
14#20273009C211yesmaximal, characteristic, complemented
15C8×SD16×C53C211yesmaximal, characteristic, complemented
16#16811177C2211yescharacteristic
17C4×D4×D53C2211yescharacteristic
18#16810955C411yescharacteristic
19#16810952C411yescharacteristic
20#16810912C411yescharacteristic
21#16810911C411yescharacteristic
22#16810827C411yescharacteristic
23#16810826C411yescharacteristic
24#16810775C411yescharacteristic
25#16810772C411yescharacteristic
26#16810754C2211yescharacteristic, complemented
27#16810753—21no—
28#16810752—21no—
29#16810750C2211yescharacteristic, complemented
30C8×Dic106C2211yescharacteristic, complemented
31C4×#11719913C2211yescharacteristic
32C4×#11719924C2211yescharacteristic
33C4×#11719923C2211yescharacteristic
34C8×D212C2211yescharacteristic
35D4×#88322C2211yescharacteristic
36#16810688C2211yescharacteristic
37C8×C22⋊D53—21no—
38#16810686—21no—
39#16810683—21no—
40#16810682—21no—
41#16810679C2211yescharacteristic
42#16810675C2211yescharacteristic
43#16810674C2211yescharacteristic
44#16810674C2211yescharacteristic
45C2×#13133129—21no—
46C22⋊C8×D53—21no—
47C4×#11719912C2211yescharacteristic
48#16810375C2211yescharacteristic, complemented
49#16810375C2211yescharacteristic, complemented
50#16810374C2211yescharacteristic, complemented
51#16810355C2211yescharacteristic, complemented
52Q8⋊#88322C2211yescharacteristic, complemented
53#16810353C2211yescharacteristic, complemented
54#16810352C2211yescharacteristic
55D4⋊#88322C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic, complemented
58#16810345C2211yescharacteristic, complemented
59#16810344C2211yescharacteristic, complemented
60C8×Q8×C53C2211yescharacteristic, complemented
61C4×SD16×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#334C2211yescharacteristic, complemented
64C53×Q8⋊C8C2211yescharacteristic, complemented
65C53×D4⋊C8C2211yescharacteristic
66C82×C53C2211yescharacteristic, complemented
67C2×D4×D53C2×C411yescharacteristic
68C22×C4×D53—21no—
69C2×#11719913C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719924C2×C411yescharacteristic
72C2×#11719912—21no—
73C22×#88322—21no—
74C2×C8×D53—21no—
75C2×C8×D53—21no—
76#13133216C2×C411yescharacteristic
77#13133215C2×C411yescharacteristic
78#13133207—21no—
79#13133197C2×C411yescharacteristic
80#13133195—21no—
81#13133193C2×C411yescharacteristic
82C4×Dic106—21no—
83C4×Dic106C2311yescharacteristic
84C4×D212C2311yescharacteristic
85D4×Dic53C2×C411yescharacteristic
86#13133179C2×C411yescharacteristic
87C4×C22⋊D53—21no—
88#13133177—21no—
89#13133174—21no—
90C4⋊C4×D53C2×C411yescharacteristic
91C22⋊C4×D53—21no—
92C42×D53C2311yescharacteristic
93#13133149C2×C411yescharacteristic
94#13133148C2×C411yescharacteristic
95#13133148C2×C411yescharacteristic
96C4⋊#88322—21no—
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133146—21no—
100#13133146—21no—
101#13133146C2×C411yescharacteristic
102#13133146C2×C411yescharacteristic
103#13133146D411yescharacteristic
104#13133146D411yescharacteristic
105#13133142C2×C411yescharacteristic
106#13133141C2×C411yescharacteristic
107#13133140C2×C411yescharacteristic
108#13133139C2×C411yescharacteristic
109#13133138C2×C411yescharacteristic
110#13133137C2×C411yescharacteristic
111C22⋊#88322—21no—
112#13133129—21no—
113#13133129—21no—
114#13133129—21no—
115#13133129—21no—
116#13133129C2×C411yescharacteristic
117#13133129C2×C411yescharacteristic
118#13133129D411yescharacteristic
119#13133129D411yescharacteristic
120#13133128—21no—
121#13133127—21no—
122#13133127—21no—
123#13133127—21no—
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C4×#88322C2311yescharacteristic
127C8×Dic53—21no—
128C8×Dic53—21no—
129C2×SD16×C53C2×C411yescharacteristic
130C22×C8×C53—21no—
131C4×Q8×C53C2311yescharacteristic
132C4×D4×C53C2311yescharacteristic
133#106×C53C2×C411yescharacteristic
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#103×C53C2×C411yescharacteristic
138#102×C53C2×C411yescharacteristic
139C22⋊C8×C53—21no—
140C4×C8×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C23×D53—21no—
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C2×C22⋊D53—21no—
149C22×Dic53—21no—
150D4×D53—22no—
151D4×D53—22no—
152C2×D212C22×C411yescharacteristic
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—81no—
157C2×C4×D53—81no—
158C2×C4×D53C22×C411yescharacteristic
159C2×C4×D53C2×D411yescharacteristic
160C2×C4×D53#4111yescharacteristic
161C2×Dic106—21no—
162C2×Dic106C22×C411yescharacteristic
163SD16×C53—22no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—81no—
168C2×C8×C53C22×C411yescharacteristic
169C2×C8×C53C22×C411yescharacteristic
170C2×C8×C53C2×D411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C4⋊C4×C53—21no—
173C4⋊C4×C53C22×C411yescharacteristic
174C4⋊C4×C53C22×C411yescharacteristic
175C22⋊C4×C53—21no—
176C42×C53—21no—
177C42×C53C2411yescharacteristic
178C22⋊Dic53—21no—
179#11719924—22no—
180#11719923—22no—
181#11719921—21no—
182#11719921—21no—
183#11719921—21no—
184#11719921—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53C22×C411yescharacteristic
187C4⋊Dic53C22×C411yescharacteristic
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194C4×Dic53—21no—
195C4×Dic53C22×C411yescharacteristic
196C4×Dic53C2×D411yescharacteristic
197C4×Dic53#4111yescharacteristic
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—81no—
202C2×#88322C22×C411yescharacteristic
203C2×#88322C22×C411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206#11719913—22no—
207#11719912—42no—
208C8×D53—81no—
209C8×D53—81no—
210C23×C53—21no—
211C22×D53—21no—
212C22×D53—24no—
213C22×D53—81no—
214C22×D53—81no—
215C22×D53C4×D411yescharacteristic
216Q8×C53—21no—
217Q8×C53—21no—
218D4×C53—21no—
219D4×C53#13112yes—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53C23×C411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C22⋊D53—24no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—81no—
233C2×Dic53C4×D411yescharacteristic
234C2×Dic53C4×D411yescharacteristic
235C2×Dic53C4×D411yescharacteristic
236D212—21no—
237D212#13112yes—
238C4×D53—21no—
239C4×D53—21no—
240C4×D53—21no—
241C4×D53—21no—
242C4×D53—41no—
243C4×D53—41no—
244C4×D53—41no—
245C4×D53—81no—
246C4×D53—81no—
247Dic106—21no—
248Dic106—21no—
249Dic106—81no—
250C8×C53—22no—
251C8×C53—21no—
252C8×C53—21no—
253C8×C53—21no—
254C8×C53—21no—
255C8×C53—41no—
256C8×C53—41no—
257#88322—21no—
258#88322—21no—
259#88322—21no—
260#88322—21no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#17924—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53C2×C4×D411yescharacteristic
269C2×D53—21no—
270C2×D53—22no—
271C2×D53—21no—
272C2×D53—21no—
273C2×D53—41no—
274C2×D53—41no—
275C2×D53—84no—
276C4×C53—21no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53C2×#13111yescharacteristic
281C4×C53C2×#13111yescharacteristic
282C4×C53#57711yescharacteristic
283C4×C53#57711yescharacteristic
284Dic53—21no—
285Dic53—21no—
286Dic53—21no—
287Dic53—41no—
288Dic53—41no—
289Dic53—41no—
290#3101—531no—
291#2985—531no—
292#2972—531no—
293#1585—531no—
294#1585—531no—
295#1582—531no—
296#1582—531no—
297SD16⋊C8—531no—
298SD16⋊C8—531no—
299C8×SD16—531no—
300C8×SD16D5311yescharacteristic, complemented
301#1572—531no—
302#1490—531no—
303#1485—531no—
304#1453—531no—
305C2×C53—42no—
306C2×C53#292911yescharacteristic
307C2×C53#294911yescharacteristic
308C2×C53#295611yescharacteristic
309D53—41no—
310D53—42no—
311#533—531no—
312C2×C4×D4—531no—
313#483—531no—
314#483—531no—
315#476—531no—
316#476—531no—
317#461—531no—
318#461—531no—
319#450—531no—
320#450—531no—
321#446—531no—
322#446—531no—
323#446—1061no—
324#446—1061no—
325C8×Q8—531no—
326C8×Q8C2×D5311yescharacteristic, complemented
327C4×SD16—531no—
328C4×SD16—531no—
329C4×SD16—531no—
330C4×SD16C2×D5311yescharacteristic
331C8×D4—531no—
332C8×D4—531no—
333C8×D4—531no—
334C8×D4—1061no—
335C8×D4—1061no—
336C8×D4C2×D5311yescharacteristic
337#433—1061no—
338#433—1061no—
339#432—531no—
340#431—531no—
341#431—531no—
342#431—531no—
343C2×C22⋊C8—1061no—
344C2×C22⋊C8—1061no—
345C4×M16—531no—
346#335—531no—
347#335—531no—
348#334—531no—
349#334C2×D5311yescharacteristic, complemented
350Q8⋊C8—531no—
351Q8⋊C8—531no—
352Q8⋊C8—531no—
353Q8⋊C8C2×D5311yescharacteristic, complemented
354D4⋊C8—531no—
355D4⋊C8—531no—
356D4⋊C8—531no—
357D4⋊C8C2×D5311yescharacteristic
358#322—531no—
359#322—531no—
360#322—531no—
361C82C2×D5311yescharacteristic, complemented
362C53#1792411yescharacteristic, complemented, Sylow
363C22×D4—531no—
364C23×C4—1061no—
365C2×SD16—531no—
366C2×SD16—531no—
367C2×SD16—531no—
368C2×SD16C4×D5311yescharacteristic
369C2×M16—1061no—
370C22×C8—21no—
371C22×C8—1061no—
372C22×C8—1061no—
373C22×C8—1061no—
374C4⋊Q8—531no—
375C4⋊Q8—531no—
376#125—1061no—
377C22⋊Q8—531no—
378C22⋊Q8—531no—
379C22⋊Q8—1061no—
380C4×Q8—531no—
381C4×Q8—1061no—
382C4×Q8C22×D5311yescharacteristic
383C4×D4—531no—
384C4×D4—531no—
385C4×D4—531no—
386C4×D4—1061no—
387C4×D4—1061no—
388C4×D4C22×D5311yescharacteristic
389#117—1061no—
390C2×C4⋊C4—531no—
391C2×C22⋊C4—1061no—
392C2×C42—531no—
393#106—531no—
394#106—531no—
395#106—531no—
396#106C4×D5311yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8C22×D5311yescharacteristic
408C4⋊C8C22×D5311yescharacteristic
409#103—531no—
410#103—531no—
411#103—531no—
412#103C4×D5311yescharacteristic
413#102—531no—
414#102—531no—
415#102—531no—
416#102C4×D5311yescharacteristic
417C22⋊C8—21no—
418C22⋊C8—531no—
419C22⋊C8—531no—
420C22⋊C8—531no—
421C22⋊C8—531no—
422C22⋊C8—1061no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426C22⋊C8—1061no—
427#97—1061no—
428#97—1061no—
429#97—1061no—
430#97—1061no—
431C4×C8—21no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—531no—
435C4×C8—1061no—
436C4×C8—1061no—
437C4×C8C22×D5311yescharacteristic
438C4×C8C22×D5311yescharacteristic
439C4×C8C22×D5311yescharacteristic
440C24—1061no—
441C2×Q8—531no—
442C2×Q8—1061no—
443C2×Q8C2×C4×D5311yescharacteristic
444C2×D4—531no—
445C2×D4—1061no—
446C2×D4—1062no—
447C2×D4—1062no—
448C2×D4C2×C4×D5311yescharacteristic
449C22×C4—21no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—4241no—
458C22×C4—4241no—
459SD16—22no—
460SD16—1062no—
461SD16—1062no—
462SD16—1062no—
463M16—2122no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—81no—
468C2×C8—531no—
469C2×C8—531no—
470C2×C8—531no—
471C2×C8—531no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—4241no—
476C2×C8—4241no—
477C2×C8—4241no—
478C2×C8C2×C4×D5311yescharacteristic
479C2×C8C2×C4×D5311yescharacteristic
480C2×C8D4×D5311yescharacteristic
481C2×C8D4×D5311yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—531no—
484C4⋊C4—531no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4C2×C4×D5311yescharacteristic
493C4⋊C4C2×C4×D5311yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1061no—
496C22⋊C4—1061no—
497C22⋊C4—1061no—
498C22⋊C4—1061no—
499C22⋊C4—1061no—
500C42—21no—
501C42—531no—
502C42—531no—
503C42—531no—
504C42—1061no—
505C42C23×D5311yescharacteristic
506C23—21no—
507C23—531no—
508C23—1064no—
509C23—1061no—
510C23—4241no—
511C23—4241no—
512Q8—21no—
513Q8—21no—
514Q8—1061no—
515Q8—1061no—
516Q8—4241no—
517D4—21no—
518D4—532no—
519D4—1064no—
520D4—1061no—
521D4#1313322312yes—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—531no—
528C2×C4—531no—
529C2×C4—531no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—4241no—
538C2×C4—4241no—
539C2×C4—4241no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4C22×C4×D5311yescharacteristic
544C2×C4C2×D4×D5311yescharacteristic
545C2×C4#41×D5311yescharacteristic
546C8—22no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—21no—
551C8—41no—
552C8—41no—
553C8—1061no—
554C8—1061no—
555C8—1061no—
556C8—1061no—
557C8—2121no—
558C8—2121no—
559C8—2121no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—1061no—
564C22—1062no—
565C22—1061no—
566C22—1061no—
567C22—4244no—
568C22—2121no—
569C22—2121no—
570C22C4×D4×D5311yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—1061no—
576C4—1061no—
577C4—1061no—
578C4—2121no—
579C4—2121no—
580C4—2121no—
581C4#131×D5311yescharacteristic
582C4C2×#1313322311yescharacteristic
583C4#1681142911yescharacteristic
584C4#1681143011yescharacteristic
585C2—42no—
586C2—2121no—
587C2—2122no—
588C2#2028340411yescharacteristic
589C2#2028356911yescharacteristic
590C2#2028362011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC8×SD16×C53C2
socleC22×C53C2×C4×D4
prime radicalC2×D4×D53C2×C4
cosocleC42×C53C24
hypercenterC8×SD16D53
hypocenterC53#17924
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. #20283806C2semidirect
4. #20273575C2semidirect
5. #20273569C2semidirect
6. #20273552C2semidirect
7. #20273550C2semidirect
8. #20273527C2semidirect
9. #20273525C2semidirect
10. C8×#11719913C2semidirect
11. #20273492C2semidirect
13. #20273134C2semidirect
14. #20273009C2semidirect
15. C8×SD16×C53C2semidirect
26. #16810754C22semidirect
29. #16810750C22semidirect
30. C8×Dic106C22semidirect
48. #16810375C22semidirect
49. #16810375C22semidirect
50. #16810374C22semidirect
51. #16810355C22semidirect
52. Q8⋊#88322C22semidirect
53. #16810353C22semidirect
57. #16810345C22semidirect
58. #16810345C22semidirect
59. #16810344C22semidirect
60. C8×Q8×C53C22semidirect
63. C53×#334C22semidirect
64. C53×Q8⋊C8C22semidirect
66. C82×C53C22semidirect
300. C8×SD16D53semidirect
326. C8×Q8C2×D53semidirect
349. #334C2×D53semidirect
353. Q8⋊C8C2×D53semidirect
361. C82C2×D53semidirect
362. C53#17924semidirect

Maximal subgroups

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

subgroupindex
#202847212
#202838062
#202837362
#202735752
#202735692
#202735522
#202735502
#202735272
#202735252
C8×#117199132
#202734922
#202731562
#202731342
#202730092
C8×SD16×C532
#1792453