Finite Group Databasebeta

← #35334773

Subgroups — #35334773

Order 21248.

1506 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24905734C212yesmaximal, complemented
2#24905725C212yesmaximal, complemented
3#24905708C211yesmaximal, characteristic
4#24905700C212yesmaximal
5#24905697C212yesmaximal, complemented
6#24905688C212yesmaximal, complemented
7#24905685C212yesmaximal
8#24905685C211yesmaximal, characteristic
9C2×#393×C83C211yesmaximal, characteristic
10C24×Dic83C2211yescharacteristic
11C22×C4⋊Dic83C2211yescharacteristic
12C22×C22⋊Dic83C2212yes—
13C22×C22⋊Dic83C2211yescharacteristic
14C22×C4×Dic83C2212yes—
15C2×C83⋊#95C2212yes—
16C2×C83⋊#95C2212yes—
17C2×C83⋊#95C2212yes—
18C2×C83⋊#95C2212yes—
19C2×C83⋊#95C2212yes—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2211yescharacteristic
26C23×Q8×C83C2211yescharacteristic
27C22×C22⋊C4×C83C2212yes—
28C22×C22⋊C4×C83C2211yescharacteristic
29#393×C83C418yescomplemented
30C2×#95×C83C2211yescharacteristic
31C2×#95×C83C2212yes—
32C23×Dic83—24no—
33C23×Dic83—22no—
34C23×Dic83—41no—
35C23×Dic83—42no—
36C23×Dic83—41no—
37C23×Dic83C2312yes—
38C23×Dic83C2312yes—
39C23×Dic83C2312yes—
40C23×Dic83C2312yes—
41C2×C4⋊Dic83—24no—
42C2×C4⋊Dic83—41no—
43C2×C22⋊Dic83—24no—
44C2×C22⋊Dic83—24no—
45C2×C22⋊Dic83—41no—
46C2×C22⋊Dic83—41no—
47C2×C22⋊Dic83—42no—
48C2×C22⋊Dic83Q814yescomplemented
49C2×C4×Dic83—24no—
50C2×C4×Dic83—42no—
51C2×C4×Dic83—42no—
52C83⋊#95—24no—
53C83⋊#95—24no—
54C83⋊#95—42no—
55C83⋊#95—41no—
56C83⋊#95—42no—
57C83⋊#95—42no—
58C83⋊#95—42no—
59C83⋊#95—42no—
60C83⋊#95—42no—
61C83⋊#95—42no—
62C83⋊#95—42no—
63C25×C83C2311yescharacteristic
64C22×Q8×C83—42no—
65C22×Q8×C83—41no—
66C22×Q8×C83—42no—
67C22×Q8×C83—41no—
68C22×Q8×C83C2×C414yescomplemented
69C23×C4×C83C2312yes—
70C23×C4×C83C2311yescharacteristic
71C23×C4×C83C2312yes—
72C23×C4×C83C2311yescharacteristic
73C2×C22⋊C4×C83—42no—
74C2×C22⋊C4×C83—41no—
75C2×C22⋊C4×C83—42no—
76C2×C22⋊C4×C83—41no—
77C2×C22⋊C4×C83C2×C418yes—
78C2×C22⋊C4×C83C2×C414yes—
79#95×C83C2×C414yescomplemented
80#95×C83C2×C418yescomplemented
81C22×Dic83—22no—
82C22×Dic83—22no—
83C22×Dic83—22no—
84C22×Dic83—22no—
85C22×Dic83—22no—
86C22×Dic83—22no—
87C22×Dic83—22no—
88C22×Dic83—22no—
89C22×Dic83—22no—
90C22×Dic83—22no—
91C22×Dic83—22no—
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—44no—
107C22×Dic83—48no—
108C22×Dic83—48no—
109C22×Dic83—44no—
110C22×Dic83—44no—
111C22×Dic83C2×Q814yes—
112C22×Dic83#4114yes—
113C22×Dic83#4114yes—
114C4⋊Dic83—44no—
115C22⋊Dic83—44no—
116C22⋊Dic83—48no—
117C4×Dic83—48no—
118C24×C83—22no—
119C24×C83—22no—
120C24×C83—21no—
121C24×C83—22no—
122C24×C83—21no—
123C24×C83—21no—
124C24×C83—22no—
125C24×C83—21no—
126C24×C83C22×C412yes—
127C24×C83C2×Q812yes—
128C24×C83#4112yes—
129C24×C83C2411yescharacteristic
130C2×Q8×C83—44no—
131C2×Q8×C83—48no—
132C2×Q8×C83—48no—
133C2×Q8×C83—44no—
134C2×Q8×C83—44no—
135C22×C4×C83—22no—
136C22×C4×C83—22no—
137C22×C4×C83—22no—
138C22×C4×C83—22no—
139C22×C4×C83—22no—
140C22×C4×C83—21no—
141C22×C4×C83—22no—
142C22×C4×C83—21no—
143C22×C4×C83—22no—
144C22×C4×C83—21no—
145C22×C4×C83—22no—
146C22×C4×C83—21no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83C22×C412yes—
156C22×C4×C83C22×C414yes—
157C22×C4×C83C22×C412yes—
158C22×C4×C83C22×C414yes—
159C22⋊C4×C83—48no—
160C22⋊C4×C83—44no—
161C23×C83—24no—
162C23×C83—22no—
163C23×C83—24no—
164C23×C83—22no—
165C23×C83—22no—
166C23×C83—22no—
167C23×C83—22no—
168C23×C83—22no—
169C23×C83—24no—
170C23×C83—24no—
171C23×C83—24no—
172C23×C83—24no—
173C23×C83—42no—
174C23×C83—42no—
175C23×C83—42no—
176C23×C83—42no—
177C23×C83—41no—
178C23×C83—41no—
179C23×C83—42no—
180C23×C83—42no—
181C23×C83—41no—
182C23×C83—41no—
183C23×C83C4×Q814yes—
184C23×C83C23×C411yescharacteristic
185C23×C83C22×Q811yescharacteristic
186C23×C83C2×#4112yes—
187C23×C83C2×#4112yes—
188C23×C83C2×#4111yescharacteristic
189C23×C83H(5,𝔽2)12yes—
190C23×C83H(5,𝔽2)12yes—
191C23×C83H(5,𝔽2)11yescharacteristic
192C23×C83H(5,𝔽2)11yescharacteristic
193C23×C83H(5,𝔽2)11yescharacteristic
194C23×C832−1+411yescharacteristic
195Q8×C83—416no—
196C2×C4×C83—24no—
197C2×C4×C83—24no—
198C2×C4×C83—24no—
199C2×C4×C83—22no—
200C2×C4×C83—22no—
201C2×C4×C83—24no—
202C2×C4×C83—24no—
203C2×C4×C83—24no—
204C2×C4×C83—24no—
205C2×C4×C83—24no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—42no—
213C2×C4×C83—42no—
214C2×C4×C83—42no—
215C2×C4×C83—42no—
216C2×C4×C83—42no—
217C2×C4×C83—41no—
218C2×C4×C83—42no—
219C2×C4×C83—41no—
220C2×Dic83—24no—
221C2×Dic83—24no—
222C2×Dic83—24no—
223C2×Dic83—24no—
224C2×Dic83—24no—
225C2×Dic83—24no—
226C2×Dic83—24no—
227C2×Dic83—24no—
228C2×Dic83—24no—
229C2×Dic83—416no—
230C2×Dic83—42no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—42no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83—42no—
237C2×Dic83—42no—
238C2×Dic83—42no—
239C2×Dic83—42no—
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—
249C22×C83—24no—
250C22×C83—28no—
251C22×C83—44no—
252C22×C83—42no—
253C22×C83—42no—
254C22×C83—42no—
255C22×C83—44no—
256C22×C83—42no—
257C22×C83—42no—
258C22×C83—42no—
259C22×C83—44no—
260C22×C83C2×C4×Q811yescharacteristic
261C22×C83C4×#4112yes—
262C22×C83#51811yescharacteristic
263C22×C83#51812yes—
264C22×C83#52011yescharacteristic
265C22×C83#53812yes—
266C22×C83#53812yes—
267C22×C83#53811yescharacteristic
268C22×C83#53911yescharacteristic
269C22×C83#54212yes—
270C22×C83#54212yes—
271C22×C83#54311yescharacteristic
272C22×C83#54411yescharacteristic
273C22×C83#55112yes—
274C22×C83#55112yes—
275C22×C83#55111yescharacteristic
276C22×C83#55111yescharacteristic
277C22×C83#55412yes—
278C22×C83#55412yes—
279C22×C83#55512yes—
280C22×C83#56012yes—
281C22×C83#56011yescharacteristic
282C22×C83H(3,𝔽4)11yescharacteristic
283C4×C83—44no—
284C4×C83—42no—
285C4×C83—42no—
286C4×C83—44no—
287Dic83—44no—
288Dic83—44no—
289Dic83—44no—
290Dic83—82no—
291#14596—831nomaximal, Sylow
292C2×C83—44no—
293C2×C83#232911yescharacteristic
294C2×C83#233712yes—
295C2×C83#236812yes—
296C2×C83#237211yescharacteristic
297C2×C83#237511yescharacteristic
298C2×C83#273611yescharacteristic
299C2×C83#273712yes—
300C2×C83#274211yescharacteristic
301C2×C83#276011yescharacteristic
302C2×C83#276311yescharacteristic
303C2×C83#278412yes—
304C2×#393D8311yescharacteristic
305#1446—832no—
306#1443—832no—
307#1439—831no—
308#1437—832no—
309#1436—832no—
310#1434—832no—
311C22⋊#95—831no—
312C22⋊#95—832no—
313C83#1459611yescharacteristic, complemented, Sylow
314C23×Q8C2×D8311yescharacteristic
315C24×C4—831no—
316C22×C4⋊C4—831no—
317C22×C22⋊C4—831no—
318C22×C22⋊C4—832no—
319C22×C22⋊C4C2×D8312yes—
320C22×C22⋊C4C2×D8311yescharacteristic
321C22×C42—832no—
322#393Dic8318yescomplemented
323C2×#95—832no—
324C2×#95—832no—
325C2×#95—832no—
326C2×#95—832no—
327C2×#95—832no—
328C2×#95—832no—
329C2×#95—832no—
330C2×#95—832no—
331C2×#95—831no—
332C2×#95—832no—
333C2×#95—832no—
334C2×#95C2×D8311yescharacteristic
335C2×#95C2×D8312yes—
336C25C22×D8311yescharacteristic
337C22×Q8—41no—
338C22×Q8—42no—
339C22×Q8—41no—
340C22×Q8—42no—
341C22×Q8C2×Dic8314yescomplemented
342C23×C4—3321no—
343C23×C4—3321no—
344C23×C4—3322no—
345C23×C4—832no—
346C23×C4—832no—
347C23×C4—832no—
348C23×C4—832no—
349C23×C4—1664no—
350C23×C4—1662no—
351C23×C4C22×D8311yescharacteristic
352C23×C4C22×D8312yes—
353C23×C4C22×D8311yescharacteristic
354C23×C4C22×D8312yes—
355C2×C4⋊C4—3321no—
356C2×C4⋊C4—1664no—
357C2×C22⋊C4—41no—
358C2×C22⋊C4—42no—
359C2×C22⋊C4—42no—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—3322no—
362C2×C22⋊C4—3321no—
363C2×C22⋊C4—3321no—
364C2×C22⋊C4—834no—
365C2×C22⋊C4—1664no—
366C2×C22⋊C4—1664no—
367C2×C22⋊C4C2×Dic8318yes—
368C2×C22⋊C4C2×Dic8314yes—
369C2×C42—3322no—
370C2×C42—3322no—
371C2×C42—1664no—
372#95—3321no—
373#95—3322no—
374#95—3322no—
375#95—3322no—
376#95—3322no—
377#95—3322no—
378#95—3322no—
379#95—3322no—
380#95—3322no—
381#95—1664no—
382#95—1664no—
383#95C2×Dic8314yescomplemented
384#95C2×Dic8318yescomplemented
385C24—22no—
386C24—22no—
387C24—21no—
388C24—22no—
389C24—21no—
390C24—22no—
391C24—21no—
392C24—21no—
393C24C22×Dic8312yes—
394C24Q8×D8312yes—
395C24#1305059112yes—
396C24C23×D8311yescharacteristic
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—44no—
401C2×Q8—44no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—3324no—
423C22×C4—3324no—
424C22×C4—3328no—
425C22×C4—3328no—
426C22×C4—3324no—
427C22×C4—834no—
428C22×C4—834no—
429C22×C4—834no—
430C22×C4—1662no—
431C22×C4—1662no—
432C22×C4—1662no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4—1662no—
442C22×C4—1662no—
443C22×C4—1662no—
444C22×C4—1662no—
445C22×C4—1662no—
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×C4C22×Dic8312yes—
456C22×C4C22×Dic8314yes—
457C22×C4C22×Dic8312yes—
458C22×C4C22×Dic8314yes—
459C4⋊C4—3324no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—3328no—
463C22⋊C4—3324no—
464C42—3328no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23Q8×Dic8314yes—
488C23C23×Dic8311yescharacteristic
489C23C2×Q8×D8311yescharacteristic
490C23#41×D8312yes—
491C23#41×D8312yes—
492C23C2×#1305059111yescharacteristic
493C23#1458183711yescharacteristic
494C23#1458183711yescharacteristic
495C23#1458183712yes—
496C23#1458183712yes—
497C23#1458183811yescharacteristic
498C23#1458183911yescharacteristic
499Q8—416no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—33216no—
525C2×C4—3322no—
526C2×C4—3322no—
527C2×C4—3322no—
528C2×C4—3322no—
529C2×C4—3322no—
530C2×C4—3322no—
531C2×C4—3322no—
532C2×C4—3322no—
533C2×C4—3322no—
534C2×C4—3322no—
535C2×C4—3322no—
536C2×C4—3322no—
537C2×C4—3322no—
538C2×C4—3322no—
539C2×C4—3322no—
540C2×C4—3322no—
541C2×C4—3322no—
542C2×C4—3322no—
543C2×C4—3322no—
544C2×C4—1664no—
545C2×C4—1664no—
546C2×C4—1664no—
547C2×C4—1664no—
548C2×C4—1664no—
549C2×C4—1664no—
550C2×C4—1664no—
551C2×C4—1664no—
552C2×C4—1664no—
553C22—24no—
554C22—28no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22C2×Q8×Dic8311yescharacteristic
565C22#41×Dic8312yes—
566C22#1912817611yescharacteristic
567C22#1912817612yes—
568C22#1912818411yescharacteristic
569C22#1912828512yes—
570C22#1912828512yes—
571C22#1912829111yescharacteristic
572C22#1912830011yescharacteristic
573C22#1912831512yes—
574C22#1912831512yes—
575C22#1912832611yescharacteristic
576C22#1912833011yescharacteristic
577C22#1912837812yes—
578C22#1912837812yes—
579C22#1912838111yescharacteristic
580C22#1912838111yescharacteristic
581C22#1912840512yes—
582C22#1912840512yes—
583C22#1912841312yes—
584C22#1912844012yes—
585C22#1912844011yescharacteristic
586C22#1912844411yescharacteristic
587C4—42no—
588C4—44no—
589C4—44no—
590C4—42no—
591C4—3324no—
592C4—3324no—
593C4—3324no—
594C4—6642no—
595C2—44no—
596C2#2491019811yescharacteristic
597C2#2491027812yes—
598C2#2491054812yes—
599C2#2491057511yescharacteristic
600C2#2491060411yescharacteristic
601C2#2491450711yescharacteristic
602C2#2491452412yes—
603C2#2491458611yescharacteristic
604C2#2491483811yescharacteristic
605C2#2491488111yescharacteristic
606C2#2491515812yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#393×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#393D83
hypocenterC83#14596
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. #24905734C2semidirect
2. #24905725C2semidirect
5. #24905697C2semidirect
6. #24905688C2semidirect
29. #393×C83C4semidirect
48. C2×C22⋊Dic83Q8semidirect
68. C22×Q8×C83C2×C4semidirect
79. #95×C83C2×C4semidirect
80. #95×C83C2×C4semidirect
313. C83#14596semidirect
322. #393Dic83semidirect
341. C22×Q8C2×Dic83semidirect
383. #95C2×Dic83semidirect
384. #95C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#249057342
#249057252
#249057082
#249057002
#249056972
#249056882
#249056852
C2×#393×C832
#1459683