Finite Group Databasebeta

← #35334497

Subgroups — #35334497

Order 21248.

1706 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127613C211yesmaximal, characteristic
2C2×#19127605C212yesmaximal, complemented
3C2×#19127574C212yesmaximal
4#24905727C212yesmaximal, complemented
5#24905721C212yesmaximal, complemented
6#24905708C211yesmaximal, characteristic
7#24905700C212yesmaximal
8#24905689C212yesmaximal, complemented
9C83×C22⋊#95C211yesmaximal, characteristic
10C23×Dic166C2211yescharacteristic
11C22×C4⋊Dic83C2211yescharacteristic
12C22×Dic83⋊C4C2212yes—
13C22×Dic83⋊C4C2212yes—
14C22×C22⋊Dic83C2212yes—
15C22×C22⋊Dic83C2212yes—
16C22×C4×Dic83C2212yes—
17#19127613C418yescomplemented
18#19127605—28no—
19#19127574—28no—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2211yescharacteristic
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2211yescharacteristic
28C2×C83⋊#95C2212yes—
29C24×C4×C83C2211yescharacteristic
30C22×C22⋊C4×C83C2211yescharacteristic
31C22×C22⋊C4×C83C2211yescharacteristic
32C2×#95×C83C2212yes—
33C2×#95×C83C2212yes—
34C22×Dic166—24no—
35C22×Dic166—24no—
36C22×Dic166—41no—
37C22×Dic166—41no—
38C22×Dic166C2×C414yescomplemented
39C23×Dic83C2312yes—
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C2×C4⋊Dic83—42no—
44C2×C4⋊Dic83—41no—
45C2×Dic83⋊C4—24no—
46C2×Dic83⋊C4—24no—
47C2×Dic83⋊C4—24no—
48C2×Dic83⋊C4—42no—
49C2×Dic83⋊C4—42no—
50C2×Dic83⋊C4—42no—
51C2×C22⋊Dic83—24no—
52C2×C22⋊Dic83—24no—
53C2×C22⋊Dic83—24no—
54C2×C22⋊Dic83—24no—
55C2×C22⋊Dic83—42no—
56C2×C22⋊Dic83C2×C418yes—
57C2×C4×Dic83—24no—
58C2×C4×Dic83—24no—
59C2×C4×Dic83—42no—
60C83⋊#95—24no—
61C83⋊#95—24no—
62C83⋊#95—24no—
63C83⋊#95—42no—
64C83⋊#95—41no—
65C83⋊#95—42no—
66C83⋊#95C2×C414yescomplemented
67C83⋊#95C2×C418yescomplemented
68C83⋊#95D418yes—
69C25×C83C2311yescharacteristic
70C23×C4×C83—22no—
71C23×C4×C83—24no—
72C23×C4×C83—42no—
73C23×C4×C83—41no—
74C23×C4×C83—41no—
75C23×C4×C83C2311yescharacteristic
76C23×C4×C83C2311yescharacteristic
77C23×C4×C83C2312yes—
78C23×C4×C83C2312yes—
79C2×C22⋊C4×C83—41no—
80C2×C22⋊C4×C83—41no—
81C2×C22⋊C4×C83—41no—
82C2×C22⋊C4×C83—41no—
83C2×C22⋊C4×C83C2×C414yes—
84C2×C22⋊C4×C83Q814yescomplemented
85#95×C83—24no—
86#95×C83—42no—
87C2×Dic166—216no—
88C2×Dic166—44no—
89C2×Dic166—44no—
90C2×Dic166—48no—
91C2×Dic166—44no—
92C22×Dic83—22no—
93C22×Dic83—22no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83C22×C414yes—
116C22×Dic83C22×C414yes—
117C22×Dic83C2×D414yes—
118C22×Dic83C2×D414yes—
119C22×Dic83#4114yes—
120C4⋊Dic83—44no—
121Dic83⋊C4—48no—
122Dic83⋊C4—48no—
123C22⋊Dic83—216no—
124C22⋊Dic83—48no—
125C4×Dic83—48no—
126C24×C83—22no—
127C24×C83—22no—
128C24×C83—21no—
129C24×C83—21no—
130C24×C83—21no—
131C24×C83—22no—
132C24×C83—22no—
133C24×C83—21no—
134C24×C83C22×C412yes—
135C24×C83C2×Q812yes—
136C24×C83#4112yes—
137C24×C83C2411yescharacteristic
138C22×C4×C83—22no—
139C22×C4×C83—22no—
140C22×C4×C83—22no—
141C22×C4×C83—22no—
142C22×C4×C83—22no—
143C22×C4×C83—22no—
144C22×C4×C83—21no—
145C22×C4×C83—21no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—21no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—21no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—44no—
159C22×C4×C83—48no—
160C22×C4×C83—48no—
161C22×C4×C83—44no—
162C22×C4×C83—44no—
163C22×C4×C83C22×C412yes—
164C22×C4×C83C22×C412yes—
165C22×C4×C83C2×D414yes—
166C22×C4×C83C2×Q814yes—
167C22⋊C4×C83—44no—
168C22⋊C4×C83—44no—
169C23×C83—24no—
170C23×C83—22no—
171C23×C83—24no—
172C23×C83—22no—
173C23×C83—22no—
174C23×C83—22no—
175C23×C83—22no—
176C23×C83—22no—
177C23×C83—24no—
178C23×C83—24no—
179C23×C83—24no—
180C23×C83—24no—
181C23×C83—42no—
182C23×C83—42no—
183C23×C83—42no—
184C23×C83—42no—
185C23×C83—41no—
186C23×C83—42no—
187C23×C83—41no—
188C23×C83—41no—
189C23×C83—42no—
190C23×C83—41no—
191C23×C83C4×Q814yes—
192C23×C83C23×C411yescharacteristic
193C23×C83C22×D412yes—
194C23×C83C22×Q811yescharacteristic
195C23×C83C2×#4112yes—
196C23×C83C2×#4111yescharacteristic
197C23×C83H(5,𝔽2)11yescharacteristic
198C23×C83H(5,𝔽2)11yescharacteristic
199C23×C83H(5,𝔽2)11yescharacteristic
200C23×C83H(5,𝔽2)12yes—
201C23×C832−1+412yes—
202C23×C832−1+411yescharacteristic
203C2×C4×C83—24no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—24no—
209C2×C4×C83—24no—
210C2×C4×C83—24no—
211C2×C4×C83—24no—
212C2×C4×C83—24no—
213C2×C4×C83—416no—
214C2×C4×C83—42no—
215C2×C4×C83—42no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—41no—
219C2×C4×C83—42no—
220C2×C4×C83—42no—
221C2×C4×C83—42no—
222C2×C4×C83—41no—
223C2×C4×C83—42no—
224C2×C4×C83—41no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83—41no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—24no—
231C2×Dic83—24no—
232C2×Dic83—24no—
233C2×Dic83—24no—
234C2×Dic83—24no—
235C2×Dic83—24no—
236C2×Dic83—24no—
237C2×Dic83—24no—
238C2×Dic83—24no—
239C2×Dic83—24no—
240C2×Dic83—42no—
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249C2×Dic83—42no—
250C2×Dic83—42no—
251C2×Dic83—42no—
252C2×Dic83—42no—
253C2×Dic83—42no—
254C2×Dic83—42no—
255C2×Dic83C4×D418yes—
256Dic166—416no—
257C22×C83—24no—
258C22×C83—28no—
259C22×C83—44no—
260C22×C83—42no—
261C22×C83—42no—
262C22×C83—42no—
263C22×C83—42no—
264C22×C83—44no—
265C22×C83—44no—
266C22×C83—42no—
267C22×C83—42no—
268C22×C83C2×C4×D412yes—
269C22×C83C2×C4×Q811yescharacteristic
270C22×C83#51811yescharacteristic
271C22×C83#52012yes—
272C22×C83#52011yescharacteristic
273C22×C83#53512yes—
274C22×C83#53612yes—
275C22×C83#53811yescharacteristic
276C22×C83#53911yescharacteristic
277C22×C83#54212yes—
278C22×C83#54212yes—
279C22×C83#54311yescharacteristic
280C22×C83#54411yescharacteristic
281C22×C83D4211yescharacteristic
282C22×C83#54612yes—
283C22×C83#54612yes—
284C22×C83D4×Q812yes—
285C22×C83#55111yescharacteristic
286C22×C83#55212yes—
287C22×C83#55412yes—
288C22×C83H(3,𝔽4)11yescharacteristic
289C22×C83#56212yes—
290C22×C83#56311yescharacteristic
291C4×C83—42no—
292C4×C83—42no—
293C4×C83—44no—
294C4×C83—44no—
295Dic83—28no—
296Dic83—44no—
297Dic83—44no—
298Dic83—82no—
299#14575—831nomaximal, Sylow
300C2×C83—44no—
301C2×C83#232612yes—
302C2×C83#232911yescharacteristic
303C2×C83#235811yescharacteristic
304C2×C83#236012yes—
305C2×C83#237911yescharacteristic
306C2×C83#267011yescharacteristic
307C2×C83#267711yescharacteristic
308C2×C83#268712yes—
309C2×C83#269212yes—
310C2×C83#276311yescharacteristic
311C2×C83#276411yescharacteristic
312C2×#393—831no—
313C2×#391—832no—
314C2×#385—832no—
315#1444—832no—
316#1442—832no—
317#1439—831no—
318#1437—832no—
319#1434—832no—
320C22⋊#95D8311yescharacteristic
321C83#1457511yescharacteristic, complemented, Sylow
322C23×Q8—831no—
323C24×C4C2×D8311yescharacteristic
324C22×C4⋊C4—831no—
325C22×C4⋊C4—832no—
326C22×C4⋊C4—832no—
327C22×C22⋊C4—832no—
328C22×C22⋊C4—832no—
329C22×C22⋊C4C2×D8311yescharacteristic
330C22×C22⋊C4C2×D8311yescharacteristic
331C22×C42—832no—
332#393—838no—
333#391—1668no—
334#385—1668no—
335C2×#95—831no—
336C2×#95—832no—
337C2×#95—832no—
338C2×#95—832no—
339C2×#95—831no—
340C2×#95—832no—
341C2×#95—832no—
342C2×#95—832no—
343C2×#95—832no—
344C2×#95C2×D8312yes—
345C2×#95C2×D8312yes—
346C25C22×D8311yescharacteristic
347C22×Q8—3321no—
348C22×Q8—3321no—
349C22×Q8—834no—
350C22×Q8—1664no—
351C22×Q8—1664no—
352C23×C4—22no—
353C23×C4—24no—
354C23×C4—42no—
355C23×C4—41no—
356C23×C4—41no—
357C23×C4—832no—
358C23×C4—832no—
359C23×C4—832no—
360C23×C4—832no—
361C23×C4C22×D8311yescharacteristic
362C23×C4C22×D8311yescharacteristic
363C23×C4C22×D8312yes—
364C23×C4C22×D8312yes—
365C2×C4⋊C4—3322no—
366C2×C4⋊C4—3322no—
367C2×C4⋊C4—3322no—
368C2×C4⋊C4—3322no—
369C2×C4⋊C4—3321no—
370C2×C4⋊C4—1664no—
371C2×C4⋊C4—1664no—
372C2×C4⋊C4—1664no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—3322no—
378C2×C22⋊C4—838no—
379C2×C22⋊C4—1664no—
380C2×C22⋊C4—1664no—
381C2×C22⋊C4—1664no—
382C2×C22⋊C4—1664no—
383C2×C22⋊C4Dic16614yescomplemented
384C2×C22⋊C4C4×D8314yes—
385C2×C42—3322no—
386C2×C42—1664no—
387C2×C42—1664no—
388#95—24no—
389#95—42no—
390#95—3322no—
391#95—3322no—
392#95—3321no—
393#95—838no—
394#95—834no—
395#95—838no—
396#95—1664no—
397#95—1664no—
398#95—1664no—
399C24—22no—
400C24—22no—
401C24—21no—
402C24—21no—
403C24—21no—
404C24—22no—
405C24—22no—
406C24—21no—
407C24C2×C4×D8312yes—
408C24C2×Dic16612yes—
409C24#1305059012yes—
410C24C23×D8311yescharacteristic
411C2×Q8—3324no—
412C2×Q8—3324no—
413C2×Q8—3328no—
414C2×Q8—3324no—
415C2×Q8—16616no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—834no—
442C22×C4—834no—
443C22×C4—834no—
444C22×C4—834no—
445C22×C4—834no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4—1662no—
449C22×C4—1662no—
450C22×C4—1662no—
451C22×C4—1662no—
452C22×C4—1662no—
453C22×C4—1662no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4—1662no—
458C22×C4—1662no—
459C22×C4—1662no—
460C22×C4—1662no—
461C22×C4—1662no—
462C22×C4—1662no—
463C22×C4—1662no—
464C22×C4—1662no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4C2×C4×D8312yes—
470C22×C4C2×C4×D8312yes—
471C22×C4D4×D8314yes—
472C22×C4C2×Dic16614yes—
473C4⋊C4—3328no—
474C4⋊C4—3328no—
475C4⋊C4—3324no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—3328no—
479C22⋊C4—16616no—
480C42—3328no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23C4×Dic16614yes—
504C23C22×C4×D8311yescharacteristic
505C23C2×D4×D8312yes—
506C23C22×Dic16611yescharacteristic
507C23#41×D8312yes—
508C23C2×#1305059011yescharacteristic
509C23#1458183712yes—
510C23#1458183711yescharacteristic
511C23#1458183711yescharacteristic
512C23#1458183711yescharacteristic
513C23#1458183912yes—
514C23#1458184011yescharacteristic
515Q8—33216no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—416no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—3322no—
542C2×C4—3322no—
543C2×C4—3322no—
544C2×C4—3322no—
545C2×C4—3322no—
546C2×C4—3322no—
547C2×C4—3322no—
548C2×C4—3322no—
549C2×C4—3322no—
550C2×C4—3322no—
551C2×C4—3322no—
552C2×C4—3322no—
553C2×C4—3322no—
554C2×C4—3322no—
555C2×C4—3322no—
556C2×C4—838no—
557C2×C4—1664no—
558C2×C4—1664no—
559C2×C4—1664no—
560C2×C4—1664no—
561C2×C4—1664no—
562C2×C4—1664no—
563C2×C4—1664no—
564C2×C4—1664no—
565C2×C4—1664no—
566C2×C4—1664no—
567C2×C4—1664no—
568C2×C4—1664no—
569C22—24no—
570C22—28no—
571C22—42no—
572C22—44no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22C4×D4×D8312yes—
581C22C2×C4×Dic16611yescharacteristic
582C22#1912817411yescharacteristic
583C22#1912818012yes—
584C22#1912818211yescharacteristic
585C22#1912826612yes—
586C22#1912827412yes—
587C22#1912828911yescharacteristic
588C22#1912829811yescharacteristic
589C22#1912831912yes—
590C22#1912832112yes—
591C22#1912832711yescharacteristic
592C22#1912833211yescharacteristic
593C22#1912833311yescharacteristic
594C22#1912834112yes—
595C22#1912834812yes—
596C22D4×Dic16612yes—
597C22#1912837911yescharacteristic
598C22#1912838412yes—
599C22#1912841012yes—
600C22#1912844411yescharacteristic
601C22#1912844612yes—
602C22#1912845411yescharacteristic
603C4—44no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—3324no—
608C4—3324no—
609C4—6642no—
610C4—1668no—
611C2—44no—
612C2#2491016912yes—
613C2#2491019611yescharacteristic
614C2#2491044611yescharacteristic
615C2#2491046912yes—
616C2#2491063311yescharacteristic
617C2#2491368911yescharacteristic
618C2#2491377811yescharacteristic
619C2#2491388512yes—
620C2#2491396012yes—
621C2#2491487711yescharacteristic
622C2#2491488611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×C22⋊#95C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC22⋊#95D83
hypocenterC83#14575
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#19127605C2semidirect
4. #24905727C2semidirect
5. #24905721C2semidirect
8. #24905689C2semidirect
17. #19127613C4semidirect
38. C22×Dic166C2×C4semidirect
66. C83⋊#95C2×C4semidirect
67. C83⋊#95C2×C4semidirect
84. C2×C22⋊C4×C83Q8semidirect
321. C83#14575semidirect
383. C2×C22⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191276132
C2×#191276052
C2×#191275742
#249057272
#249057212
#249057082
#249057002
#249056892
C83×C22⋊#952
#1457583