Finite Group Databasebeta

← #35477522

Subgroups — #35477522

Order 21248.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910488C211yesmaximal, characteristic
2#24908882C211yesmaximal, characteristic, complemented
3#24908841C211yesmaximal, characteristic, complemented
4#24908666C211yesmaximal, characteristic
5#24908659C211yesmaximal, characteristic
6#24908229C211yesmaximal, characteristic, complemented
7#24908198C211yesmaximal, characteristic, complemented
8#24908078C211yesmaximal, characteristic
9#24908052C211yesmaximal, characteristic, complemented
10#24908048C211yesmaximal, characteristic, complemented
11#24907716C211yesmaximal, characteristic, complemented
12#24907660C211yesmaximal, characteristic
13#24907555C211yesmaximal, characteristic
14#24907296C211yesmaximal, characteristic, complemented
15C83×#2494C211yesmaximal, characteristic
16C2×#14581771C2211yescharacteristic
17C2×C4×Dic166C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24C2×#14581715C2211yescharacteristic
25C2×#14581715C2211yescharacteristic
26C2×#14581715C2211yescharacteristic
27C2×#14581715C2211yescharacteristic
28C2×C22⋊(C83⋊C8)C2211yescharacteristic
29C2×C22⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31#19127607C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127595—21no—
34#19127589—21no—
35#19127583—21no—
36C4×C4⋊Dic83C2211yescharacteristic
37#19127552—21no—
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48#401×C83—21no—
49#397×C83—21no—
50#389×C83C2211yescharacteristic
51#389×C83C2211yescharacteristic
52#388×C83—21no—
53#387×C83—21no—
54#382×C83C2211yescharacteristic
55#380×C83C2211yescharacteristic
56#380×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59C22×Dic166C2311yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581771—22no—
65#14581771—22no—
66C4×Dic166—22no—
67C4×Dic166—41no—
68#14581751—22no—
69#14581751—41no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×C22⋊Dic83—21no—
77C2×C22⋊Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83—21no—
80C2×C4×Dic83—21no—
81#14581725—22no—
82#14581725—22no—
83#14581723—22no—
84#14581723—41no—
85C4⋊(C83⋊C8)—22no—
86#14581715—22no—
87#14581715—22no—
88#14581715—22no—
89#14581715—41no—
90C22⋊(C83⋊C8)—22no—
91C22⋊(C83⋊C8)—41no—
92#14581703—41no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C23×C4×C83C2311yescharacteristic
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83C2311yescharacteristic
101C2×C4⋊C4×C83C2311yescharacteristic
102C2×C4⋊C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C22⋊C4×C83—21no—
105C2×C22⋊C4×C83—21no—
106C2×C22⋊C4×C83D411yescharacteristic
107C2×C22⋊C4×C83D411yescharacteristic
108C2×C42×C83—21no—
109C2×C42×C83C2311yescharacteristic
110C2×C42×C83C2311yescharacteristic
111#95×C83—21no—
112#95×C83—21no—
113#95×C83—21no—
114#95×C83—21no—
115#95×C83—21no—
116#95×C83—21no—
117#95×C83—21no—
118#95×C83D411yescharacteristic
119#95×C83D411yescharacteristic
120C2×Dic166—21no—
121C2×Dic166—21no—
122C2×Dic166—21no—
123C2×Dic166—21no—
124C2×Dic166—21no—
125C2×Dic166#4112yes—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8#4112yes—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83#4112yes—
147C4⋊Dic83#4112yes—
148Dic83⋊C4—24no—
149Dic83⋊C4—42no—
150Dic83⋊C4—42no—
151C22⋊Dic83—24no—
152C22⋊Dic83—42no—
153C4×Dic83—42no—
154C4×Dic83—42no—
155C4×Dic83—42no—
156C24×C83C2×D411yescharacteristic
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2411yescharacteristic
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—42no—
182C4⋊C4×C83—81no—
183C22⋊C4×C83—41no—
184C22⋊C4×C83—41no—
185C22⋊C4×C83—42no—
186C22⋊C4×C83—81no—
187C42×C83—21no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—81no—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83—21no—
199C23×C83C22×D411yescharacteristic
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83C2×#4111yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83H(5,𝔽2)11yescharacteristic
236C2×C4×C832−1+411yescharacteristic
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—22no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249Dic166—22no—
250Dic166—22no—
251Dic166—41no—
252Dic166—41no—
253C83⋊C8—22no—
254C83⋊C8—41no—
255C83⋊C8—41no—
256C83⋊C8—41no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83—41no—
261C22×C83—41no—
262C22×C83—41no—
263C22×C83—41no—
264C22×C83#52511yescharacteristic
265C22×C83#54611yescharacteristic
266C22×C83#54711yescharacteristic
267C22×C83#57511yescharacteristic
268C22×C83#57511yescharacteristic
269C22×C83#57711yescharacteristic
270C22×C83#57811yescharacteristic
271C4×C83—41no—
272C4×C83—41no—
273C4×C83—41no—
274C4×C83—41no—
275C4×C83—41no—
276C4×C83—81no—
277C4×C83—81no—
278C4×C83—81no—
279C4×C83—81no—
280C4×C83#52911yescharacteristic
281C4×C83#53911yescharacteristic
282C4×C83#55311yescharacteristic
283C4×C83#55611yescharacteristic
284Dic83—42no—
285Dic83—42no—
286Dic83—42no—
287Dic83—81no—
288#27061—831nomaximal, Sylow
289C2×C83—81no—
290C2×C83#246611yescharacteristic
291C2×C83#311111yescharacteristic
292C2×C83#312111yescharacteristic
293C2×C83#330711yescharacteristic
294C2×C83#332711yescharacteristic
295C2×C83#334911yescharacteristic
296C2×C83#335011yescharacteristic
297#2494D8311yescharacteristic
298#2362—831no—
299#2074—831no—
300#2066—831no—
301#2036—831no—
302#2035—831no—
303#1949—831no—
304#1943—831no—
305#1924—831no—
306#1921—831no—
307#1920—831no—
308#1863—831no—
309#1854—831no—
310#1825—831no—
311#1763—831no—
312C83#2706111yescharacteristic, complemented, Sylow
313C2×C22⋊Q8—831no—
314C2×C4×Q8—831no—
315C2×#117—831no—
316C2×#107—831no—
317C2×#107—831no—
318C2×#106—831no—
319C2×#106—831no—
320C2×C4⋊C8—831no—
321C2×#103—831no—
322C2×#103—831no—
323C2×#103—831no—
324C2×#103—831no—
325C2×C22⋊C8—831no—
326C2×C22⋊C8—831no—
327C2×#97—831no—
328#401—21no—
329#397—21no—
330#391—831no—
331#389—831no—
332#389C2×D8311yescharacteristic
333#389C2×D8311yescharacteristic
334#388—21no—
335#388—1661no—
336#388—1661no—
337#387—21no—
338#387—1661no—
339#382C2×D8311yescharacteristic
340#380C2×D8311yescharacteristic
341#380C2×D8311yescharacteristic
342C4×C4⋊C4—831no—
343C4×C4⋊C4—1661no—
344C4×C4⋊C4C2×D8311yescharacteristic
345C4×C22⋊C4C2×D8311yescharacteristic
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#336—831no—
355#336—831no—
356C22×Q8—831no—
357C23×C4C22×D8311yescharacteristic
358C22×C8—831no—
359C22×C8—831no—
360C22×C8—831no—
361C22×C8—831no—
362C22⋊Q8—1662no—
363C22⋊Q8—1662no—
364C4×Q8—3321no—
365C4×Q8—1662no—
366#117—3321no—
367#117—1662no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4C22×D8311yescharacteristic
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1661no—
383C2×C22⋊C4—1661no—
384C2×C22⋊C4C22⋊D8311yescharacteristic
385C2×C22⋊C4C22⋊D8311yescharacteristic
386C2×C42—21no—
387C2×C42—1661no—
388C2×C42—1661no—
389C2×C42—1661no—
390C2×C42C22×D8311yescharacteristic
391C2×C42C22×D8311yescharacteristic
392#107—1662no—
393#107—1662no—
394#106—3321no—
395#106—1662no—
396C4⋊C8—1662no—
397#103—3321no—
398#103—1662no—
399#103—1662no—
400#103—1662no—
401C22⋊C8—3321no—
402C22⋊C8—1662no—
403#97—3321no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414#95—1661no—
415#95C22⋊D8311yescharacteristic
416#95C22⋊D8311yescharacteristic
417C24C2×C22⋊D8311yescharacteristic
418C2×Q8—832no—
419C2×Q8—1661no—
420C2×Q8—1661no—
421C2×Q8—1661no—
422C2×Q8—1661no—
423C2×Q8—1661no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1661no—
437C22×C4—1661no—
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4C2×C22⋊D8311yescharacteristic
440C22×C4C2×C22⋊D8311yescharacteristic
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4C2×C22⋊D8311yescharacteristic
443C22×C4C23×D8311yescharacteristic
444C2×C8—832no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8—1661no—
451C2×C8—1661no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—3322no—
467C4⋊C4—3322no—
468C4⋊C4—832no—
469C4⋊C4—832no—
470C4⋊C4—1664no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—3322no—
480C22⋊C4—1664no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—3322no—
487C42—3322no—
488C42—3322no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C22×C22⋊D8311yescharacteristic
497Q8—3321no—
498Q8—3321no—
499Q8—1662no—
500Q8—1662no—
501C2×C4—21no—
502C2×C4—21no—
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—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—1662no—
542C2×C4—1662no—
543C2×C4—1662no—
544C2×C4#41×D8311yescharacteristic
545C2×C4#41×D8311yescharacteristic
546C2×C4#41×D8311yescharacteristic
547C2×C4#41×D8311yescharacteristic
548C2×C4#1458183811yescharacteristic
549C2×C4#1458183911yescharacteristic
550C8—3321no—
551C8—3321no—
552C8—3321no—
553C8—1662no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1912820811yescharacteristic
562C22#1912833911yescharacteristic
563C22#1912835011yescharacteristic
564C22#1912849911yescharacteristic
565C22#1912849911yescharacteristic
566C22#1912851211yescharacteristic
567C22#1912851611yescharacteristic
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—3322no—
578C4—3322no—
579C4—3322no—
580C4—6641no—
581C4#1912823411yescharacteristic
582C4#1912829211yescharacteristic
583C4#1912840211yescharacteristic
584C4#1912841811yescharacteristic
585C2—81no—
586C2#2491149511yescharacteristic
587C2#2491756611yescharacteristic
588C2#2491768611yescharacteristic
589C2#2491943311yescharacteristic
590C2#2491972511yescharacteristic
591C2#2491998711yescharacteristic
592C2#2492000211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2494C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2494D83
hypocenterC83#27061
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2494

Complements

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

NH ≅ G/Ntype
2. #24908882C2semidirect
3. #24908841C2semidirect
6. #24908229C2semidirect
7. #24908198C2semidirect
9. #24908052C2semidirect
10. #24908048C2semidirect
11. #24907716C2semidirect
14. #24907296C2semidirect
312. C83#27061semidirect

Maximal subgroups

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

subgroupindex
#249104882
#249088822
#249088412
#249086662
#249086592
#249082292
#249081982
#249080782
#249080522
#249080482
#249077162
#249076602
#249075552
#249072962
C83×#24942
#2706183