Finite Group Databasebeta

← #35375593

Subgroups — #35375593

Order 21248.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917539C211yesmaximal, characteristic
2#24916550C211yesmaximal, characteristic, complemented
3#24906358C211yesmaximal, characteristic, complemented
4#24906351C211yesmaximal, characteristic, complemented
5#24906339C211yesmaximal, characteristic, complemented
6#24906334C211yesmaximal, characteristic, complemented
7#24906332C211yesmaximal, characteristic, complemented
8#24906320C211yesmaximal, characteristic, complemented
9C8×#13050580C211yesmaximal, characteristic, complemented
10C8×#13050578C211yesmaximal, characteristic, complemented
11#24906246C211yesmaximal, characteristic, complemented
12#24905923C211yesmaximal, characteristic
13C2×#19127431C211yesmaximal, characteristic, complemented
14#24905747C211yesmaximal, characteristic, complemented
15C83×#2994C211yesmaximal, characteristic
16C2×C4×Dic166C2211yescharacteristic
17#19128029C411yescharacteristic
18#19128012C411yescharacteristic
19#19128005C411yescharacteristic
20#19128005C411yescharacteristic
21#19127920C411yescharacteristic
22#19127920C411yescharacteristic
23#19127856C411yescharacteristic
24#19127839C411yescharacteristic
25#19127831C2211yescharacteristic
26#19127831C2211yescharacteristic
27#19127830—21no—
28#19127830—21no—
29C8×Dic166C2211yescharacteristic
30C8×Dic166C2211yescharacteristic
31C4×#13050580C2211yescharacteristic
32C4×#13050580C2211yescharacteristic
33C4×#13050578C2211yescharacteristic
34C4×#13050578C2211yescharacteristic
35#19127758C2211yescharacteristic
36#19127755—21no—
37#19127754C2211yescharacteristic
38#19127751—21no—
39C2×C4×C83⋊C8C2211yescharacteristic
40#19127453C2211yescharacteristic
41#19127453C2211yescharacteristic
42#19127451C2211yescharacteristic
43#19127451C2211yescharacteristic
44Q8⋊(C83⋊C8)C2211yescharacteristic
45Q8⋊(C83⋊C8)C2211yescharacteristic
46#19127431C2211yescharacteristic
47#19127431C2211yescharacteristic
48#19127431C2211yescharacteristic
49#19127431C2211yescharacteristic
50D4⋊(C83⋊C8)C2211yescharacteristic
51D4⋊(C83⋊C8)C2211yescharacteristic
52#19127423C2211yescharacteristic
53#19127423C2211yescharacteristic
54C8×C83⋊C8C2211yescharacteristic
55C8×C83⋊C8C2211yescharacteristic
56#529×C83C2211yescharacteristic
57#446×C83C2211yescharacteristic
58C8×Q8×C83C2211yescharacteristic
59#436×C83C2211yescharacteristic
60#435×C83—21no—
61C8×D4×C83—21no—
62C8×D4×C83C2211yescharacteristic
63#433×C83—21no—
64#431×C83C2211yescharacteristic
65#424×C83—21no—
66C2×C4⋊C8×C83C2211yescharacteristic
67C22×Dic166C2×C411yescharacteristic
68C2×#13050580C2×C411yescharacteristic
69C2×#13050580C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C2×#13050578C2×C411yescharacteristic
72C22×C83⋊C8—21no—
73C4×Dic166—21no—
74C4×Dic166—21no—
75C4×Dic166C2×C411yescharacteristic
76C4×Dic166C2×C411yescharacteristic
77C4×Dic166C2311yescharacteristic
78C4×Dic166C2311yescharacteristic
79C2×C4⋊Dic83C2×C411yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×C4×Dic83—21no—
82#14581725C2×C411yescharacteristic
83#14581725C2×C411yescharacteristic
84#14581723C2×C411yescharacteristic
85#14581723C2×C411yescharacteristic
86C4⋊(C83⋊C8)C2×C411yescharacteristic
87C4⋊(C83⋊C8)C2×C411yescharacteristic
88C4⋊(C83⋊C8)C2311yescharacteristic
89C4⋊(C83⋊C8)C2311yescharacteristic
90Dic83⋊C8—21no—
91Dic83⋊C8—21no—
92Dic83⋊C8—21no—
93Dic83⋊C8—21no—
94#14581716C2×C411yescharacteristic
95#14581716C2×C411yescharacteristic
96#14581715C2×C411yescharacteristic
97#14581715C2×C411yescharacteristic
98#14581715C2×C411yescharacteristic
99#14581715C2×C411yescharacteristic
100#14581713C2×C411yescharacteristic
101#14581713C2×C411yescharacteristic
102C22⋊(C83⋊C8)—21no—
103C22⋊(C83⋊C8)C2×C411yescharacteristic
104C22⋊(C83⋊C8)C2×C411yescharacteristic
105#14581703—21no—
106#14581702—21no—
107#14581702—21no—
108C4×C83⋊C8—21no—
109C4×C83⋊C8C2311yescharacteristic
110C4×C83⋊C8C2311yescharacteristic
111C4×C83⋊C8C2311yescharacteristic
112C4×C83⋊C8C2311yescharacteristic
113C8×Dic83—21no—
114C8×Dic83—21no—
115C2×M16×C83—21no—
116C22×C8×C83—21no—
117C22×C8×C83—21no—
118#126×C83—21no—
119#125×C83C2×C411yescharacteristic
120#124×C83C2×C411yescharacteristic
121#123×C83—21no—
122C22⋊Q8×C83C2×C411yescharacteristic
123#121×C83C2×C411yescharacteristic
124C4×Q8×C83C2311yescharacteristic
125C4×D4×C83—21no—
126C4×D4×C83C2311yescharacteristic
127#117×C83—21no—
128C2×C42×C83C2311yescharacteristic
129C4⋊C8×C83—21no—
130C4⋊C8×C83—21no—
131C4⋊C8×C83—21no—
132C4⋊C8×C83D411yescharacteristic
133C4⋊C8×C83D411yescharacteristic
134C4⋊C8×C83C2311yescharacteristic
135C4⋊C8×C83C2311yescharacteristic
136C22⋊C8×C83—21no—
137C22⋊C8×C83—21no—
138C22⋊C8×C83D411yescharacteristic
139C22⋊C8×C83D411yescharacteristic
140#97×C83—21no—
141C4×C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Dic166—22no—
145C2×Dic166—21no—
146C2×Dic166—41no—
147C2×Dic166C22×C411yescharacteristic
148C2×Dic166C22×C411yescharacteristic
149C22×Dic83—21no—
150C22×Dic83—21no—
151#13050580—22no—
152#13050580—41no—
153#13050578—22no—
154#13050578—41no—
155C2×C83⋊C8—24no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8—21no—
159C2×C83⋊C8—21no—
160C2×C83⋊C8C22×C411yescharacteristic
161C2×C83⋊C8C22×C411yescharacteristic
162C2×C83⋊C8C22×C411yescharacteristic
163C2×C83⋊C8C22×C411yescharacteristic
164C4⋊Dic83—21no—
165C4⋊Dic83C22×C411yescharacteristic
166C4⋊Dic83C22×C411yescharacteristic
167Dic83⋊C4—21no—
168Dic83⋊C4—21no—
169Dic83⋊C4—21no—
170Dic83⋊C4—21no—
171C4×Dic83—21no—
172C4×Dic83—21no—
173C4×Dic83—21no—
174C4×Dic83—21no—
175C2×Q8×C83C22×C411yescharacteristic
176C2×D4×C83—21no—
177C2×D4×C83C22×C411yescharacteristic
178C22×C4×C83—21no—
179C22×C4×C83C22×C411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83#4111yescharacteristic
182M16×C83—81no—
183C2×C8×C83—21no—
184C2×C8×C83—21no—
185C2×C8×C83—81no—
186C2×C8×C83—81no—
187C2×C8×C83C2×D411yescharacteristic
188C2×C8×C83C2×D411yescharacteristic
189C2×C8×C83C2×D411yescharacteristic
190C2×C8×C83C2×D411yescharacteristic
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83C22×C411yescharacteristic
196C4⋊C4×C83C22×C411yescharacteristic
197C22⋊C4×C83—21no—
198C22⋊C4×C83—21no—
199C22⋊C4×C83—21no—
200C22⋊C4×C83—21no—
201C22⋊C4×C83—21no—
202C42×C83—21no—
203C42×C83C22×C411yescharacteristic
204C42×C83C2×D411yescharacteristic
205C42×C83#4111yescharacteristic
206C42×C83C2411yescharacteristic
207C23×C83—21no—
208C23×C83C4×D411yescharacteristic
209Q8×C83—21no—
210Q8×C83—21no—
211D4×C83—21no—
212D4×C83—21no—
213D4×C83—42no—
214C2×C4×C83—22no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83C4×D411yescharacteristic
225C2×C4×C83C4×D411yescharacteristic
226C2×C4×C83C4×D411yescharacteristic
227C2×C4×C83C23×C411yescharacteristic
228C2×C4×C83C22×D411yescharacteristic
229C2×C4×C83C2×#4111yescharacteristic
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236Dic166—21no—
237Dic166—21no—
238Dic166—21no—
239Dic166—42no—
240Dic166#13112yes—
241C8×C83—21no—
242C8×C83—21no—
243C8×C83—21no—
244C8×C83—21no—
245C8×C83—41no—
246C8×C83—41no—
247C83⋊C8—22no—
248C83⋊C8—22no—
249C83⋊C8—21no—
250C83⋊C8—21no—
251C83⋊C8—21no—
252C83⋊C8—21no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C22×C83—21no—
256C22×C83—21no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83#44312yes—
261C22×C83C2×C4×D411yescharacteristic
262C4×C83—21no—
263C4×C83—21no—
264C4×C83—21no—
265C4×C83—21no—
266C4×C83—41no—
267C4×C83—41no—
268C4×C83—41no—
269C4×C83C2×#13111yescharacteristic
270C4×C83#56811yescharacteristic
271C4×C83#57711yescharacteristic
272C4×C83#57811yescharacteristic
273Dic83—22no—
274Dic83—41no—
275Dic83—41no—
276Dic83—41no—
277#18017—831nomaximal, Sylow
278C2×C83—22no—
279C2×C83—81no—
280C2×C83#293311yescharacteristic
281C2×C83#294711yescharacteristic
282C2×C83C2×#44311yescharacteristic
283#3108—831no—
284#2994D8311yescharacteristic
285#2984—831no—
286#1591—831no—
287#1590—831no—
288#1588—831no—
289#1587—831no—
290#1587—831no—
291#1585—831no—
292C8×Q16—831no—
293C8×SD16—831no—
294#1573—831no—
295#1494—831no—
296C2×Q8⋊C8—831no—
297#1450—831no—
298C83#1801711yescharacteristic, complemented, Sylow
299#529C2×D8311yescharacteristic
300C2×C4×Q8—831no—
301#483—831no—
302#480—831no—
303#479—831no—
304#479—831no—
305#464—831no—
306#464—831no—
307#451—831no—
308#448—831no—
309#446—831no—
310#446—831no—
311#446—1661no—
312#446—1661no—
313#446C2×D8311yescharacteristic
314C8×Q8—831no—
315C8×Q8—831no—
316C8×Q8C2×D8311yescharacteristic
317C4×Q16—831no—
318C4×Q16—831no—
319C4×SD16—831no—
320C4×SD16—831no—
321#436C2×D8311yescharacteristic
322#435—21no—
323C8×D4—21no—
324C8×D4C2×D8311yescharacteristic
325#433—21no—
326#432—831no—
327#432—1661no—
328#431—831no—
329#431—1661no—
330#431C2×D8311yescharacteristic
331#424—21no—
332C2×C4⋊C8C2×D8311yescharacteristic
333C2×C4×C8—831no—
334#335—831no—
335#335—831no—
336#334—831no—
337#334—831no—
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8—831no—
341Q8⋊C8—831no—
342Q8⋊C8—831no—
343Q8⋊C8—831no—
344D4⋊C8—831no—
345D4⋊C8—831no—
346#322—831no—
347#322—831no—
348C82—831no—
349C82—831no—
350C22×Q8—831no—
351C2×Q16—831no—
352C2×Q16—831no—
353C2×SD16—831no—
354C2×SD16—831no—
355C2×M16—21no—
356C22×C8—21no—
357C22×C8—21no—
358C22×C8—1661no—
359#126—21no—
360#125C4×D8311yescharacteristic
361#124C4×D8311yescharacteristic
362#123—21no—
363C22⋊Q8C4×D8311yescharacteristic
364#121C4×D8311yescharacteristic
365C4×Q8—831no—
366C4×Q8—831no—
367C4×Q8—831no—
368C4×Q8—831no—
369C4×Q8—1661no—
370C4×Q8—1661no—
371C4×Q8C22×D8311yescharacteristic
372C4×D4—21no—
373C4×D4C22×D8311yescharacteristic
374#117—21no—
375C2×C4⋊C4—831no—
376C2×C4⋊C4—1661no—
377C2×C42—1661no—
378C2×C42C22×D8311yescharacteristic
379#107—831no—
380#107—831no—
381#106—831no—
382#106—831no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—831no—
387C4⋊C8—831no—
388C4⋊C8—831no—
389C4⋊C8—831no—
390C4⋊C8—1661no—
391C4⋊C8—1661no—
392C4⋊C8—1661no—
393C4⋊C8—1661no—
394C4⋊C8C22⋊D8311yescharacteristic
395C4⋊C8C22⋊D8311yescharacteristic
396C4⋊C8C22×D8311yescharacteristic
397C4⋊C8C22×D8311yescharacteristic
398#103—831no—
399#103—831no—
400#103—831no—
401#103—831no—
402#103—831no—
403#103—831no—
404#102—831no—
405#102—831no—
406C22⋊C8—21no—
407C22⋊C8—21no—
408C22⋊C8—831no—
409C22⋊C8—831no—
410C22⋊C8—1661no—
411C22⋊C8C22⋊D8311yescharacteristic
412C22⋊C8C22⋊D8311yescharacteristic
413#97—21no—
414#97—1661no—
415#97—1661no—
416#97—1661no—
417C4×C8—21no—
418C4×C8—831no—
419C4×C8—831no—
420C4×C8—831no—
421C4×C8—831no—
422C4×C8—1661no—
423C4×C8—1661no—
424C4×C8—1661no—
425C4×C8C22×D8311yescharacteristic
426C4×C8C22×D8311yescharacteristic
427C2×Q8—3321no—
428C2×Q8—831no—
429C2×Q8—831no—
430C2×Q8—1662no—
431C2×Q8—1661no—
432C2×Q8C2×C4×D8311yescharacteristic
433C2×D4—21no—
434C2×D4C2×C4×D8311yescharacteristic
435C22×C4—21no—
436C22×C4—1661no—
437C22×C4—1661no—
438C22×C4C2×C4×D8311yescharacteristic
439C22×C4C2×C22⋊D8311yescharacteristic
440C22×C4#1305059011yescharacteristic
441Q16—3321no—
442Q16—1662no—
443SD16—3321no—
444SD16—1662no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—81no—
450C2×C8—831no—
451C2×C8—831no—
452C2×C8—831no—
453C2×C8—831no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1664no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8C2×C22⋊D8311yescharacteristic
460C2×C8C2×C22⋊D8311yescharacteristic
461C2×C8C2×C22⋊D8311yescharacteristic
462C2×C8C2×C22⋊D8311yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—831no—
468C4⋊C4—831no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4C2×C4×D8311yescharacteristic
475C4⋊C4C2×C4×D8311yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—1661no—
483C42—1661no—
484C42—1661no—
485C42—1661no—
486C42C2×C4×D8311yescharacteristic
487C42C2×C22⋊D8311yescharacteristic
488C42#1305059011yescharacteristic
489C42C23×D8311yescharacteristic
490C23—21no—
491C23C4×C22⋊D8311yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—3322no—
495Q8—832no—
496Q8—1661no—
497Q8—1661no—
498Q8—1661no—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—3322no—
513C2×C4—3322no—
514C2×C4—1661no—
515C2×C4—1661no—
516C2×C4—1661no—
517C2×C4—1661no—
518C2×C4C4×C22⋊D8311yescharacteristic
519C2×C4C4×C22⋊D8311yescharacteristic
520C2×C4C4×C22⋊D8311yescharacteristic
521C2×C4C22×C4×D8311yescharacteristic
522C2×C4C22×C22⋊D8311yescharacteristic
523C2×C4C2×#1305059011yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—41no—
529C8—41no—
530C8—3321no—
531C8—3321no—
532C8—1662no—
533C8—1662no—
534C8—1661no—
535C8—1661no—
536C8—1661no—
537C8—1661no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1912781612yes—
544C22C2×C4×C22⋊D8311yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—3321no—
553C4—3321no—
554C4—3321no—
555C4—1662no—
556C4#131×D8311yescharacteristic
557C4#1912846611yescharacteristic
558C4#1912851211yescharacteristic
559C4#1912851611yescharacteristic
560C2—22no—
561C2—81no—
562C2#2491618511yescharacteristic
563C2#2491630211yescharacteristic
564C2C2×#1912781611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#2994C2
socleC22×C83C2×C4×D4
prime radical#121×C83C2×C4
cosocleC42×C83C24
hypercenter#2994D83
hypocenterC83#18017
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2994

Complements

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

NH ≅ G/Ntype
2. #24916550C2semidirect
3. #24906358C2semidirect
4. #24906351C2semidirect
5. #24906339C2semidirect
6. #24906334C2semidirect
7. #24906332C2semidirect
8. #24906320C2semidirect
9. C8×#13050580C2semidirect
10. C8×#13050578C2semidirect
11. #24906246C2semidirect
13. C2×#19127431C2semidirect
14. #24905747C2semidirect
298. C83#18017semidirect

Maximal subgroups

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

subgroupindex
#249175392
#249165502
#249063582
#249063512
#249063392
#249063342
#249063322
#249063202
C8×#130505802
C8×#130505782
#249062462
#249059232
C2×#191274312
#249057472
C83×#29942
#1801783