Finite Group Databasebeta

← #35375530

Subgroups — #35375530

Order 21248.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917543C211yesmaximal, characteristic
2#24916467C211yesmaximal, characteristic
3#24906370C211yesmaximal, characteristic, complemented
4#24906353C211yesmaximal, characteristic, complemented
5#24906334C211yesmaximal, characteristic, complemented
6#24906318C211yesmaximal, characteristic, complemented
7#24906311C211yesmaximal, characteristic, complemented
8#24906306C211yesmaximal, characteristic, complemented
9#24906273C211yesmaximal, characteristic, complemented
10C8×#13050580C211yesmaximal, characteristic, complemented
11#24906254C211yesmaximal, characteristic, complemented
12#24905926C211yesmaximal, characteristic
13#24905858C211yesmaximal, characteristic, complemented
14#24905760C211yesmaximal, characteristic, complemented
15Q8×M16×C83C211yesmaximal, characteristic, complemented
16#19128235C2211yescharacteristic
17#19128031C411yescharacteristic
18#19128014C411yescharacteristic
19#19128008C411yescharacteristic
20#19128008C411yescharacteristic
21#19127923C411yescharacteristic
22#19127923C411yescharacteristic
23#19127857C411yescharacteristic
24#19127840C411yescharacteristic
25#19127831C2211yescharacteristic
26C8×Dic166C2211yescharacteristic
27C4×#13050580C2211yescharacteristic
28C4×#13050580C2211yescharacteristic
29C4×#13050577C2211yescharacteristic
30C4×#13050577C2211yescharacteristic
31#19127780C2211yescharacteristic
32#19127777—21no—
33#19127770—21no—
34C8×D332C2211yescharacteristic
35#19127763—21no—
36#19127754C2211yescharacteristic
37#19127754C2211yescharacteristic
38#19127754C2211yescharacteristic
39#19127658—21no—
40#19127453C2211yescharacteristic
41#19127453C2211yescharacteristic
42#19127451C2211yescharacteristic
43#19127451C2211yescharacteristic
44Q8⋊(C83⋊C8)C2211yescharacteristic
45Q8⋊(C83⋊C8)C2211yescharacteristic
46Q8⋊(C83⋊C8)C2211yescharacteristic
47Q8⋊(C83⋊C8)C2211yescharacteristic
48#19127431C2211yescharacteristic
49#19127431C2211yescharacteristic
50#19127428C2211yescharacteristic
51#19127428C2211yescharacteristic
52#19127423C2211yescharacteristic
53#19127423C2211yescharacteristic
54#19127422C2211yescharacteristic
55C8×C83⋊C8C2211yescharacteristic
56C2×C4×Q8×C83C2211yescharacteristic
57#446×C83—21no—
58#446×C83—21no—
59#446×C83C2211yescharacteristic
60#446×C83C2211yescharacteristic
61C8×Q8×C83C2211yescharacteristic
62C8×Q8×C83C2211yescharacteristic
63#423×C83—21no—
64#423×C83C2211yescharacteristic
65C4×M16×C83—21no—
66C4×M16×C83C2211yescharacteristic
67C2×#13050580C2×C411yescharacteristic
68C2×#13050580C2×C411yescharacteristic
69C2×#13050577C2×C411yescharacteristic
70C2×#13050577C2×C411yescharacteristic
71C2×#13050573—21no—
72C2×C8×D83—21no—
73#14581786—21no—
74#14581784C2×C411yescharacteristic
75#14581780C2×C411yescharacteristic
76#14581775—21no—
77#14581771C2×C411yescharacteristic
78#14581766C2×C411yescharacteristic
79C4×Dic166C2311yescharacteristic
80C4×D332C2311yescharacteristic
81C4×C22⋊D83—21no—
82#14581749—21no—
83#14581725C2×C411yescharacteristic
84#14581725C2×C411yescharacteristic
85#14581723C2×C411yescharacteristic
86#14581723C2×C411yescharacteristic
87C4⋊(C83⋊C8)—21no—
88C4⋊(C83⋊C8)C2×C411yescharacteristic
89C4⋊(C83⋊C8)C2×C411yescharacteristic
90C4⋊(C83⋊C8)C2311yescharacteristic
91C4⋊(C83⋊C8)C2311yescharacteristic
92Dic83⋊C8—21no—
93Dic83⋊C8—21no—
94#14581716C2×C411yescharacteristic
95#14581716C2×C411yescharacteristic
96#14581716C2×C411yescharacteristic
97#14581716C2×C411yescharacteristic
98#14581715C2×C411yescharacteristic
99#14581715C2×C411yescharacteristic
100#14581712C2×C411yescharacteristic
101#14581712C2×C411yescharacteristic
102C22⋊(C83⋊C8)—21no—
103C22⋊(C83⋊C8)—21no—
104C22⋊(C83⋊C8)C2×C411yescharacteristic
105C22⋊(C83⋊C8)C2×C411yescharacteristic
106(C2×D83)⋊C8—21no—
107(C2×D83)⋊C8—21no—
108#14581702—21no—
109C4×C83⋊C8—21no—
110C4×C83⋊C8C2311yescharacteristic
111C4×C83⋊C8C2311yescharacteristic
112C4×C83⋊C8C2311yescharacteristic
113C4×C83⋊C8C2311yescharacteristic
114C8×Dic83—21no—
115C22×Q8×C83C2×C411yescharacteristic
116C2×M16×C83—21no—
117C2×M16×C83D411yescharacteristic
118C2×M16×C83D411yescharacteristic
119C4×Q8×C83—21no—
120C4×Q8×C83—21no—
121C4×Q8×C83C2×C411yescharacteristic
122C4×Q8×C83C2×C411yescharacteristic
123C4×Q8×C83C2311yescharacteristic
124C4×Q8×C83C2311yescharacteristic
125C2×C4⋊C4×C83—21no—
126C2×C4⋊C4×C83C2×C411yescharacteristic
127C2×C42×C83—21no—
128C2×C42×C83C2311yescharacteristic
129C4⋊C8×C83—21no—
130C4⋊C8×C83—21no—
131C4⋊C8×C83—21no—
132C4⋊C8×C83—21no—
133C4⋊C8×C83—21no—
134C4⋊C8×C83C2311yescharacteristic
135C4⋊C8×C83C2311yescharacteristic
136#97×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#13050580—41no—
150#13050577—22no—
151#13050577—41no—
152#13050573—42no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8C22×C411yescharacteristic
158C2×C83⋊C8C22×C411yescharacteristic
159C2×C83⋊C8C22×C411yescharacteristic
160C2×C83⋊C8C22×C411yescharacteristic
161C8×D83—42no—
162C4⋊Dic83C22×C411yescharacteristic
163C4⋊Dic83C22×C411yescharacteristic
164Dic83⋊C4—21no—
165Dic83⋊C4—21no—
166Dic83⋊C4—21no—
167Dic83⋊C4—21no—
168C22⋊Dic83—21no—
169(C2×D83)⋊C4—21no—
170(C2×D83)⋊C4—21no—
171(C2×D83)⋊C4—21no—
172(C2×D83)⋊C4—21no—
173C4×Dic83—21no—
174C2×Q8×C83—21no—
175C2×Q8×C83—22no—
176C2×Q8×C83—41no—
177C2×Q8×C83C22×C411yescharacteristic
178C2×Q8×C83C22×C411yescharacteristic
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83C22×C411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83#4111yescharacteristic
184M16×C83—22no—
185M16×C83—41no—
186M16×C83—42no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
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×C83—21no—
198C4⋊C4×C83C22×C411yescharacteristic
199C4⋊C4×C83C22×C411yescharacteristic
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—21no—
204C42×C83C22×C411yescharacteristic
205C42×C83C2×D411yescharacteristic
206C42×C83#4111yescharacteristic
207C42×C83C2411yescharacteristic
208C23×C83C4×D411yescharacteristic
209C22×D83—21no—
210Q8×C83—21no—
211Q8×C83—21no—
212Q8×C83—21no—
213Q8×C83—42no—
214Q8×C83#13112yes—
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—21no—
223C2×C4×C83—41no—
224C2×C4×C83—42no—
225C2×C4×C83—81no—
226C2×C4×C83C4×D411yescharacteristic
227C2×C4×C83C4×D411yescharacteristic
228C2×C4×C83C4×D411yescharacteristic
229C2×C4×C83C23×C411yescharacteristic
230C2×C4×C83C22×D411yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C22⋊D83—81no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236D332—21no—
237D332—21no—
238C4×D83—41no—
239C4×D83—41no—
240Dic166—21no—
241Dic166—21no—
242C8×C83—21no—
243C8×C83—21no—
244C8×C83—21no—
245C8×C83—41no—
246C8×C83—41no—
247C8×C83#13712yes—
248C83⋊C8—21no—
249C83⋊C8—21no—
250C83⋊C8—21no—
251C83⋊C8—21no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C83⋊C8—41no—
256C22×C83—21no—
257C22×C83—21no—
258C22×C83—21no—
259C22×C83C2×C4×D411yescharacteristic
260C2×D83—41no—
261C2×D83—41no—
262C2×D83—41no—
263C4×C83—21no—
264C4×C83—22no—
265C4×C83—21no—
266C4×C83—21no—
267C4×C83—21no—
268C4×C83—41no—
269C4×C83—41no—
270C4×C83—41no—
271C4×C83C2×#13111yescharacteristic
272C4×C83#56811yescharacteristic
273C4×C83C2×#13711yescharacteristic
274C4×C83#57511yescharacteristic
275Dic83—41no—
276Dic83—41no—
277Dic83—41no—
278#18012—831nomaximal, Sylow
279C2×C83—41no—
280C2×C83#293311yescharacteristic
281C2×C83C4×#13711yescharacteristic
282C2×C83#295711yescharacteristic
283D83—81no—
284#3108—831no—
285#2969—831no—
286Q8×M16D8311yescharacteristic, complemented
287#1593—831no—
288#1590—831no—
289#1587—831no—
290#1585—831no—
291#1584—831no—
292#1583—831no—
293SD16⋊C8—831no—
294C8×Q16—831no—
295#1574—831no—
296#1495—831no—
297#1479—831no—
298#1453—831no—
299C83#1801211yescharacteristic, complemented, Sylow
300#529—831no—
301C2×C4×Q8C2×D8311yescharacteristic
302#483—831no—
303#480—831no—
304#479—831no—
305#479—831no—
306#464—831no—
307#464—831no—
308#451—831no—
309#448—831no—
310#446—21no—
311#446—21no—
312#446—831no—
313#446C2×D8311yescharacteristic
314#446C2×D8311yescharacteristic
315C8×Q8—831no—
316C8×Q8C2×D8311yescharacteristic
317C8×Q8C2×D8311yescharacteristic
318C4×Q16—831no—
319C4×Q16—831no—
320C4×SD16—831no—
321C4×SD16—831no—
322#436—831no—
323#436—1661no—
324#435—1661no—
325C8×D4—831no—
326#433—1661no—
327#431—831no—
328#431—831no—
329#431—831no—
330#423—21no—
331#423C2×D8311yescharacteristic
332#405—1661no—
333C4×M16—21no—
334C4×M16C2×D8311yescharacteristic
335#335—831no—
336#335—831no—
337#334—831no—
338#334—831no—
339Q8⋊C8—831no—
340Q8⋊C8—831no—
341Q8⋊C8—831no—
342Q8⋊C8—831no—
343Q8⋊C8—831no—
344Q8⋊C8—831no—
345D4⋊C8—831no—
346D4⋊C8—831no—
347#322—831no—
348#322—831no—
349#322—831no—
350C82—831no—
351C22×Q8C4×D8311yescharacteristic
352C2×Q16—831no—
353C2×Q16—831no—
354C2×SD16—831no—
355C2×SD16—831no—
356C2×M16—21no—
357C2×M16—1661no—
358C2×M16C22⋊D8311yescharacteristic
359C2×M16C22⋊D8311yescharacteristic
360C22×C8—1661no—
361#126—1661no—
362#125—831no—
363#124—831no—
364#123—1661no—
365C22⋊Q8—831no—
366#121—831no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—831no—
370C4×Q8C4×D8311yescharacteristic
371C4×Q8C4×D8311yescharacteristic
372C4×Q8C22×D8311yescharacteristic
373C4×Q8C22×D8311yescharacteristic
374C4×D4—831no—
375C4×D4—1661no—
376#117—1661no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C4×D8311yescharacteristic
379C2×C42—21no—
380C2×C42C22×D8311yescharacteristic
381#107—831no—
382#107—831no—
383#106—831no—
384#106—831no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—831no—
391C4⋊C8—831no—
392C4⋊C8—831no—
393C4⋊C8—831no—
394C4⋊C8—1661no—
395C4⋊C8—1661no—
396C4⋊C8—1661no—
397C4⋊C8C22×D8311yescharacteristic
398C4⋊C8C22×D8311yescharacteristic
399#103—831no—
400#103—831no—
401#103—831no—
402#103—831no—
403#103—831no—
404#103—831no—
405#102—831no—
406#102—831no—
407C22⋊C8—831no—
408C22⋊C8—831no—
409C22⋊C8—1661no—
410C22⋊C8—1661no—
411C22⋊C8—1661no—
412C22⋊C8—1661no—
413#97—21no—
414#97—21no—
415#97—1661no—
416#97C22⋊D8311yescharacteristic
417#97C22⋊D8311yescharacteristic
418C4×C8—21no—
419C4×C8—21no—
420C4×C8—831no—
421C4×C8—831no—
422C4×C8—831no—
423C4×C8—831no—
424C4×C8—1661no—
425C4×C8—1661no—
426C4×C8C22×D8311yescharacteristic
427C4×C8C22×D8311yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—831no—
432C2×Q8C2×C4×D8311yescharacteristic
433C2×Q8C2×C4×D8311yescharacteristic
434C2×D4—831no—
435C2×D4—1661no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—1661no—
439C22×C4C2×C4×D8311yescharacteristic
440C22×C4C2×C22⋊D8311yescharacteristic
441C22×C4#1305059011yescharacteristic
442Q16—3321no—
443Q16—1662no—
444SD16—3321no—
445SD16—1662no—
446M16—22no—
447M16—41no—
448M16—42no—
449M16—3322no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—3322no—
453C2×C8—831no—
454C2×C8—831no—
455C2×C8—831no—
456C2×C8—831no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1661no—
461C2×C8C2×C22⋊D8311yescharacteristic
462C2×C8C2×C22⋊D8311yescharacteristic
463C2×C8C2×C22⋊D8311yescharacteristic
464C2×C8C2×C22⋊D8311yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—831no—
471C4⋊C4—831no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
475C4⋊C4—1661no—
476C4⋊C4C2×C4×D8311yescharacteristic
477C4⋊C4C2×C4×D8311yescharacteristic
478C22⋊C4—1661no—
479C22⋊C4—1661no—
480C22⋊C4—1661no—
481C22⋊C4—1661no—
482C22⋊C4—1661no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—1661no—
488C42C2×C4×D8311yescharacteristic
489C42C2×C22⋊D8311yescharacteristic
490C42#1305059011yescharacteristic
491C42C23×D8311yescharacteristic
492C23—1661no—
493C23C4×C22⋊D8311yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—1661no—
499Q8—1661no—
500Q8#1458180012yes—
501D4—6641no—
502D4—1661no—
503D4—1661no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—3321no—
516C2×C4—3321no—
517C2×C4—1661no—
518C2×C4—1661no—
519C2×C4—1661no—
520C2×C4C4×C22⋊D8311yescharacteristic
521C2×C4C4×C22⋊D8311yescharacteristic
522C2×C4C4×C22⋊D8311yescharacteristic
523C2×C4C22×C4×D8311yescharacteristic
524C2×C4C22×C22⋊D8311yescharacteristic
525C2×C4C2×#1305059011yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—3321no—
532C8—3321no—
533C8—3321no—
534C8—3321no—
535C8—1661no—
536C8—1661no—
537C8—1661no—
538C8—1661no—
539C8#1458182312yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—3321no—
544C22—3321no—
545C22—3321no—
546C22C2×C4×C22⋊D8311yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—3321no—
556C4—3321no—
557C4—3321no—
558C4C2×#1458180011yescharacteristic
559C4#1912846711yescharacteristic
560C4C2×#1458182311yescharacteristic
561C4#1912849911yescharacteristic
562C2—41no—
563C2—6641no—
564C2#2491618811yescharacteristic
565C2C4×#1458182311yescharacteristic
566C2#2491637811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingQ8×M16×C83C2
socleC22×C83C2×C4×D4
prime radical#14581766C2×C4
cosocleC42×C83C24
hypercenterQ8×M16D83
hypocenterC83#18012
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. #24906370C2semidirect
4. #24906353C2semidirect
5. #24906334C2semidirect
6. #24906318C2semidirect
7. #24906311C2semidirect
8. #24906306C2semidirect
9. #24906273C2semidirect
10. C8×#13050580C2semidirect
11. #24906254C2semidirect
13. #24905858C2semidirect
14. #24905760C2semidirect
15. Q8×M16×C83C2semidirect
286. Q8×M16D83semidirect
299. C83#18012semidirect

Maximal subgroups

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

subgroupindex
#249175432
#249164672
#249063702
#249063532
#249063342
#249063182
#249063112
#249063062
#249062732
C8×#130505802
#249062542
#249059262
#249058582
#249057602
Q8×M16×C832
#1801283