Finite Group Databasebeta

← #35374475

Subgroups — #35374475

Order 21248.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917539C211yesmaximal, characteristic
2#24916543C211yesmaximal, characteristic, complemented
3#24906339C211yesmaximal, characteristic, complemented
4#24906332C211yesmaximal, characteristic, complemented
5#24906311C211yesmaximal, characteristic, complemented
6#24906304C211yesmaximal, characteristic, complemented
7#24906280C211yesmaximal, characteristic, complemented
8#24906275C211yesmaximal, characteristic, complemented
9C8×#13050580C211yesmaximal, characteristic, complemented
10C8×#13050578C211yesmaximal, characteristic, complemented
11#24906254C211yesmaximal, characteristic, complemented
12#24905891C211yesmaximal, characteristic, complemented
13#24905871C211yesmaximal, characteristic
14#24905763C211yesmaximal, characteristic, complemented
15C83×#2993C211yesmaximal, characteristic
16C2×C4×Dic166C2211yescharacteristic
17#19128029C411yescharacteristic
18#19128012C411yescharacteristic
19#19128005C411yescharacteristic
20#19128005C411yescharacteristic
21#19127920C411yescharacteristic
22#19127920C411yescharacteristic
23#19127856C411yescharacteristic
24#19127839C411yescharacteristic
25C8×Dic166C2211yescharacteristic
26C8×Dic166C2211yescharacteristic
27C8×Dic166C2211yescharacteristic
28C8×Dic166C2211yescharacteristic
29#19127825—21no—
30#19127825—21no—
31C4×#13050580C2211yescharacteristic
32C4×#13050580C2211yescharacteristic
33C4×#13050578C2211yescharacteristic
34C4×#13050578C2211yescharacteristic
35#19127754C2211yescharacteristic
36#19127754C2211yescharacteristic
37#19127754C2211yescharacteristic
38#19127751—21no—
39#19127751—21no—
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
54#19127422C2211yescharacteristic
55C8×C83⋊C8C2211yescharacteristic
56#529×C83C2211yescharacteristic
57#446×C83C2211yescharacteristic
58C8×Q8×C83C2211yescharacteristic
59#436×C83—21no—
60#436×C83C2211yescharacteristic
61#435×C83—21no—
62C8×D4×C83C2211yescharacteristic
63#433×C83—21no—
64#431×C83C2211yescharacteristic
65#424×C83—21no—
66#423×C83C2211yescharacteristic
67C22×Dic166C2×C411yescharacteristic
68C2×#13050580C2×C411yescharacteristic
69C2×#13050580C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C2×#13050578C2×C411yescharacteristic
72C4×Dic166—21no—
73C4×Dic166—21no—
74C4×Dic166C2×C411yescharacteristic
75C4×Dic166C2×C411yescharacteristic
76C4×Dic166C2311yescharacteristic
77C4×Dic166C2311yescharacteristic
78C2×C4⋊Dic83C2×C411yescharacteristic
79C2×Dic83⋊C4—21no—
80C2×C4×Dic83—21no—
81#14581725C2×C411yescharacteristic
82#14581725C2×C411yescharacteristic
83#14581723C2×C411yescharacteristic
84#14581723C2×C411yescharacteristic
85C4⋊(C83⋊C8)—21no—
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)—21no—
104C22⋊(C83⋊C8)C2×C411yescharacteristic
105C22⋊(C83⋊C8)C2×C411yescharacteristic
106C4×C83⋊C8—21no—
107C4×C83⋊C8C2311yescharacteristic
108C4×C83⋊C8C2311yescharacteristic
109C4×C83⋊C8C2311yescharacteristic
110C4×C83⋊C8C2311yescharacteristic
111C8×Dic83—21no—
112C8×Dic83—21no—
113C8×Dic83—21no—
114C8×Dic83—21no—
115C2×M16×C83—21no—
116C2×M16×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—21no—
145C2×Dic166—22no—
146C2×Dic166—22no—
147C2×Dic166C22×C411yescharacteristic
148C2×Dic166C22×C411yescharacteristic
149C22×Dic83—21no—
150C22×Dic83—21no—
151#13050580—22no—
152#13050580—22no—
153#13050578—22no—
154#13050578—22no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8—21no—
159C2×C83⋊C8C22×C411yescharacteristic
160C2×C83⋊C8C22×C411yescharacteristic
161C2×C83⋊C8C22×C411yescharacteristic
162C2×C83⋊C8C22×C411yescharacteristic
163C4⋊Dic83—21no—
164C4⋊Dic83C22×C411yescharacteristic
165C4⋊Dic83C22×C411yescharacteristic
166Dic83⋊C4—21no—
167Dic83⋊C4—21no—
168Dic83⋊C4—21no—
169Dic83⋊C4—21no—
170C4×Dic83—21no—
171C4×Dic83—21no—
172C4×Dic83—21no—
173C4×Dic83—21no—
174C2×Q8×C83C22×C411yescharacteristic
175C2×D4×C83—21no—
176C2×D4×C83C22×C411yescharacteristic
177C22×C4×C83—21no—
178C22×C4×C83C22×C411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83#4111yescharacteristic
181M16×C83—81no—
182M16×C83—81no—
183C2×C8×C83—21no—
184C2×C8×C83—21no—
185C2×C8×C83—81no—
186C2×C8×C83C2×D411yescharacteristic
187C2×C8×C83C2×D411yescharacteristic
188C2×C8×C83C2×D411yescharacteristic
189C2×C8×C83C2×D411yescharacteristic
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83C22×C411yescharacteristic
195C4⋊C4×C83C22×C411yescharacteristic
196C22⋊C4×C83—21no—
197C22⋊C4×C83—21no—
198C22⋊C4×C83—21no—
199C22⋊C4×C83—21no—
200C22⋊C4×C83—21no—
201C42×C83—21no—
202C42×C83C22×C411yescharacteristic
203C42×C83C2×D411yescharacteristic
204C42×C83#4111yescharacteristic
205C42×C83C2411yescharacteristic
206C23×C83—21no—
207C23×C83C4×D411yescharacteristic
208Q8×C83—21no—
209Q8×C83—21no—
210D4×C83—21no—
211D4×C83—21no—
212D4×C83—81no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83C4×D411yescharacteristic
224C2×C4×C83C4×D411yescharacteristic
225C2×C4×C83C4×D411yescharacteristic
226C2×C4×C83C23×C411yescharacteristic
227C2×C4×C83C22×D411yescharacteristic
228C2×C4×C83C2×#4111yescharacteristic
229C2×Dic83—24no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—81no—
235Dic166—24no—
236Dic166—21no—
237Dic166—21no—
238Dic166#13112yes—
239Dic166#13112yes—
240C8×C83—21no—
241C8×C83—21no—
242C8×C83—21no—
243C8×C83—21no—
244C8×C83—41no—
245C8×C83—41no—
246C83⋊C8—21no—
247C83⋊C8—21no—
248C83⋊C8—21no—
249C83⋊C8—21no—
250C83⋊C8—41no—
251C83⋊C8—41no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C22×C83—21no—
255C22×C83—21no—
256C22×C83—21no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83C2×C4×D411yescharacteristic
261C4×C83—21no—
262C4×C83—21no—
263C4×C83—21no—
264C4×C83—21no—
265C4×C83—41no—
266C4×C83—41no—
267C4×C83—41no—
268C4×C83C2×#13111yescharacteristic
269C4×C83C2×#13111yescharacteristic
270C4×C83#57711yescharacteristic
271C4×C83#57811yescharacteristic
272Dic83—22no—
273Dic83—22no—
274Dic83—41no—
275Dic83—41no—
276#17919—831nomaximal, Sylow
277C2×C83—41no—
278C2×C83—81no—
279C2×C83#292911yescharacteristic
280C2×C83#294711yescharacteristic
281C2×C83#295711yescharacteristic
282#3108—831no—
283#2993D8311yescharacteristic
284#2982—831no—
285#1588—831no—
286#1587—831no—
287#1584—831no—
288#1583—831no—
289#1579—831no—
290SD16⋊C8—831no—
291C8×Q16—831no—
292C8×SD16—831no—
293#1574—831no—
294#1487—831no—
295#1482—831no—
296#1454—831no—
297C83#1791911yescharacteristic, complemented, Sylow
298#529C2×D8311yescharacteristic
299C2×C4×Q8—831no—
300#483—831no—
301#480—831no—
302#479—831no—
303#479—831no—
304#464—831no—
305#464—831no—
306#451—831no—
307#448—831no—
308#446C2×D8311yescharacteristic
309C8×Q8—831no—
310C8×Q8—831no—
311C8×Q8—831no—
312C8×Q8—831no—
313C8×Q8—1661no—
314C8×Q8—1661no—
315C8×Q8C2×D8311yescharacteristic
316C4×Q16—831no—
317C4×Q16—831no—
318C4×SD16—831no—
319C4×SD16—831no—
320#436—21no—
321#436C2×D8311yescharacteristic
322#435—21no—
323C8×D4C2×D8311yescharacteristic
324#433—21no—
325#431—831no—
326#431—831no—
327#431—831no—
328#431—1661no—
329#431—1661no—
330#431C2×D8311yescharacteristic
331#424—21no—
332#423C2×D8311yescharacteristic
333#335—831no—
334#335—831no—
335#334—831no—
336#334—831no—
337Q8⋊C8—831no—
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8—831no—
341Q8⋊C8—831no—
342Q8⋊C8—831no—
343D4⋊C8—831no—
344D4⋊C8—831no—
345#322—831no—
346#322—831no—
347#322—831no—
348C82—831no—
349C22×Q8—831no—
350C2×Q16—831no—
351C2×Q16—831no—
352C2×SD16—831no—
353C2×SD16—831no—
354C2×M16—21no—
355C2×M16—21no—
356C22×C8—21no—
357#126—21no—
358#125C4×D8311yescharacteristic
359#124C4×D8311yescharacteristic
360#123—21no—
361C22⋊Q8C4×D8311yescharacteristic
362#121C4×D8311yescharacteristic
363C4×Q8—831no—
364C4×Q8—831no—
365C4×Q8—831no—
366C4×Q8—831no—
367C4×Q8—1661no—
368C4×Q8—1661no—
369C4×Q8C22×D8311yescharacteristic
370C4×D4—21no—
371C4×D4C22×D8311yescharacteristic
372#117—21no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—1661no—
375C2×C42—1661no—
376C2×C42C22×D8311yescharacteristic
377#107—831no—
378#107—831no—
379#106—831no—
380#106—831no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—831no—
385C4⋊C8—831no—
386C4⋊C8—831no—
387C4⋊C8—831no—
388C4⋊C8—1661no—
389C4⋊C8—1661no—
390C4⋊C8—1661no—
391C4⋊C8—1661no—
392C4⋊C8—1661no—
393C4⋊C8C22⋊D8311yescharacteristic
394C4⋊C8C22⋊D8311yescharacteristic
395C4⋊C8C22×D8311yescharacteristic
396C4⋊C8C22×D8311yescharacteristic
397#103—831no—
398#103—831no—
399#103—831no—
400#103—831no—
401#103—831no—
402#103—831no—
403#102—831no—
404#102—831no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—831no—
408C22⋊C8—831no—
409C22⋊C8—1661no—
410C22⋊C8—1661no—
411C22⋊C8C22⋊D8311yescharacteristic
412C22⋊C8C22⋊D8311yescharacteristic
413#97—21no—
414C4×C8—21no—
415C4×C8—831no—
416C4×C8—831no—
417C4×C8—831no—
418C4×C8—831no—
419C4×C8—1661no—
420C4×C8—1661no—
421C4×C8—1661no—
422C4×C8—1661no—
423C4×C8—1661no—
424C4×C8C22×D8311yescharacteristic
425C4×C8C22×D8311yescharacteristic
426C2×Q8—831no—
427C2×Q8—831no—
428C2×Q8—1662no—
429C2×Q8—1662no—
430C2×Q8—1661no—
431C2×Q8C2×C4×D8311yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D8311yescharacteristic
434C22×C4—21no—
435C22×C4—1661no—
436C22×C4—1661no—
437C22×C4C2×C4×D8311yescharacteristic
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4#1305059011yescharacteristic
440Q16—1662no—
441Q16—1662no—
442SD16—1662no—
443SD16—1662no—
444M16—81no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—831no—
450C2×C8—831no—
451C2×C8—831no—
452C2×C8—831no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8C2×C22⋊D8311yescharacteristic
458C2×C8C2×C22⋊D8311yescharacteristic
459C2×C8C2×C22⋊D8311yescharacteristic
460C2×C8C2×C22⋊D8311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—831no—
466C4⋊C4—831no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4C2×C4×D8311yescharacteristic
473C4⋊C4C2×C4×D8311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—21no—
480C42—1661no—
481C42—1661no—
482C42—1661no—
483C42—1661no—
484C42C2×C4×D8311yescharacteristic
485C42C2×C22⋊D8311yescharacteristic
486C42#1305059011yescharacteristic
487C42C23×D8311yescharacteristic
488C23—21no—
489C23C4×C22⋊D8311yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—832no—
493Q8—832no—
494Q8—1664no—
495Q8—1661no—
496Q8—1661no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—6641no—
511C2×C4—1661no—
512C2×C4—1661no—
513C2×C4—1664no—
514C2×C4—1661no—
515C2×C4—1661no—
516C2×C4C4×C22⋊D8311yescharacteristic
517C2×C4C4×C22⋊D8311yescharacteristic
518C2×C4C4×C22⋊D8311yescharacteristic
519C2×C4C22×C4×D8311yescharacteristic
520C2×C4C22×C22⋊D8311yescharacteristic
521C2×C4C2×#1305059011yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—3321no—
529C8—3321no—
530C8—3321no—
531C8—3321no—
532C8—1661no—
533C8—1661no—
534C8—1661no—
535C8—1661no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×C4×C22⋊D8311yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—3321no—
551C4—3321no—
552C4—1662no—
553C4—1662no—
554C4#131×D8311yescharacteristic
555C4#131×D8311yescharacteristic
556C4#1912851211yescharacteristic
557C4#1912851611yescharacteristic
558C2—41no—
559C2—81no—
560C2#2491615511yescharacteristic
561C2#2491630211yescharacteristic
562C2#2491637811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#2993C2
socleC22×C83C2×C4×D4
prime radical#121×C83C2×C4
cosocleC42×C83C24
hypercenter#2993D83
hypocenterC83#17919
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
2. #24916543C2semidirect
3. #24906339C2semidirect
4. #24906332C2semidirect
5. #24906311C2semidirect
6. #24906304C2semidirect
7. #24906280C2semidirect
8. #24906275C2semidirect
9. C8×#13050580C2semidirect
10. C8×#13050578C2semidirect
11. #24906254C2semidirect
12. #24905891C2semidirect
14. #24905763C2semidirect
297. C83#17919semidirect

Maximal subgroups

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

subgroupindex
#249175392
#249165432
#249063392
#249063322
#249063112
#249063042
#249062802
#249062752
C8×#130505802
C8×#130505782
#249062542
#249058912
#249058712
#249057632
C83×#29932
#1791983