Finite Group Databasebeta

← #35376187

Subgroups — #35376187

Order 21248.

600 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918027C211yesmaximal, characteristic
2#24916467C211yesmaximal, characteristic, complemented
3#24906358C211yesmaximal, characteristic, complemented
4#24906346C211yesmaximal, characteristic, complemented
5#24906306C211yesmaximal, characteristic, complemented
6#24906304C211yesmaximal, characteristic, complemented
7#24906299C211yesmaximal, characteristic, complemented
8#24906297C211yesmaximal, characteristic, complemented
9C8×#13050580C211yesmaximal, characteristic, complemented
10C8×#13050575C211yesmaximal, characteristic, complemented
11#24906209C211yesmaximal, characteristic, complemented
12#24905871C211yesmaximal, characteristic, complemented
13#24905869C211yesmaximal, characteristic, complemented
14M16×C83⋊C8C211yesmaximal, characteristic, complemented
15C83×#2969C211yesmaximal, characteristic, complemented
16#19128235C2211yescharacteristic
17#19128100C411yescharacteristic
18#19128083C411yescharacteristic
19#19128058C411yescharacteristic
20#19128055C411yescharacteristic
21#19127940C411yescharacteristic
22#19127935C411yescharacteristic
23#19127932C411yescharacteristic
24#19127928C411yescharacteristic
25#19127831C2211yescharacteristic
26C8×Dic166C2211yescharacteristic
27C4×#13050580C2211yescharacteristic
28C4×#13050577C2211yescharacteristic
29C4×#13050578C2211yescharacteristic
30C4×#13050575C2211yescharacteristic
31#19127780C2211yescharacteristic
32#19127777—21no—
33#19127770—21no—
34C8×D332C2211yescharacteristic
35#19127763—21no—
36#19127754C2211yescharacteristic
37#19127754C2211yescharacteristic
38#19127658—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
48D4⋊(C83⋊C8)C2211yescharacteristic
49D4⋊(C83⋊C8)C2211yescharacteristic
50#19127428C2211yescharacteristic
51#19127428C2211yescharacteristic
52#19127422C2211yescharacteristic
53#19127422C2211yescharacteristic
54C8×C83⋊C8C2211yescharacteristic
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#405×C83—21no—
66C4×M16×C83C2211yescharacteristic
67C2×#13050580C2×C411yescharacteristic
68C2×#13050578C2×C411yescharacteristic
69C2×#13050577C2×C411yescharacteristic
70C2×#13050575C2×C411yescharacteristic
71C2×#13050573—21no—
72C22×C83⋊C8C2×C411yescharacteristic
73C22×C83⋊C8C2×C411yescharacteristic
74C2×C8×D83—21no—
75#14581786—21no—
76#14581784C2×C411yescharacteristic
77#14581780C2×C411yescharacteristic
78#14581775—21no—
79#14581771C2×C411yescharacteristic
80#14581766C2×C411yescharacteristic
81C4×Dic166C2311yescharacteristic
82C4×D332C2311yescharacteristic
83C4×C22⋊D83—21no—
84#14581749—21no—
85#14581725C2×C411yescharacteristic
86#14581725C2×C411yescharacteristic
87#14581723C2×C411yescharacteristic
88#14581723C2×C411yescharacteristic
89C4⋊(C83⋊C8)—21no—
90C4⋊(C83⋊C8)C2311yescharacteristic
91C4⋊(C83⋊C8)C2311yescharacteristic
92Dic83⋊C8—21no—
93Dic83⋊C8—21no—
94#14581716C2×C411yescharacteristic
95#14581716C2×C411yescharacteristic
96#14581715C2×C411yescharacteristic
97#14581715C2×C411yescharacteristic
98#14581713C2×C411yescharacteristic
99#14581713C2×C411yescharacteristic
100#14581712C2×C411yescharacteristic
101#14581712C2×C411yescharacteristic
102C22⋊(C83⋊C8)—21no—
103C22⋊(C83⋊C8)—21no—
104(C2×D83)⋊C8—21no—
105(C2×D83)⋊C8—21no—
106#14581702—21no—
107C4×C83⋊C8—21no—
108C4×C83⋊C8C2×C411yescharacteristic
109C4×C83⋊C8C2×C411yescharacteristic
110C4×C83⋊C8C2311yescharacteristic
111C4×C83⋊C8C2311yescharacteristic
112C4×C83⋊C8C2311yescharacteristic
113C4×C83⋊C8C2311yescharacteristic
114C8×Dic83—21no—
115C2×M16×C83—21no—
116C2×M16×C83D411yescharacteristic
117C2×M16×C83D411yescharacteristic
118C22×C8×C83—21no—
119#126×C83—21no—
120#125×C83C2×C411yescharacteristic
121#124×C83C2×C411yescharacteristic
122#123×C83—21no—
123C22⋊Q8×C83C2×C411yescharacteristic
124#121×C83C2×C411yescharacteristic
125C4×Q8×C83C2311yescharacteristic
126C4×D4×C83—21no—
127C4×D4×C83C2311yescharacteristic
128#117×C83—21no—
129C2×C42×C83C2311yescharacteristic
130C4⋊C8×C83—21no—
131C4⋊C8×C83—21no—
132C4⋊C8×C83C2311yescharacteristic
133C4⋊C8×C83C2311yescharacteristic
134C22⋊C8×C83—21no—
135C22⋊C8×C83—21no—
136C22⋊C8×C83—21no—
137#97×C83—21no—
138#97×C83D411yescharacteristic
139#97×C83D411yescharacteristic
140C4×C8×C83—21no—
141C4×C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Dic166C22×C411yescharacteristic
145C2×D332C22×C411yescharacteristic
146C2×C22⋊D83—21no—
147C2×C4×D83—21no—
148#13050580—22no—
149#13050578—41no—
150#13050577—41no—
151#13050575—22no—
152#13050573—42no—
153C2×C83⋊C8—22no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8—41no—
159C2×C83⋊C8C22×C411yescharacteristic
160C2×C83⋊C8C22×C411yescharacteristic
161C2×C83⋊C8C22×C411yescharacteristic
162C2×C83⋊C8C22×C411yescharacteristic
163C8×D83—42no—
164C4⋊Dic83C22×C411yescharacteristic
165C4⋊Dic83C22×C411yescharacteristic
166Dic83⋊C4—21no—
167Dic83⋊C4—21no—
168Dic83⋊C4—21no—
169Dic83⋊C4—21no—
170C22⋊Dic83—21no—
171(C2×D83)⋊C4—21no—
172(C2×D83)⋊C4—21no—
173(C2×D83)⋊C4—21no—
174(C2×D83)⋊C4—21no—
175C4×Dic83—21no—
176C2×Q8×C83C22×C411yescharacteristic
177C2×D4×C83—21no—
178C2×D4×C83C22×C411yescharacteristic
179C22×C4×C83—21no—
180C22×C4×C83C22×C411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83#4111yescharacteristic
183M16×C83—22no—
184M16×C83—22no—
185M16×C83—42no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—42no—
189C2×C8×C83C2×D411yescharacteristic
190C2×C8×C83C2×D411yescharacteristic
191C2×C8×C83C2×D411yescharacteristic
192C2×C8×C83C2×D411yescharacteristic
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83C22×C411yescharacteristic
198C4⋊C4×C83C22×C411yescharacteristic
199C22⋊C4×C83—21no—
200C22⋊C4×C83—21no—
201C22⋊C4×C83—21no—
202C22⋊C4×C83—21no—
203C22⋊C4×C83—21no—
204C42×C83—21no—
205C42×C83C22×C411yescharacteristic
206C42×C83C2×D411yescharacteristic
207C42×C83#4111yescharacteristic
208C42×C83C2411yescharacteristic
209C23×C83—21no—
210C23×C83C4×D411yescharacteristic
211C22×D83—21no—
212Q8×C83—21no—
213Q8×C83—21no—
214D4×C83—21no—
215D4×C83—21no—
216D4×C83—81no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83C4×D411yescharacteristic
228C2×C4×C83C4×D411yescharacteristic
229C2×C4×C83C4×D411yescharacteristic
230C2×C4×C83C23×C411yescharacteristic
231C2×C4×C83C22×D411yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C22⋊D83—81no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237D332—21no—
238D332—21no—
239C4×D83—41no—
240C4×D83—41no—
241Dic166—21no—
242Dic166—21no—
243C8×C83—21no—
244C8×C83—21no—
245C8×C83—41no—
246C8×C83—41no—
247C8×C83#13512yes—
248C8×C83#13512yes—
249C83⋊C8—21no—
250C83⋊C8—21no—
251C83⋊C8—21no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C83⋊C8—41no—
256C83⋊C8#13112yes—
257C22×C83—21no—
258C22×C83—21no—
259C22×C83—21no—
260C22×C83—41no—
261C22×C83—41no—
262C22×C83—41no—
263C22×C83C2×C4×D411yescharacteristic
264C2×D83—41no—
265C2×D83—41no—
266C2×D83—41no—
267C4×C83—21no—
268C4×C83—21no—
269C4×C83—21no—
270C4×C83—21no—
271C4×C83—41no—
272C4×C83—41no—
273C4×C83—41no—
274C4×C83C2×#13111yescharacteristic
275C4×C83#56811yescharacteristic
276C4×C83C2×#13511yescharacteristic
277C4×C83C2×#13511yescharacteristic
278Dic83—41no—
279Dic83—41no—
280Dic83—41no—
281#18069—831nomaximal, Sylow
282C2×C83—41no—
283C2×C83—81no—
284C2×C83#293511yescharacteristic
285C2×C83C4×#13511yescharacteristic
286C2×C83#295711yescharacteristic
287D83—81no—
288#3168—831no—
289#2969—831no—
290#2969D8311yescharacteristic, complemented
291#1591—831no—
292#1589—831no—
293#1583—831no—
294#1583—831no—
295#1582—831no—
296#1582—831no—
297C8×Q16—831no—
298C8×D8—831no—
299#1564—831no—
300#1482—831no—
301#1482—831no—
302C8×M16—831no—
303C83#1806911yescharacteristic, complemented, Sylow
304#529—831no—
305#529C2×D8311yescharacteristic
306#499—831no—
307#495—831no—
308#488—831no—
309#488—831no—
310#467—831no—
311#466—831no—
312#465—831no—
313#465—831no—
314#446—831no—
315#446C2×D8311yescharacteristic
316C8×Q8—831no—
317C8×Q8C2×D8311yescharacteristic
318C4×Q16—831no—
319C4×SD16—831no—
320C4×SD16—831no—
321C4×D8—831no—
322#436—21no—
323#436—831no—
324#436—1661no—
325#436C2×D8311yescharacteristic
326#435—21no—
327#435—1661no—
328C8×D4—831no—
329C8×D4C2×D8311yescharacteristic
330#433—21no—
331#433—1661no—
332#431—831no—
333#431—831no—
334#431C2×D8311yescharacteristic
335#405—21no—
336#405—1661no—
337C4×M16C2×D8311yescharacteristic
338C2×C4×C8—831no—
339#335—831no—
340#335—831no—
341#334—831no—
342#334—831no—
343Q8⋊C8—831no—
344Q8⋊C8—831no—
345Q8⋊C8—831no—
346Q8⋊C8—831no—
347D4⋊C8—831no—
348D4⋊C8—831no—
349D4⋊C8—831no—
350D4⋊C8—831no—
351#322—831no—
352#322—831no—
353C82—831no—
354C82—831no—
355C2×Q16—831no—
356C2×SD16—831no—
357C2×SD16—831no—
358C2×D8—831no—
359C2×M16—21no—
360C2×M16—1661no—
361C2×M16C22⋊D8311yescharacteristic
362C2×M16C22⋊D8311yescharacteristic
363C22×C8—21no—
364C22×C8—831no—
365C22×C8—831no—
366C22×C8—1661no—
367#126—21no—
368#126—1661no—
369#125—831no—
370#125C4×D8311yescharacteristic
371#124—831no—
372#124C4×D8311yescharacteristic
373#123—21no—
374#123—1661no—
375C22⋊Q8—831no—
376C22⋊Q8C4×D8311yescharacteristic
377#121—831no—
378#121C4×D8311yescharacteristic
379C4×Q8—831no—
380C4×Q8C22×D8311yescharacteristic
381C4×D4—21no—
382C4×D4—831no—
383C4×D4—1661no—
384C4×D4C22×D8311yescharacteristic
385#117—21no—
386#117—1661no—
387C2×C42C22×D8311yescharacteristic
388#107—831no—
389#107—831no—
390#106—831no—
391#106—831no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—831no—
395C4⋊C8—831no—
396C4⋊C8—1661no—
397C4⋊C8—1661no—
398C4⋊C8—1661no—
399C4⋊C8C22×D8311yescharacteristic
400C4⋊C8C22×D8311yescharacteristic
401#103—831no—
402#103—831no—
403#103—831no—
404#103—831no—
405#102—831no—
406#102—831no—
407#102—831no—
408#102—831no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—1661no—
413C22⋊C8—1661no—
414C22⋊C8—1661no—
415C22⋊C8—1661no—
416#97—21no—
417#97—1661no—
418#97C22⋊D8311yescharacteristic
419#97C22⋊D8311yescharacteristic
420C4×C8—21no—
421C4×C8—21no—
422C4×C8—831no—
423C4×C8—831no—
424C4×C8—831no—
425C4×C8—831no—
426C4×C8—831no—
427C4×C8—831no—
428C4×C8—1661no—
429C4×C8—1661no—
430C4×C8C22×D8311yescharacteristic
431C4×C8C22×D8311yescharacteristic
432C2×Q8—831no—
433C2×Q8C2×C4×D8311yescharacteristic
434C2×D4—21no—
435C2×D4—831no—
436C2×D4—1661no—
437C2×D4C2×C4×D8311yescharacteristic
438C22×C4—21no—
439C22×C4—1661no—
440C22×C4C2×C4×D8311yescharacteristic
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4#1305059011yescharacteristic
443Q16—1662no—
444SD16—3321no—
445SD16—3321no—
446D8—1662no—
447M16—22no—
448M16—22no—
449M16—42no—
450M16—3322no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—3322no—
455C2×C8—3321no—
456C2×C8—831no—
457C2×C8—831no—
458C2×C8—831no—
459C2×C8—831no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1662no—
465C2×C8C2×C22⋊D8311yescharacteristic
466C2×C8C2×C22⋊D8311yescharacteristic
467C2×C8C2×C22⋊D8311yescharacteristic
468C2×C8C2×C22⋊D8311yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—831no—
474C4⋊C4—831no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4C2×C4×D8311yescharacteristic
480C4⋊C4C2×C4×D8311yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—1661no—
487C22⋊C4—1661no—
488C22⋊C4—1661no—
489C22⋊C4—1661no—
490C22⋊C4—1661no—
491C42—21no—
492C42—1661no—
493C42C2×C4×D8311yescharacteristic
494C42C2×C22⋊D8311yescharacteristic
495C42#1305059011yescharacteristic
496C42C23×D8311yescharacteristic
497C23—21no—
498C23—1661no—
499C23C4×C22⋊D8311yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—1661no—
503Q8—1661no—
504D4—21no—
505D4—21no—
506D4—81no—
507D4—6641no—
508D4—1661no—
509D4—1661no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—1661no—
523C2×C4—1661no—
524C2×C4—1661no—
525C2×C4C4×C22⋊D8311yescharacteristic
526C2×C4C4×C22⋊D8311yescharacteristic
527C2×C4C4×C22⋊D8311yescharacteristic
528C2×C4C22×C4×D8311yescharacteristic
529C2×C4C22×C22⋊D8311yescharacteristic
530C2×C4C2×#1305059011yescharacteristic
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—3321no—
536C8—3321no—
537C8—3321no—
538C8—3321no—
539C8—832no—
540C8—1661no—
541C8—1661no—
542C8—1661no—
543C8#1458181212yes—
544C8#1458181212yes—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—3321no—
552C22—3321no—
553C22—3321no—
554C22C2×C4×C22⋊D8311yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—3321no—
563C4—3321no—
564C4—3321no—
565C4#131×D8311yescharacteristic
566C4#1912846611yescharacteristic
567C4C2×#1458181211yescharacteristic
568C4C2×#1458181211yescharacteristic
569C2—41no—
570C2—81no—
571C2—6641no—
572C2#2491620511yescharacteristic
573C2C4×#1458181211yescharacteristic
574C2#2491637811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#2969C2
socleC22×C83C2×C4×D4
prime radical#19127935C4
cosocleC42×C83C24
hypercenter#2969D83
hypocenterC83#18069
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. #24916467C2semidirect
3. #24906358C2semidirect
4. #24906346C2semidirect
5. #24906306C2semidirect
6. #24906304C2semidirect
7. #24906299C2semidirect
8. #24906297C2semidirect
9. C8×#13050580C2semidirect
10. C8×#13050575C2semidirect
11. #24906209C2semidirect
12. #24905871C2semidirect
13. #24905869C2semidirect
14. M16×C83⋊C8C2semidirect
15. C83×#2969C2semidirect
290. #2969D83semidirect
303. C83#18069semidirect

Maximal subgroups

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

subgroupindex
#249180272
#249164672
#249063582
#249063462
#249063062
#249063042
#249062992
#249062972
C8×#130505802
C8×#130505752
#249062092
#249058712
#249058692
M16×C83⋊C82
C83×#29692
#1806983