Finite Group Databasebeta

← #35463275

Subgroups — #35463275

Order 21248.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128108C211yesmaximal, characteristic, complemented
2C2×#19127744C211yesmaximal, characteristic
3#24911188C211yesmaximal, characteristic
4#24911148C211yesmaximal, characteristic
5#24908950C211yesmaximal, characteristic, complemented
6#24908948C211yesmaximal, characteristic, complemented
7#24908799C211yesmaximal, characteristic, complemented
8#24908790C211yesmaximal, characteristic, complemented
9#24908200C211yesmaximal, characteristic, complemented
10#24908195C211yesmaximal, characteristic, complemented
11#24907653C211yesmaximal, characteristic
12#24907626C211yesmaximal, characteristic, complemented
13#24907494C211yesmaximal, characteristic
14#24907249C211yesmaximal, characteristic
15C83×#1854C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581791C2211yescharacteristic
18C2×#14581783C2211yescharacteristic
19C22×C4⋊Dic83C2211yescharacteristic
20#19128108—24no—
21#19127744—24no—
22C2×#14581726C2211yescharacteristic
23C2×#14581726C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581725C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×#14581723C2211yescharacteristic
29C2×#14581723C2211yescharacteristic
30C2×#14581702C2211yescharacteristic
31C2×C8×Dic83C2211yescharacteristic
32#19127643—21no—
33#19127633—21no—
34#19127624—21no—
35#19127613—21no—
36#19127607—21no—
37#19127600C2211yescharacteristic
38#19127600C2211yescharacteristic
39#19127597C2211yescharacteristic
40#19127597C2211yescharacteristic
41#19127585—21no—
42#19127576—21no—
43#19127566C2211yescharacteristic
44#19127566C2211yescharacteristic
45#19127561C2211yescharacteristic
46#19127559—21no—
47C4×C22⋊Dic83C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53#19127455C2211yescharacteristic
54C2×#117×C83C2211yescharacteristic
55C2×#107×C83C2211yescharacteristic
56C2×#106×C83C2211yescharacteristic
57C2×C22⋊C8×C83C2211yescharacteristic
58#380×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60#340×C83C2211yescharacteristic
61C22×Dic166—21no—
62C23×Dic83—21no—
63C2×#13050574—22no—
64C2×#13050574—22no—
65C2×#13050574Q814yescomplemented
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68#14581791—22no—
69#14581791—41no—
70#14581783—22no—
71#14581783—41no—
72C2×C4⋊Dic83—22no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—41no—
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×C22⋊Dic83—21no—
87C2×C22⋊Dic83D411yescharacteristic
88C2×C22⋊Dic83D411yescharacteristic
89C2×C4×Dic83C2311yescharacteristic
90C2×C4×Dic83C2311yescharacteristic
91#14581726—22no—
92#14581726—41no—
93#14581725—22no—
94#14581725—22no—
95#14581725D414yes—
96#14581723—22no—
97#14581723—22no—
98#14581723—22no—
99#14581702—22no—
100C8×Dic83—22no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95D411yescharacteristic
111C83⋊#95D411yescharacteristic
112C23×C4×C83C2311yescharacteristic
113C22×C8×C83C2311yescharacteristic
114C22×C8×C83C2311yescharacteristic
115#117×C83—22no—
116#117×C83—41no—
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C22⋊C4×C83—21no—
122C2×C22⋊C4×C83—21no—
123C2×C42×C83—21no—
124#107×C83—41no—
125#106×C83—22no—
126C22⋊C8×C83Q814yescomplemented
127#95×C83—21no—
128C2×Dic166—42no—
129C2×Dic166—42no—
130C2×Dic166—81no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—42no—
136C22×Dic83—42no—
137C22×Dic83—81no—
138C22×Dic83C2×D411yescharacteristic
139C22×Dic83C2×D411yescharacteristic
140C22×Dic83C2×D411yescharacteristic
141C22×Dic83C2×D411yescharacteristic
142#13050574—28no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8C2×D412yes—
146C2×C83⋊C8C2×Q812yes—
147C2×C83⋊C8C2×Q812yes—
148C2×C83⋊C8#4112yes—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—42no—
157C4⋊Dic83—42no—
158C4⋊Dic83—81no—
159C4⋊Dic83—81no—
160C4⋊Dic83C2×D412yes—
161Dic83⋊C4—42no—
162Dic83⋊C4—42no—
163Dic83⋊C4—42no—
164Dic83⋊C4—81no—
165C22⋊Dic83—41no—
166C22⋊Dic83—41no—
167C22⋊Dic83—81no—
168C4×Dic83—21no—
169C4×Dic83—21no—
170C4×Dic83—21no—
171C4×Dic83—21no—
172C24×C83C2×D411yescharacteristic
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×Q812yes—
183C22×C4×C83C2×Q812yes—
184C22×C4×C83C2411yescharacteristic
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83C2×Q812yes—
190C2×C8×C83C2×Q812yes—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83C2×D412yes—
199C22⋊C4×C83—42no—
200C22⋊C4×C83—42no—
201C42×C83—42no—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83C4⋊Q812yes—
209C23×C83C22×D411yescharacteristic
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—22no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
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×C83C4⋊Q812yes—
229C2×C4×C83C22×D411yescharacteristic
230C2×C4×C83C22×Q811yescharacteristic
231C2×C4×C83C22×Q811yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83H(5,𝔽2)11yescharacteristic
234C2×C4×C832−1+411yescharacteristic
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—22no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—84no—
260Dic166—84no—
261C8×C83—22no—
262C8×C83—22no—
263C83⋊C8—22no—
264C83⋊C8C22⋊Q814yes—
265C22×C83—22no—
266C22×C83—22no—
267C22×C83—22no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83C2×C4⋊Q811yescharacteristic
273C22×C83D4211yescharacteristic
274C22×C83#54711yescharacteristic
275C22×C83#57511yescharacteristic
276C22×C83#57511yescharacteristic
277C22×C83#57611yescharacteristic
278C22×C83#57811yescharacteristic
279C4×C83—22no—
280C4×C83—42no—
281C4×C83—81no—
282C4×C83—81no—
283C4×C83—81no—
284C4×C83C2×C22⋊Q811yescharacteristic
285C4×C83#54411yescharacteristic
286C4×C83D4×Q811yescharacteristic
287C4×C83#55411yescharacteristic
288Dic83—41no—
289Dic83—41no—
290Dic83—41no—
291Dic83—42no—
292Dic83—41no—
293Dic83—81no—
294Dic83—81no—
295Dic83—81no—
296#26079—831nomaximal, Sylow
297C2×C83—42no—
298C2×C83#244011yescharacteristic
299C2×C83#316411yescharacteristic
300C2×C83#316611yescharacteristic
301C2×C83#327311yescharacteristic
302C2×C83#327711yescharacteristic
303C2×C83#334711yescharacteristic
304C2×C83#334911yescharacteristic
305C2×#501—831no—
306C2×#428—831no—
307#2442—831no—
308#2439—831no—
309#2085—831no—
310#2085—831no—
311#2059—831no—
312#2057—831no—
313#1943—831no—
314#1942—831no—
315#1854D8311yescharacteristic
316#1852—831no—
317#1846—831no—
318#1809—831no—
319#1753—831no—
320C83#2607911yescharacteristic, complemented, Sylow
321C22×M16—831no—
322C2×C4⋊Q8—831no—
323C2×#125—831no—
324C2×#117C2×D8311yescharacteristic
325C22×C4⋊C4—831no—
326#501—1664no—
327#428—1664no—
328C2×#107—831no—
329C2×#107—831no—
330C2×#107—831no—
331C2×#107—831no—
332C2×#107—831no—
333C2×#107C2×D8311yescharacteristic
334C2×#106—831no—
335C2×#106—831no—
336C2×#106—831no—
337C2×#106C2×D8311yescharacteristic
338C2×C22⋊C8C2×D8311yescharacteristic
339C2×#97—831no—
340C2×C4×C8—831no—
341#400—1661no—
342#398—1661no—
343#396—1661no—
344#393—1661no—
345#391—1661no—
346#389—831no—
347#389—831no—
348#389—831no—
349#389—831no—
350#387—1661no—
351#385—1661no—
352#382—831no—
353#382—831no—
354#380—831no—
355#380—1661no—
356#380C2×D8311yescharacteristic
357C4×C22⋊C4—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340C2×D8311yescharacteristic
363#340C2×D8311yescharacteristic
364#336—831no—
365#336—831no—
366C22×Q8—1661no—
367C23×C4—1661no—
368C23×C4C22×D8311yescharacteristic
369C2×M16—834no—
370C2×M16—1662no—
371C2×M16—1662no—
372C22×C8—831no—
373C22×C8—831no—
374C22×C8C22×D8311yescharacteristic
375C22×C8C22×D8311yescharacteristic
376C4⋊Q8—3321no—
377C4⋊Q8—1662no—
378#125—3321no—
379#125—1662no—
380#117—22no—
381#117—41no—
382C2×C4⋊C4—3321no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—1661no—
388C2×C4⋊C4—1661no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1662no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—831no—
403C2×C22⋊C4—831no—
404C2×C22⋊C4—1661no—
405C2×C42—21no—
406C2×C42—831no—
407C2×C42—831no—
408#107—41no—
409#107—3321no—
410#107—834no—
411#107—1662no—
412#107—1662no—
413#107—1662no—
414#106—22no—
415#106—1662no—
416#106—1662no—
417#106—1662no—
418C22⋊C8Dic16614yescomplemented
419#97—1662no—
420C4×C8—1662no—
421#95—21no—
422#95—831no—
423#95—831no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433C24D4×D8311yescharacteristic
434C2×Q8—3322no—
435C2×Q8—3322no—
436C2×Q8—6641no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—831no—
448C22×C4—831no—
449C22×C4—831no—
450C22×C4—831no—
451C22×C4—6641no—
452C22×C4—1661no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4D4×D8311yescharacteristic
457C22×C4Q8×D8312yes—
458C22×C4C2×Dic16612yes—
459C22×C4C23×D8311yescharacteristic
460M16—1668no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—832no—
466C2×C8—832no—
467C2×C8—832no—
468C2×C8—832no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8C2×Dic16612yes—
472C2×C8C2×Dic16612yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—3322no—
481C4⋊C4—3322no—
482C4⋊C4—3322no—
483C4⋊C4—3322no—
484C4⋊C4—3322no—
485C4⋊C4—832no—
486C4⋊C4—6641no—
487C4⋊C4—6641no—
488C4⋊C4—6641no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4D4×D8312yes—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—3321no—
500C22⋊C4—3321no—
501C22⋊C4—6641no—
502C42—42no—
503C42—1661no—
504C42—1661no—
505C42—1661no—
506C42—1661no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1458179112yes—
514C23C2×D4×D8311yescharacteristic
515Q8—6644no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
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—41no—
533C2×C4—41no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—6644no—
546C2×C4—1661no—
547C2×C4—1661no—
548C2×C4—1662no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4#1458179112yes—
560C2×C4C2×D4×D8311yescharacteristic
561C2×C4C2×Q8×D8311yescharacteristic
562C2×C4C22×Dic16611yescharacteristic
563C2×C4#41×D8311yescharacteristic
564C2×C4#1458183711yescharacteristic
565C2×C4#1458183911yescharacteristic
566C8—22no—
567C8—22no—
568C8—834no—
569C8—1662no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1458179111yescharacteristic
578C22#1912833311yescharacteristic
579C22#1912834911yescharacteristic
580C22#1912849711yescharacteristic
581C22#1912850011yescharacteristic
582C22#1912850311yescharacteristic
583C22#1912851511yescharacteristic
584C4—22no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—3322no—
590C4—3321no—
591C4—3321no—
592C4—3321no—
593C4—3321no—
594C4—6641no—
595C4—6641no—
596C4—6641no—
597C4C22⋊Q8×D8311yescharacteristic
598C4#1912833011yescharacteristic
599C4D4×Dic16611yescharacteristic
600C4#1912841011yescharacteristic
601C2—42no—
602C2#2491116411yescharacteristic
603C2#2491800311yescharacteristic
604C2#2491801311yescharacteristic
605C2#2491906711yescharacteristic
606C2#2491911311yescharacteristic
607C2#2491995611yescharacteristic
608C2#2491999611yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1854C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1854D83
hypocenterC83#26079
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. C2×#19128108C2semidirect
5. #24908950C2semidirect
6. #24908948C2semidirect
7. #24908799C2semidirect
8. #24908790C2semidirect
9. #24908200C2semidirect
10. #24908195C2semidirect
12. #24907626C2semidirect
65. C2×#13050574Q8semidirect
126. C22⋊C8×C83Q8semidirect
320. C83#26079semidirect
418. C22⋊C8Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191281082
C2×#191277442
#249111882
#249111482
#249089502
#249089482
#249087992
#249087902
#249082002
#249081952
#249076532
#249076262
#249074942
#249072492
C83×#18542
#2607983