Finite Group Databasebeta

← #35376186

Subgroups — #35376186

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#24918029C211yesmaximal, characteristic
2#24916470C211yesmaximal, characteristic, complemented
3#24916466C211yesmaximal, characteristic, complemented
4#24906357C211yesmaximal, characteristic, complemented
5#24906346C211yesmaximal, characteristic, complemented
6#24906308C211yesmaximal, characteristic, complemented
7#24906300C211yesmaximal, characteristic, complemented
8#24906298C211yesmaximal, characteristic, complemented
9C8×#13050580C211yesmaximal, characteristic, complemented
10#24906255C211yesmaximal, characteristic, complemented
11#24906208C211yesmaximal, characteristic, complemented
12#24905870C211yesmaximal, characteristic, complemented
13#24905868C211yesmaximal, characteristic, complemented
14#24905742C211yesmaximal, characteristic, complemented
15C83×#1583C211yesmaximal, characteristic, complemented
16#19128238C2211yescharacteristic
17#19128233C2211yescharacteristic
18#19128101C411yescharacteristic
19#19128082C411yescharacteristic
20#19128057C411yescharacteristic
21#19128054C411yescharacteristic
22#19127941C411yescharacteristic
23#19127936C411yescharacteristic
24#19127931C411yescharacteristic
25#19127927C411yescharacteristic
26#19127832C2211yescharacteristic
27#19127828C2211yescharacteristic
28C8×Dic166C2211yescharacteristic
29C4×#13050580C2211yescharacteristic
30C4×#13050579C2211yescharacteristic
31C4×#13050575C2211yescharacteristic
32#19127780—21no—
33#19127780C2211yescharacteristic
34#19127777—21no—
35#19127771—21no—
36#19127769—21no—
37D4×C83⋊C8C2211yescharacteristic
38#19127766C2211yescharacteristic
39#19127761—21no—
40#19127759—21no—
41#19127753C2211yescharacteristic
42#19127753C2211yescharacteristic
43#19127752C2211yescharacteristic
44#19127659—21no—
45#19127655—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127453C2211yescharacteristic
49#19127453C2211yescharacteristic
50#19127452C2211yescharacteristic
51#19127433C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53#19127431C2211yescharacteristic
54#19127430C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic
58C8×C83⋊C8C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60C8×Q8×C83C2211yescharacteristic
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×#13050580C2×C411yescharacteristic
68C2×#13050579C2×C411yescharacteristic
69C2×#13050575C2×C411yescharacteristic
70C2×#13050573—21no—
71C2×#13050573D411yescharacteristic
72C2×#13050573D411yescharacteristic
73C22×C83⋊C8—21no—
74C2×C8×D83—21no—
75C2×C8×D83C2×C411yescharacteristic
76C2×C8×D83C2×C411yescharacteristic
77#14581787—21no—
78#14581785—21no—
79#14581783C2×C411yescharacteristic
80#14581783C2×C411yescharacteristic
81#14581781C2×C411yescharacteristic
82#14581779C2×C411yescharacteristic
83#14581777—21no—
84#14581774—21no—
85#14581772C2×C411yescharacteristic
86#14581768C2×C411yescharacteristic
87#14581767C2×C411yescharacteristic
88#14581763C2×C411yescharacteristic
89C4×Dic166C2311yescharacteristic
90C4×D332—21no—
91C4×D332C2311yescharacteristic
92C4×C22⋊D83—21no—
93#14581751—21no—
94#14581749—21no—
95C42×D83C2311yescharacteristic
96#14581725C2×C411yescharacteristic
97#14581725C2×C411yescharacteristic
98#14581724C2×C411yescharacteristic
99C4⋊(C83⋊C8)—21no—
100C4⋊(C83⋊C8)C2311yescharacteristic
101C4⋊(C83⋊C8)C2311yescharacteristic
102Dic83⋊C8—21no—
103Dic83⋊C8—21no—
104Dic83⋊C8—21no—
105#14581717C2×C411yescharacteristic
106#14581716C2×C411yescharacteristic
107#14581715C2×C411yescharacteristic
108#14581714C2×C411yescharacteristic
109#14581713C2×C411yescharacteristic
110#14581712C2×C411yescharacteristic
111C22⋊(C83⋊C8)—21no—
112(C2×D83)⋊C8—21no—
113(C2×D83)⋊C8—21no—
114(C2×D83)⋊C8—21no—
115(C2×D83)⋊C8—21no—
116(C2×D83)⋊C8—21no—
117#14581703—21no—
118#14581702—21no—
119#14581702D411yescharacteristic
120#14581702D411yescharacteristic
121C4×C83⋊C8C2311yescharacteristic
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C8×Dic83—21no—
125C8×Dic83—21no—
126C8×Dic83—21no—
127C8×Dic83C2×C411yescharacteristic
128C8×Dic83C2×C411yescharacteristic
129C2×SD16×C83C2×C411yescharacteristic
130C2×M16×C83—21no—
131C4×Q8×C83C2311yescharacteristic
132C4×D4×C83C2311yescharacteristic
133#106×C83C2×C411yescharacteristic
134C4⋊C8×C83—21no—
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#103×C83C2×C411yescharacteristic
138#102×C83C2×C411yescharacteristic
139C22⋊C8×C83—21no—
140C4×C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Dic166C22×C411yescharacteristic
145C2×D332—21no—
146C2×D332C22×C411yescharacteristic
147C2×C22⋊D83—21no—
148C2×C4×D83—21no—
149C2×C4×D83C22×C411yescharacteristic
150C2×C4×D83C2×D411yescharacteristic
151C2×C4×D83#4111yescharacteristic
152#13050580—22no—
153#13050579—41no—
154#13050575—22no—
155#13050573—22no—
156#13050573—22no—
157#13050573—42no—
158C2×C83⋊C8—21no—
159C2×C83⋊C8—21no—
160C2×C83⋊C8—21no—
161C2×C83⋊C8—42no—
162C2×C83⋊C8C22×C411yescharacteristic
163C2×C83⋊C8C22×C411yescharacteristic
164C2×C83⋊C8C2×D411yescharacteristic
165C2×C83⋊C8C2×D411yescharacteristic
166C8×D83—22no—
167C8×D83—41no—
168C8×D83—42no—
169C4⋊Dic83—21no—
170C4⋊Dic83—21no—
171C4⋊Dic83—21no—
172C4⋊Dic83C22×C411yescharacteristic
173C4⋊Dic83C22×C411yescharacteristic
174Dic83⋊C4—21no—
175Dic83⋊C4—21no—
176Dic83⋊C4—21no—
177Dic83⋊C4—21no—
178C22⋊Dic83—21no—
179C22⋊Dic83—21no—
180C22⋊Dic83—21no—
181(C2×D83)⋊C4—21no—
182(C2×D83)⋊C4—21no—
183(C2×D83)⋊C4—21no—
184(C2×D83)⋊C4—21no—
185(C2×D83)⋊C4—21no—
186(C2×D83)⋊C4—21no—
187C4×Dic83—21no—
188C4×Dic83C22×C411yescharacteristic
189C4×Dic83C2×D411yescharacteristic
190C4×Dic83#4111yescharacteristic
191C2×Q8×C83C22×C411yescharacteristic
192C2×D4×C83C22×C411yescharacteristic
193C22×C4×C83—21no—
194SD16×C83—41no—
195M16×C83—42no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83—21no—
199C2×C8×C83C22×C411yescharacteristic
200C2×C8×C83C22×C411yescharacteristic
201C2×C8×C83C2×D411yescharacteristic
202C2×C8×C83C2×D411yescharacteristic
203C4⋊C4×C83—21no—
204C4⋊C4×C83C22×C411yescharacteristic
205C4⋊C4×C83C22×C411yescharacteristic
206C22⋊C4×C83—21no—
207C42×C83—21no—
208C42×C83C2411yescharacteristic
209C23×C83—21no—
210C22×D83—21no—
211C22×D83C4×D411yescharacteristic
212Q8×C83—21no—
213Q8×C83—21no—
214D4×C83—21no—
215D4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83C23×C411yescharacteristic
222C2×C4×C83C22×D411yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C22⋊D83—81no—
225C2×Dic83—21no—
226C2×Dic83—21no—
227C2×Dic83—21no—
228C2×Dic83C4×D411yescharacteristic
229C2×Dic83C4×D411yescharacteristic
230C2×Dic83C4×D411yescharacteristic
231D332—21no—
232D332—21no—
233D332—81no—
234C4×D83—21no—
235C4×D83—21no—
236C4×D83—21no—
237C4×D83—21no—
238C4×D83—41no—
239C4×D83—41no—
240C4×D83—41no—
241Dic166—21no—
242Dic166—21no—
243C8×C83—21no—
244C8×C83—21no—
245C8×C83—21no—
246C8×C83—41no—
247C8×C83—41no—
248C8×C83—41no—
249C8×C83#13512yes—
250C83⋊C8—21no—
251C83⋊C8—21no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C83⋊C8#13112yes—
256C83⋊C8#13512yes—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83C2×C4×D411yescharacteristic
261C2×D83—21no—
262C2×D83—21no—
263C2×D83—21no—
264C2×D83—41no—
265C2×D83—41no—
266C2×D83—41no—
267C4×C83—21no—
268C4×C83—41no—
269C4×C83—41no—
270C4×C83—41no—
271C4×C83C2×#13111yescharacteristic
272C4×C83#56811yescharacteristic
273C4×C83C2×#13511yescharacteristic
274C4×C83C2×#13511yescharacteristic
275Dic83—21no—
276Dic83—21no—
277Dic83—21no—
278Dic83—41no—
279Dic83—41no—
280Dic83—41no—
281#18069—831nomaximal, Sylow
282C2×C83—81no—
283C2×C83#293511yescharacteristic
284C2×C83C4×#13511yescharacteristic
285C2×C83#295711yescharacteristic
286D83—41no—
287D83—81no—
288#3168—831no—
289#2969—831no—
290#2969—831no—
291#1591—831no—
292#1589—831no—
293#1583—831no—
294#1583D8311yescharacteristic, complemented
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#529—831no—
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#446—831no—
316C8×Q8—831no—
317C8×Q8C2×D8311yescharacteristic
318C4×Q16—831no—
319C4×SD16—831no—
320C4×SD16C2×D8311yescharacteristic
321C4×D8—831no—
322#436—831no—
323#436—1661no—
324#436—1661no—
325#436C2×D8311yescharacteristic
326#435—1661no—
327#435—1661no—
328C8×D4—831no—
329C8×D4—831no—
330#433—1661no—
331#433—1661no—
332#431—831no—
333#431—831no—
334#431—831no—
335#405—1661no—
336#405—1661no—
337C4×M16—831no—
338C2×C4×C8—831no—
339#335—831no—
340#335—831no—
341#334—831no—
342#334C2×D8311yescharacteristic
343Q8⋊C8—831no—
344Q8⋊C8—831no—
345Q8⋊C8—831no—
346Q8⋊C8C2×D8311yescharacteristic
347D4⋊C8—831no—
348D4⋊C8—831no—
349D4⋊C8—831no—
350D4⋊C8C2×D8311yescharacteristic
351#322—831no—
352#322C2×D8311yescharacteristic
353C82—831no—
354C82—831no—
355C2×Q16—831no—
356C2×SD16—831no—
357C2×SD16C4×D8311yescharacteristic
358C2×D8—831no—
359C2×M16—21no—
360C2×M16—831no—
361C2×M16—831no—
362C2×M16—1661no—
363C22×C8—831no—
364C22×C8—831no—
365C22×C8—1661no—
366C22×C8—1661no—
367#126—1661no—
368#126—1661no—
369#125—831no—
370#125—831no—
371#124—831no—
372#124—831no—
373#123—1661no—
374#123—1661no—
375C22⋊Q8—831no—
376C22⋊Q8—831no—
377#121—831no—
378#121—831no—
379C4×Q8—831no—
380C4×Q8C22×D8311yescharacteristic
381C4×D4—831no—
382C4×D4—1661no—
383C4×D4—1661no—
384C4×D4C22×D8311yescharacteristic
385#117—1661no—
386#117—1661no—
387C2×C42—831no—
388#107—831no—
389#107—831no—
390#106—831no—
391#106C4×D8311yescharacteristic
392C4⋊C8—21no—
393C4⋊C8—831no—
394C4⋊C8—831no—
395C4⋊C8—1661no—
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#103C4×D8311yescharacteristic
405#102—831no—
406#102—831no—
407#102—831no—
408#102C4×D8311yescharacteristic
409C22⋊C8—21no—
410C22⋊C8—1661no—
411C22⋊C8—1661no—
412C22⋊C8—1661no—
413C22⋊C8—1661no—
414C22⋊C8—1661no—
415C22⋊C8—1661no—
416#97—831no—
417#97—831no—
418#97—1661no—
419#97—1661no—
420C4×C8—21no—
421C4×C8—831no—
422C4×C8—831no—
423C4×C8—831no—
424C4×C8—831no—
425C4×C8—831no—
426C4×C8—1661no—
427C4×C8—1661no—
428C4×C8—1661no—
429C4×C8C22×D8311yescharacteristic
430C4×C8C22×D8311yescharacteristic
431C4×C8C22×D8311yescharacteristic
432C2×Q8—831no—
433C2×Q8C2×C4×D8311yescharacteristic
434C2×D4—831no—
435C2×D4—1661no—
436C2×D4—1661no—
437C2×D4C2×C4×D8311yescharacteristic
438C22×C4—21no—
439C22×C4—831no—
440C22×C4—831no—
441C22×C4—831no—
442C22×C4—1661no—
443Q16—1662no—
444SD16—41no—
445SD16—3321no—
446D8—1662no—
447M16—42no—
448M16—3322no—
449M16—1662no—
450M16—1662no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—3321no—
455C2×C8—3322no—
456C2×C8—3322no—
457C2×C8—831no—
458C2×C8—831no—
459C2×C8—831no—
460C2×C8—831no—
461C2×C8—1662no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8C2×C4×D8311yescharacteristic
466C2×C8C2×C4×D8311yescharacteristic
467C2×C8D4×D8311yescharacteristic
468C2×C8D4×D8311yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—831no—
471C4⋊C4—831no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
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—1661no—
483C22⋊C4—1661no—
484C22⋊C4—1661no—
485C22⋊C4—1661no—
486C22⋊C4—1661no—
487C22⋊C4—1661no—
488C22⋊C4—1661no—
489C22⋊C4—1661no—
490C22⋊C4—1661no—
491C42—21no—
492C42—831no—
493C42—831no—
494C42—831no—
495C42—1661no—
496C42C23×D8311yescharacteristic
497C23—21no—
498C23—831no—
499C23—1661no—
500Q8—21no—
501Q8—21no—
502Q8—1661no—
503Q8—1661no—
504D4—21no—
505D4—21no—
506D4—6641no—
507D4—6641no—
508D4—1661no—
509D4—1661no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—3321no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—831no—
519C2×C4—831no—
520C2×C4—831no—
521C2×C4—1661no—
522C2×C4—1661no—
523C2×C4—1661no—
524C2×C4—1661no—
525C2×C4—1661no—
526C2×C4—1661no—
527C2×C4—1661no—
528C2×C4C22×C4×D8311yescharacteristic
529C2×C4C2×D4×D8311yescharacteristic
530C2×C4#41×D8311yescharacteristic
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—3321no—
538C8—3321no—
539C8—3321no—
540C8—832no—
541C8—832no—
542C8—1661no—
543C8—1661no—
544C8#1458181312yes—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—3321no—
549C22—3321no—
550C22—3321no—
551C22—1661no—
552C22—1661no—
553C22—1661no—
554C22C4×D4×D8311yescharacteristic
555C4—21no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—3321no—
560C4—3321no—
561C4—3321no—
562C4—1661no—
563C4—1661no—
564C4—1661no—
565C4#131×D8311yescharacteristic
566C4#1912846711yescharacteristic
567C4#135×D8311yescharacteristic
568C4C2×#1458181311yescharacteristic
569C2—81no—
570C2—3321no—
571C2—6641no—
572C2#2491620411yescharacteristic
573C2C4×#1458181311yescharacteristic
574C2#2491637711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1583C2
socleC22×C83C2×C4×D4
prime radical#19127936C4
cosocleC42×C83C24
hypercenter#1583D83
hypocenterC83#18069
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
2. #24916470C2semidirect
3. #24916466C2semidirect
4. #24906357C2semidirect
5. #24906346C2semidirect
6. #24906308C2semidirect
7. #24906300C2semidirect
8. #24906298C2semidirect
9. C8×#13050580C2semidirect
10. #24906255C2semidirect
11. #24906208C2semidirect
12. #24905870C2semidirect
13. #24905868C2semidirect
14. #24905742C2semidirect
15. C83×#1583C2semidirect
294. #1583D83semidirect
303. C83#18069semidirect

Maximal subgroups

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

subgroupindex
#249180292
#249164702
#249164662
#249063572
#249063462
#249063082
#249063002
#249062982
C8×#130505802
#249062552
#249062082
#249058702
#249058682
#249057422
C83×#15832
#1806983