Finite Group Databasebeta

← #35375591

Subgroups — #35375591

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#24917543C211yesmaximal, characteristic
2#24916619C211yesmaximal, characteristic
3#24916552C211yesmaximal, characteristic, complemented
4#24906357C211yesmaximal, characteristic, complemented
5#24906350C211yesmaximal, characteristic, complemented
6#24906338C211yesmaximal, characteristic, complemented
7#24906337C211yesmaximal, characteristic, complemented
8#24906331C211yesmaximal, characteristic, complemented
9#24906323C211yesmaximal, characteristic, complemented
10#24906267C211yesmaximal, characteristic, complemented
11#24906260C211yesmaximal, characteristic, complemented
12#24906246C211yesmaximal, characteristic, complemented
13#24905924C211yesmaximal, characteristic
14#24905746C211yesmaximal, characteristic, complemented
15C2×Q8⋊C8×C83C211yesmaximal, characteristic, complemented
16#19128235C2211yescharacteristic
17#19128031C411yescharacteristic
18#19128014C411yescharacteristic
19#19128008C411yescharacteristic
20#19128008C411yescharacteristic
21#19127923C411yescharacteristic
22#19127923C411yescharacteristic
23#19127857C411yescharacteristic
24#19127840C411yescharacteristic
25#19127832—21no—
26#19127832—21no—
27#19127832C2211yescharacteristic
28#19127832C2211yescharacteristic
29#19127828C2211yescharacteristic
30Q8×C83⋊C8C2211yescharacteristic
31Q8×C83⋊C8C2211yescharacteristic
32#19127825C2211yescharacteristic
33C4×#13050580C2211yescharacteristic
34C4×#13050580C2211yescharacteristic
35C4×#13050577C2211yescharacteristic
36C4×#13050577C2211yescharacteristic
37#19127778C2211yescharacteristic
38#19127772—21no—
39#19127766C2211yescharacteristic
40C8×C22⋊D83—21no—
41#19127761—21no—
42#19127758—21no—
43#19127758C2211yescharacteristic
44#19127754—21no—
45#19127754C2211yescharacteristic
46#19127754C2211yescharacteristic
47#19127726—21no—
48#19127453C2211yescharacteristic
49#19127453C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127451C2211yescharacteristic
52#19127433C2211yescharacteristic
53#19127433C2211yescharacteristic
54#19127430C2211yescharacteristic
55#19127430C2211yescharacteristic
56#19127422C2211yescharacteristic
57#19127422C2211yescharacteristic
58C8×C83⋊C8C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60C2×C4×Q8×C83C2211yescharacteristic
61C2×C4⋊C8×C83C2211yescharacteristic
62C2×C4×C8×C83C2211yescharacteristic
63Q8⋊C8×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65Q8⋊C8×C83C2211yescharacteristic
66Q8⋊C8×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)—21no—
89C4⋊(C83⋊C8)—21no—
90C4⋊(C83⋊C8)—21no—
91C4⋊(C83⋊C8)C2×C411yescharacteristic
92C4⋊(C83⋊C8)C2×C411yescharacteristic
93C4⋊(C83⋊C8)D411yescharacteristic
94C4⋊(C83⋊C8)D411yescharacteristic
95C4⋊(C83⋊C8)C2311yescharacteristic
96C4⋊(C83⋊C8)C2311yescharacteristic
97Dic83⋊C8—21no—
98Dic83⋊C8—21no—
99#14581716C2×C411yescharacteristic
100#14581716C2×C411yescharacteristic
101#14581716C2×C411yescharacteristic
102#14581716C2×C411yescharacteristic
103#14581715C2×C411yescharacteristic
104#14581715C2×C411yescharacteristic
105#14581712C2×C411yescharacteristic
106#14581712C2×C411yescharacteristic
107C22⋊(C83⋊C8)—21no—
108C22⋊(C83⋊C8)C2×C411yescharacteristic
109C22⋊(C83⋊C8)C2×C411yescharacteristic
110C22⋊(C83⋊C8)D411yescharacteristic
111C22⋊(C83⋊C8)D411yescharacteristic
112(C2×D83)⋊C8—21no—
113(C2×D83)⋊C8—21no—
114#14581703—21no—
115#14581703—21no—
116#14581703—21no—
117#14581702—21no—
118C4×C83⋊C8—21no—
119C4×C83⋊C8—21no—
120C4×C83⋊C8C2311yescharacteristic
121C4×C83⋊C8C2311yescharacteristic
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C8×Dic83—21no—
125C22×Q8×C83C2×C411yescharacteristic
126C22×C8×C83—21no—
127C22×C8×C83—21no—
128C4×Q8×C83—21no—
129C4×Q8×C83—21no—
130C4×Q8×C83C2×C411yescharacteristic
131C4×Q8×C83C2×C411yescharacteristic
132C4×Q8×C83C2311yescharacteristic
133C4×Q8×C83C2311yescharacteristic
134C2×C4⋊C4×C83—21no—
135C2×C4⋊C4×C83C2×C411yescharacteristic
136C2×C42×C83—21no—
137C2×C42×C83C2311yescharacteristic
138C4⋊C8×C83—21no—
139C4⋊C8×C83C2311yescharacteristic
140C4⋊C8×C83C2311yescharacteristic
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—81no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8C22×C411yescharacteristic
156C2×C83⋊C8C22×C411yescharacteristic
157C2×C83⋊C8C22×C411yescharacteristic
158C2×C83⋊C8C22×C411yescharacteristic
159C2×C83⋊C8C2×D411yescharacteristic
160C2×C83⋊C8C2×D411yescharacteristic
161C2×C83⋊C8C2×D411yescharacteristic
162C2×C83⋊C8C2×D411yescharacteristic
163C8×D83—81no—
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×C83—21no—
177C2×Q8×C83—22no—
178C2×Q8×C83—41no—
179C2×Q8×C83C22×C411yescharacteristic
180C2×Q8×C83C22×C411yescharacteristic
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83C22×C411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83#4111yescharacteristic
186C2×C8×C83—24no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83—81no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83C22×C411yescharacteristic
198C4⋊C4×C83C22×C411yescharacteristic
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83C22×C411yescharacteristic
204C42×C83C2×D411yescharacteristic
205C42×C83#4111yescharacteristic
206C42×C83C2411yescharacteristic
207C23×C83C4×D411yescharacteristic
208C22×D83—21no—
209Q8×C83—21no—
210Q8×C83—21no—
211Q8×C83—21no—
212Q8×C83—42no—
213Q8×C83#13112yes—
214C2×C4×C83—21no—
215C2×C4×C83—22no—
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—42no—
224C2×C4×C83—42no—
225C2×C4×C83C4×D411yescharacteristic
226C2×C4×C83C4×D411yescharacteristic
227C2×C4×C83C4×D411yescharacteristic
228C2×C4×C83C23×C411yescharacteristic
229C2×C4×C83C22×D411yescharacteristic
230C2×C4×C83C2×#4111yescharacteristic
231C22⋊D83—42no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235D332—21no—
236D332—21no—
237C4×D83—41no—
238C4×D83—41no—
239Dic166—21no—
240Dic166—21no—
241C8×C83—22no—
242C8×C83—22no—
243C8×C83—41no—
244C8×C83—41no—
245C83⋊C8—21no—
246C83⋊C8—21no—
247C83⋊C8—21no—
248C83⋊C8—21no—
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#44312yes—
258C22×C83C2×C4×D411yescharacteristic
259C2×D83—41no—
260C2×D83—41no—
261C2×D83—41no—
262C4×C83—21no—
263C4×C83—22no—
264C4×C83—21no—
265C4×C83—21no—
266C4×C83—21no—
267C4×C83—41no—
268C4×C83—41no—
269C4×C83—41no—
270C4×C83C2×#13111yescharacteristic
271C4×C83#56811yescharacteristic
272C4×C83#57711yescharacteristic
273C4×C83#57811yescharacteristic
274Dic83—41no—
275Dic83—41no—
276Dic83—41no—
277#18017—831nomaximal, Sylow
278C2×C83—22no—
279C2×C83#293311yescharacteristic
280C2×C83#294711yescharacteristic
281C2×C83C2×#44311yescharacteristic
282D83—81no—
283#3108—831no—
284#2994—831no—
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⋊C8D8311yescharacteristic, complemented
297#1450—831no—
298C83#1801711yescharacteristic, complemented, Sylow
299#529—831no—
300C2×C4×Q8C2×D8311yescharacteristic
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—831no—
312#446—1661no—
313#446—1661no—
314C8×Q8—831no—
315C8×Q8—831no—
316C8×Q8—831no—
317C4×Q16—831no—
318C4×Q16—831no—
319C4×SD16—831no—
320C4×SD16—831no—
321#436—831no—
322#435—1661no—
323C8×D4—831no—
324C8×D4—1661no—
325#433—1661no—
326#432—831no—
327#432—1661no—
328#431—831no—
329#431—831no—
330#431—1661no—
331#424—1661no—
332C2×C4⋊C8C2×D8311yescharacteristic
333C2×C4×C8C2×D8311yescharacteristic
334#335—831no—
335#335—831no—
336#334—831no—
337#334—831no—
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8C2×D8311yescharacteristic
341Q8⋊C8C2×D8311yescharacteristic
342Q8⋊C8C2×D8311yescharacteristic
343Q8⋊C8C2×D8311yescharacteristic
344D4⋊C8—831no—
345D4⋊C8—831no—
346#322—831no—
347#322—831no—
348C82—831no—
349C82—831no—
350C22×Q8C4×D8311yescharacteristic
351C2×Q16—831no—
352C2×Q16—831no—
353C2×SD16—831no—
354C2×SD16—831no—
355C2×M16—1661no—
356C22×C8—21no—
357C22×C8—21no—
358C22×C8—1661no—
359#126—1661no—
360#125—831no—
361#124—831no—
362#123—1661no—
363C22⋊Q8—831no—
364#121—831no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—831no—
368C4×Q8C4×D8311yescharacteristic
369C4×Q8C4×D8311yescharacteristic
370C4×Q8C22×D8311yescharacteristic
371C4×Q8C22×D8311yescharacteristic
372C4×D4—831no—
373C4×D4—1661no—
374#117—1661no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C4×D8311yescharacteristic
377C2×C42—21no—
378C2×C42C22×D8311yescharacteristic
379#107—831no—
380#107—831no—
381#106—831no—
382#106—831no—
383C4⋊C8—21no—
384C4⋊C8—831no—
385C4⋊C8—831no—
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⋊C8—1661no—
395C4⋊C8—1661no—
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—831no—
407C22⋊C8—831no—
408C22⋊C8—831no—
409C22⋊C8—831no—
410C22⋊C8—1661no—
411C22⋊C8—1661no—
412C22⋊C8—1661no—
413#97—1661no—
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—21no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8—831no—
431C2×Q8C2×C4×D8311yescharacteristic
432C2×Q8C2×C4×D8311yescharacteristic
433C2×D4—831no—
434C2×D4—1661no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—1661no—
438C22×C4C2×C4×D8311yescharacteristic
439C22×C4D4×D8311yescharacteristic
440C22×C4#1305058911yescharacteristic
441Q16—3321no—
442Q16—1662no—
443SD16—3321no—
444SD16—1662no—
445M16—6641no—
446C2×C8—24no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—81no—
452C2×C8—831no—
453C2×C8—831no—
454C2×C8—831no—
455C2×C8—831no—
456C2×C8—831no—
457C2×C8—831no—
458C2×C8—831no—
459C2×C8—831no—
460C2×C8—6641no—
461C2×C8—1661no—
462C2×C8—1661no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—831no—
469C4⋊C4—831no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4C2×C4×D8311yescharacteristic
475C4⋊C4C2×C4×D8311yescharacteristic
476C22⋊C4—1661no—
477C22⋊C4—1661no—
478C22⋊C4—1661no—
479C22⋊C4—1661no—
480C22⋊C4—1661no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—1661no—
486C42C2×C4×D8311yescharacteristic
487C42D4×D8311yescharacteristic
488C42#1305058911yescharacteristic
489C42C23×D8311yescharacteristic
490C23—1661no—
491C23(C22⋊D83)⋊C411yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—42no—
496Q8—1661no—
497Q8—1661no—
498Q8#1458179812yes—
499D4—3322no—
500D4—1661no—
501D4—1661no—
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—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—1661no—
516C2×C4—1661no—
517C2×C4—1661no—
518C2×C4(C22⋊D83)⋊C411yescharacteristic
519C2×C4(C22⋊D83)⋊C411yescharacteristic
520C2×C4(C22⋊D83)⋊C411yescharacteristic
521C2×C4C22×C4×D8311yescharacteristic
522C2×C4C2×D4×D8311yescharacteristic
523C2×C4C2×#1305058911yescharacteristic
524C8—22no—
525C8—22no—
526C8—41no—
527C8—41no—
528C8—3321no—
529C8—3321no—
530C8—1661no—
531C8—1661no—
532C8—1661no—
533C8—1661no—
534C8—1661no—
535C8—1661no—
536C8—1661no—
537C8—1661no—
538C22—21no—
539C22—21no—
540C22—3321no—
541C22—3321no—
542C22—3321no—
543C22#1912781512yes—
544C22C2×(C22⋊D83)⋊C411yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—3321no—
554C4—3321no—
555C4—3321no—
556C4C2×#1458179811yescharacteristic
557C4#1912846711yescharacteristic
558C4#1912850711yescharacteristic
559C4#1912851411yescharacteristic
560C2—22no—
561C2—6641no—
562C2#2491618411yescharacteristic
563C2#2491630011yescharacteristic
564C2C2×#1912781511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC2×Q8⋊C8×C83C2
socleC22×C83C2×C4×D4
prime radical#14581766C2×C4
cosocleC42×C83C24
hypercenterC2×Q8⋊C8D83
hypocenterC83#18017
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #24916552C2semidirect
4. #24906357C2semidirect
5. #24906350C2semidirect
6. #24906338C2semidirect
7. #24906337C2semidirect
8. #24906331C2semidirect
9. #24906323C2semidirect
10. #24906267C2semidirect
11. #24906260C2semidirect
12. #24906246C2semidirect
14. #24905746C2semidirect
15. C2×Q8⋊C8×C83C2semidirect
296. C2×Q8⋊C8D83semidirect
298. C83#18017semidirect

Maximal subgroups

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

subgroupindex
#249175432
#249166192
#249165522
#249063572
#249063502
#249063382
#249063372
#249063312
#249063232
#249062672
#249062602
#249062462
#249059242
#249057462
C2×Q8⋊C8×C832
#1801783