Finite Group Databasebeta

← #35326002

Subgroups — #35326002

Order 21248.

2022 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127595C212yesmaximal
2C2×#19127573C211yesmaximal, characteristic
3C2×C4×Dic83⋊C4C212yesmaximal, complemented
4C2×C22⋊C4×Dic83C212yesmaximal
5#24905700C211yesmaximal, characteristic
6#24905698C212yesmaximal, complemented
7#24905694C212yesmaximal, complemented
8#95×Dic83C212yesmaximal, complemented
9C83×#1437C211yesmaximal, characteristic
10C24×Dic83C2211yescharacteristic
11C22×Dic83⋊C4C2212yes—
12C22×Dic83⋊C4C2212yes—
13C22×C22⋊Dic83C2212yes—
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C4×Dic83C2211yescharacteristic
16C22×C4×Dic83C2212yes—
17C22×C4×Dic83C2212yes—
18C22×C4×Dic83C2212yes—
19#19127595C4116yescomplemented
20#19127573C418yescomplemented
21C4×Dic83⋊C4—28no—
22C22⋊C4×Dic83—28no—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C22×C22⋊C4×C83C2212yes—
31C22×C22⋊C4×C83C2211yescharacteristic
32C22×C42×C83C2211yescharacteristic
33C2×#95×C83C2212yes—
34C2×#95×C83C2211yescharacteristic
35C23×Dic83—24no—
36C23×Dic83—24no—
37C23×Dic83—41no—
38C23×Dic83—41no—
39C23×Dic83C2×C414yes—
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43C23×Dic83C2312yes—
44C2×Dic83⋊C4—24no—
45C2×Dic83⋊C4—24no—
46C2×Dic83⋊C4—24no—
47C2×Dic83⋊C4—24no—
48C2×Dic83⋊C4C2×C418yes—
49C2×Dic83⋊C4C2×C418yescomplemented
50C2×C22⋊Dic83—24no—
51C2×C22⋊Dic83—24no—
52C2×C22⋊Dic83C2×C418yes—
53C2×C22⋊Dic83C2×C414yes—
54C2×C22⋊Dic83C2×C418yes—
55C2×C4×Dic83—24no—
56C2×C4×Dic83—24no—
57C2×C4×Dic83—24no—
58C2×C4×Dic83—24no—
59C2×C4×Dic83—24no—
60C2×C4×Dic83—24no—
61C2×C4×Dic83—24no—
62C2×C4×Dic83—42no—
63C2×C4×Dic83—42no—
64C2×C4×Dic83—41no—
65C2×C4×Dic83C2×C418yescomplemented
66C83⋊#95—24no—
67C83⋊#95—24no—
68C83⋊#95—41no—
69C83⋊#95C2×C418yes—
70C83⋊#95C2×C418yescomplemented
71C83⋊#95C2×C418yescomplemented
72C83⋊#95D418yes—
73C25×C83C2311yescharacteristic
74C23×C4×C83C2311yescharacteristic
75C23×C4×C83C2312yes—
76C23×C4×C83C2311yescharacteristic
77C23×C4×C83C2312yes—
78C2×C22⋊C4×C83—24no—
79C2×C22⋊C4×C83—24no—
80C2×C22⋊C4×C83—41no—
81C2×C22⋊C4×C83—41no—
82C2×C22⋊C4×C83C2×C418yes—
83C2×C22⋊C4×C83C2×C414yes—
84C2×C42×C83—24no—
85C2×C42×C83—41no—
86#95×C83—24no—
87#95×C83—41no—
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—216no—
107C22×Dic83—44no—
108C22×Dic83—44no—
109C22×Dic83—48no—
110C22×Dic83—44no—
111C22×Dic83C22×C414yes—
112C22×Dic83C22×C414yes—
113C22×Dic83C22×C414yes—
114C22×Dic83C22×C414yes—
115C22×Dic83C22×C414yes—
116C22×Dic83C22×C414yes—
117C22×Dic83C2×D414yes—
118C22×Dic83C2×D414yes—
119C22×Dic83#4114yes—
120C22×Dic83#4114yes—
121Dic83⋊C4—216no—
122Dic83⋊C4—216no—
123C22⋊Dic83—216no—
124C22⋊Dic83C42116yescomplemented
125C4×Dic83—216no—
126C4×Dic83—48no—
127C4×Dic83—44no—
128C4×Dic83—48no—
129C24×C83—22no—
130C24×C83—22no—
131C24×C83—22no—
132C24×C83—22no—
133C24×C83—21no—
134C24×C83—21no—
135C24×C83—22no—
136C24×C83C22×C412yes—
137C24×C83C22×C414yes—
138C24×C83C2411yescharacteristic
139C22×C4×C83—22no—
140C22×C4×C83—22no—
141C22×C4×C83—22no—
142C22×C4×C83—22no—
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—21no—
152C22×C4×C83—22no—
153C22×C4×C83—21no—
154C22×C4×C83—21no—
155C22×C4×C83—21no—
156C22×C4×C83—22no—
157C22×C4×C83C22×C412yes—
158C22×C4×C83C22×C414yes—
159C22×C4×C83C22×C412yes—
160C22×C4×C83C22×C414yes—
161C22×C4×C83C2×D414yes—
162C22×C4×C83#4114yes—
163C22⋊C4×C83—216no—
164C22⋊C4×C83—44no—
165C42×C83—44no—
166C23×C83—24no—
167C23×C83—24no—
168C23×C83—24no—
169C23×C83—24no—
170C23×C83—24no—
171C23×C83—24no—
172C23×C83—22no—
173C23×C83—22no—
174C23×C83—24no—
175C23×C83—24no—
176C23×C83—41no—
177C23×C83—42no—
178C23×C83—42no—
179C23×C83—42no—
180C23×C83—41no—
181C23×C83—41no—
182C23×C83—42no—
183C23×C83—42no—
184C23×C83—42no—
185C23×C83—41no—
186C23×C83C2×C4214yes—
187C23×C83C23×C411yescharacteristic
188C23×C83C23×C412yes—
189C23×C83C22×D412yes—
190C23×C83C2×#4112yes—
191C23×C83C2×#4112yes—
192C23×C83C2×#4112yes—
193C23×C83H(5,𝔽2)11yescharacteristic
194C23×C83H(5,𝔽2)11yescharacteristic
195C23×C832−1+412yes—
196C2×C4×C83—24no—
197C2×C4×C83—24no—
198C2×C4×C83—24no—
199C2×C4×C83—24no—
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—24no—
203C2×C4×C83—24no—
204C2×C4×C83—24no—
205C2×C4×C83—24no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—24no—
209C2×C4×C83—41no—
210C2×C4×C83—42no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—42no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—41no—
219C2×C4×C83C4×D418yes—
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—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—24no—
241C2×Dic83—416no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83C2×C4218yes—
247C2×Dic83C4×D418yes—
248C2×Dic83C4×D418yes—
249C22×C83—28no—
250C22×C83—24no—
251C22×C83—42no—
252C22×C83—44no—
253C22×C83—44no—
254C22×C83—44no—
255C22×C83—44no—
256C22×C83—44no—
257C22×C83—42no—
258C22×C83C22×C4211yescharacteristic
259C22×C83C2×C4×D412yes—
260C22×C83C2×C4×D412yes—
261C22×C83C2×C4×D412yes—
262C22×C83C4×#4112yes—
263C22×C83C4×#4112yes—
264C22×C83C4×#4112yes—
265C22×C83#51811yescharacteristic
266C22×C83#51911yescharacteristic
267C22×C83#52012yes—
268C22×C83#52012yes—
269C22×C83D4211yescharacteristic
270C22×C83#54612yes—
271C22×C83#54712yes—
272C22×C83#54812yes—
273C22×C83#55111yescharacteristic
274C22×C83#55111yescharacteristic
275C22×C83#55212yes—
276C22×C83#55312yes—
277C22×C83#55612yes—
278C22×C83#55911yescharacteristic
279C4×C83—28no—
280C4×C83—42no—
281C4×C83—44no—
282C4×C83—42no—
283Dic83—28no—
284Dic83—28no—
285Dic83—28no—
286Dic83—44no—
287#13823—831nomaximal, Sylow
288C2×C83—44no—
289C2×C83C42×D412yes—
290C2×C83#227411yescharacteristic
291C2×C83#235811yescharacteristic
292C2×C83#235912yes—
293C2×C83#236112yes—
294C2×C83#236212yes—
295C2×C83#236712yes—
296C2×C83#237312yes—
297C2×C83#237411yescharacteristic
298C2×#388—832no—
299C2×#385—831no—
300C2×C4×C4⋊C4—832no—
301C2×C4×C22⋊C4—832no—
302#1437—831no—
303#1437D8311yescharacteristic
304#1436—832no—
305#1436—832no—
306C4×#95—832no—
307C83#1382311yescharacteristic, complemented, Sylow
308C24×C4—831no—
309C22×C4⋊C4—832no—
310C22×C4⋊C4—832no—
311C22×C22⋊C4—831no—
312C22×C22⋊C4—832no—
313C22×C22⋊C4C2×D8312yes—
314C22×C22⋊C4C2×D8311yescharacteristic
315C22×C42—831no—
316C22×C42—832no—
317C22×C42—832no—
318C22×C42—832no—
319C22×C42C2×D8311yescharacteristic
320#388—8316no—
321#385—838no—
322C4×C4⋊C4—1668no—
323C4×C22⋊C4—1668no—
324C2×#95—831no—
325C2×#95—832no—
326C2×#95—832no—
327C2×#95—832no—
328C2×#95—832no—
329C2×#95—832no—
330C2×#95—832no—
331C2×#95C2×D8312yes—
332C2×#95C2×D8311yescharacteristic
333C25C22×D8311yescharacteristic
334C23×C4—3321no—
335C23×C4—3321no—
336C23×C4—834no—
337C23×C4—832no—
338C23×C4—832no—
339C23×C4—832no—
340C23×C4—832no—
341C23×C4—1664no—
342C23×C4—1664no—
343C23×C4C22×D8311yescharacteristic
344C23×C4C22×D8312yes—
345C23×C4C22×D8311yescharacteristic
346C23×C4C22×D8312yes—
347C2×C4⋊C4—838no—
348C2×C4⋊C4—838no—
349C2×C4⋊C4—1664no—
350C2×C4⋊C4—1664no—
351C2×C4⋊C4—1664no—
352C2×C4⋊C4—1664no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—41no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—834no—
358C2×C22⋊C4—838no—
359C2×C22⋊C4—838no—
360C2×C22⋊C4—1664no—
361C2×C22⋊C4—1664no—
362C2×C22⋊C4C4×D8318yes—
363C2×C22⋊C4C4×D8314yes—
364C2×C42—24no—
365C2×C42—41no—
366C2×C42—3322no—
367C2×C42—3322no—
368C2×C42—3321no—
369C2×C42—838no—
370C2×C42—1664no—
371C2×C42—1664no—
372C2×C42—1664no—
373C2×C42—1664no—
374C2×C42—1664no—
375C2×C42—1664no—
376C2×C42—1664no—
377#95—24no—
378#95—41no—
379#95—3321no—
380#95—838no—
381#95—838no—
382#95—838no—
383#95—838no—
384#95—1664no—
385#95—1664no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24—21no—
392C24—22no—
393C24C2×C4×D8314yes—
394C24C2×C4×D8312yes—
395C24C23×D8311yescharacteristic
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—3324no—
415C22×C4—3324no—
416C22×C4—3328no—
417C22×C4—3324no—
418C22×C4—834no—
419C22×C4—834no—
420C22×C4—834no—
421C22×C4—834no—
422C22×C4—834no—
423C22×C4—834no—
424C22×C4—834no—
425C22×C4—834no—
426C22×C4—834no—
427C22×C4—834no—
428C22×C4—16616no—
429C22×C4—1662no—
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×C4C2×C4×D8314yes—
448C22×C4C2×C4×D8312yes—
449C22×C4C2×C4×D8312yes—
450C22×C4C2×C4×D8314yes—
451C22×C4D4×D8314yes—
452C22×C4#1305058914yes—
453C4⋊C4—16616no—
454C4⋊C4—16616no—
455C22⋊C4—216no—
456C22⋊C4—44no—
457C22⋊C4—8316no—
458C22⋊C4—16616no—
459C42—44no—
460C42—3328no—
461C42—3328no—
462C42—3324no—
463C42—16616no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23C42×D8314yes—
485C23C22×C4×D8311yescharacteristic
486C23C22×C4×D8312yes—
487C23C2×D4×D8312yes—
488C23#41×D8312yes—
489C23#41×D8312yes—
490C23C2×#1305058912yes—
491C23#1458183711yescharacteristic
492C23#1458183711yescharacteristic
493C23#1458183912yes—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—3322no—
518C2×C4—3322no—
519C2×C4—3322no—
520C2×C4—3322no—
521C2×C4—33216no—
522C2×C4—838no—
523C2×C4—838no—
524C2×C4—838no—
525C2×C4—1664no—
526C2×C4—1664no—
527C2×C4—1664no—
528C2×C4—1664no—
529C2×C4—1664no—
530C2×C4—1664no—
531C2×C4—1664no—
532C2×C4—1664no—
533C2×C4—1664no—
534C2×C4—1664no—
535C2×C4—1664no—
536C2×C4—1664no—
537C2×C4—1664no—
538C2×C4—1664no—
539C2×C4—1664no—
540C2×C4—1664no—
541C2×C4—1664no—
542C2×C4—1664no—
543C2×C4—1664no—
544C2×C4—1664no—
545C2×C4—1664no—
546C2×C4(C22⋊D83)⋊C418yes—
547C22—28no—
548C22—24no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22C2×C42×D8311yescharacteristic
557C22C4×D4×D8312yes—
558C22C4×D4×D8312yes—
559C22C2×(C22⋊D83)⋊C412yes—
560C22C4×#1305058912yes—
561C22C4×#1305058912yes—
562C22#1912816912yes—
563C22#1912817411yescharacteristic
564C22#1912817811yescharacteristic
565C22#1912818012yes—
566C22#1912818012yes—
567C22#1912833311yescharacteristic
568C22#1912834812yes—
569C22#1912834912yes—
570C22#1912835712yes—
571C22#1912837911yescharacteristic
572C22#1912838111yescharacteristic
573C22#1912838712yes—
574C22#1912839412yes—
575C22#1912842012yes—
576C22#1912843711yescharacteristic
577C4—28no—
578C4—42no—
579C4—42no—
580C4—44no—
581C4—3324no—
582C4—1668no—
583C4—1668no—
584C4—1668no—
585C2—44no—
586C2C4×(C22⋊D83)⋊C412yes—
587C2#2490981811yescharacteristic
588C2#2491044611yescharacteristic
589C2#2491045212yes—
590C2#2491047412yes—
591C2#2491049312yes—
592C2#2491053312yes—
593C2#2491058112yes—
594C2#2491059811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC22×C83C22×C42
FittingC83×#1437C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenter#1437D83
hypocenterC83#13823
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic83⋊C4C2semidirect
6. #24905698C2semidirect
7. #24905694C2semidirect
8. #95×Dic83C2semidirect
19. #19127595C4semidirect
20. #19127573C4semidirect
49. C2×Dic83⋊C4C2×C4semidirect
65. C2×C4×Dic83C2×C4semidirect
70. C83⋊#95C2×C4semidirect
71. C83⋊#95C2×C4semidirect
124. C22⋊Dic83C42semidirect
307. C83#13823semidirect

Maximal subgroups

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

subgroupindex
C2×#191275952
C2×#191275732
C2×C4×Dic83⋊C42
C2×C22⋊C4×Dic832
#249057002
#249056982
#249056942
#95×Dic832
C83×#14372
#1382383