Finite Group Databasebeta

← #35404127

Subgroups — #35404127

Order 21248.

1486 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D83C211yesmaximal, characteristic
2C2×#19127664C211yesmaximal, characteristic
3C22×(C2×D83)⋊C8C211yesmaximal, characteristic
4#24907533C214yesmaximal, complemented
5#24907530C214yesmaximal, complemented
6#24907161C211yesmaximal, characteristic, complemented
7#24907160C211yesmaximal, characteristic, complemented
8#24907159C211yesmaximal, characteristic, complemented
9C83×#1729C211yesmaximal, characteristic, complemented
10C23×C4×D83C2211yescharacteristic
11C22×#13050574C2211yescharacteristic
12C23×C83⋊C8C2211yescharacteristic
13C2×#14581751C2211yescharacteristic
14C2×#14581750C411yescharacteristic
15C2×#14581750C411yescharacteristic
16C2×#14581749C2212yes—
17C2×#14581748C412yes—
18#117×D83—24no—
19#117×D83—24no—
20C2×C4⋊C4×D83C411yescharacteristic
21C2×C4⋊C4×D83C411yescharacteristic
22C2×C22⋊C4×D83C412yes—
23C2×C42×D83C2212yes—
24#19127664—24no—
25C2×(C2×D83)⋊C8—24no—
26C2×(C2×D83)⋊C8C2212yes—
27C2×(C2×D83)⋊C8C2212yes—
28C2×(C2×D83)⋊C8C2212yes—
29C2×(C2×D83)⋊C8C2212yes—
30#19127456C2214yescomplemented
31#19127455C2214yescomplemented
32#19127455C2214yescomplemented
33C22×M16×C83C2211yescharacteristic
34C23×C8×C83C2211yescharacteristic
35C2×#117×C83C2211yescharacteristic
36#336×C83C2214yescomplemented
37C24×D83C2×C411yescharacteristic
38C23×Dic83C2×C411yescharacteristic
39C22×C4×D83—22no—
40C22×C4×D83—22no—
41C22×C4×D83—22no—
42C22×C4×D83—22no—
43C22×C4×D83C2×C412yes—
44C22×C4×D83C2×C411yescharacteristic
45C22×C4×D83C2×C411yescharacteristic
46C22×C4×D83C2×C412yes—
47C22×C4×D83D414yes—
48C22×C4×D83D414yes—
49C22×C4×D83C2311yescharacteristic
50C22×C4×D83C2311yescharacteristic
51C2×#13050574—22no—
52C2×#13050574—22no—
53C2×#13050574—22no—
54C22×C83⋊C8—22no—
55C22×C83⋊C8—22no—
56C22×C83⋊C8—22no—
57C22×C83⋊C8C2312yes—
58C22×C83⋊C8C2312yes—
59#14581751—22no—
60#14581751—22no—
61#14581750—24no—
62#14581750—24no—
63#14581749—24no—
64#14581749—24no—
65#14581748—28no—
66C2×C4⋊Dic83C2×C411yescharacteristic
67C2×C4⋊Dic83C2×C411yescharacteristic
68C2×Dic83⋊C4C2×C412yes—
69C2×Dic83⋊C4C2×C412yes—
70C4⋊C4×D83—24no—
71C4⋊C4×D83—24no—
72C2×C22⋊Dic83C2×C412yes—
73C2×(C2×D83)⋊C4C2×C412yes—
74C2×(C2×D83)⋊C4C2×C412yes—
75C22⋊C4×D83—28no—
76C2×C4×Dic83C2×C412yes—
77C2×C4×Dic83C2×C412yes—
78C2×C4×Dic83C2312yes—
79C42×D83—24no—
80C42×D83—24no—
81(C2×D83)⋊C8—24no—
82(C2×D83)⋊C8—24no—
83(C2×D83)⋊C8—24no—
84(C2×D83)⋊C8—24no—
85C23×C4×C83C2311yescharacteristic
86C2×M16×C83—22no—
87C2×M16×C83—22no—
88C2×M16×C83—22no—
89C22×C8×C83—22no—
90C22×C8×C83—22no—
91C22×C8×C83—22no—
92C22×C8×C83C2312yes—
93C22×C8×C83C2312yes—
94#117×C83—22no—
95#117×C83—22no—
96C2×C4⋊C4×C83C2×C411yescharacteristic
97C2×C4⋊C4×C83C2×C411yescharacteristic
98C2×C22⋊C4×C83C2×C412yes—
99C2×C42×C83C2312yes—
100C23×D83—22no—
101C23×D83—22no—
102C23×D83—24no—
103C23×D83—42no—
104C23×D83C22⋊C414yes—
105C23×D83C22×C411yescharacteristic
106C23×D83C22×C411yescharacteristic
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83C22⋊C414yes—
110C22×Dic83C22×C412yes—
111C22×Dic83C22×C412yes—
112C22×Dic83C22×C411yescharacteristic
113C22×Dic83C22×C411yescharacteristic
114C2×C4×D83—22no—
115C2×C4×D83—22no—
116C2×C4×D83—22no—
117C2×C4×D83—22no—
118C2×C4×D83—28no—
119C2×C4×D83—28no—
120C2×C4×D83—28no—
121C2×C4×D83—28no—
122C2×C4×D83—21no—
123C2×C4×D83—21no—
124C2×C4×D83—21no—
125C2×C4×D83—21no—
126C2×C4×D83—21no—
127C2×C4×D83—21no—
128C2×C4×D83—21no—
129C2×C4×D83—21no—
130C2×C4×D83—24no—
131C2×C4×D83—24no—
132C2×C4×D83—28no—
133C2×C4×D83—44no—
134C2×C4×D83C22⋊C414yes—
135C2×C4×D83C22⋊C414yes—
136C2×C4×D83C2×D412yes—
137C2×C4×D83C2×D412yes—
138C2×C4×D83C2×D412yes—
139C2×C4×D83C2×D412yes—
140#13050574—44no—
141C2×C83⋊C8—22no—
142C2×C83⋊C8—22no—
143C2×C83⋊C8—22no—
144C2×C83⋊C8—22no—
145C2×C83⋊C8—22no—
146C2×C83⋊C8—22no—
147C2×C83⋊C8—44no—
148C4⋊Dic83—22no—
149C4⋊Dic83—22no—
150Dic83⋊C4—24no—
151Dic83⋊C4—24no—
152C22⋊Dic83—24no—
153(C2×D83)⋊C4—24no—
154(C2×D83)⋊C4—24no—
155C4×Dic83—22no—
156C4×Dic83—22no—
157C4×Dic83—24no—
158C4×Dic83—24no—
159C24×C83C22×C411yescharacteristic
160C22×C4×C83C22×C412yes—
161C22×C4×C83C22×C412yes—
162C22×C4×C83C22×C411yescharacteristic
163C22×C4×C83C2×D412yes—
164C22×C4×C83C2×D412yes—
165C22×C4×C83#4112yes—
166C22×C4×C83#4112yes—
167C22×C4×C83#4112yes—
168C22×C4×C83#4112yes—
169C22×C4×C83C2411yescharacteristic
170M16×C83—44no—
171C2×C8×C83—22no—
172C2×C8×C83—22no—
173C2×C8×C83—22no—
174C2×C8×C83—22no—
175C2×C8×C83—22no—
176C2×C8×C83—22no—
177C2×C8×C83—44no—
178C4⋊C4×C83—22no—
179C4⋊C4×C83—22no—
180C22⋊C4×C83—24no—
181C42×C83—22no—
182C42×C83—22no—
183C23×C83—22no—
184C23×C83—22no—
185C23×C83C2×C22⋊C412yes—
186C23×C83#11712yes—
187C23×C83#11712yes—
188C23×C83C23×C411yescharacteristic
189C22×D83—21no—
190C22×D83—21no—
191C22×D83—22no—
192C22×D83—21no—
193C22×D83—21no—
194C22×D83—22no—
195C22×D83—22no—
196C22×D83—28no—
197C22×D83—28no—
198C22×D83—44no—
199C22×D83—48no—
200C22×D83—48no—
201C22×D83C2×C22⋊C412yes—
202C22×D83C2×C22⋊C412yes—
203C22×D83#13114yes—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83C2×C22⋊C412yes—
209C2×C4×C83#11712yes—
210C2×C4×C83#11712yes—
211C2×C4×C83#12314yes—
212C2×C4×C83#12314yes—
213C2×C4×C83#12314yes—
214C2×C4×C83#12314yes—
215C2×C4×C83C22×D411yescharacteristic
216C2×C4×C83C22×D411yescharacteristic
217C2×C4×C83C2×#4111yescharacteristic
218C2×C4×C83C2×#4111yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×Dic83—21no—
222C2×Dic83—21no—
223C2×Dic83—21no—
224C2×Dic83—21no—
225C2×Dic83—28no—
226C2×Dic83—22no—
227C2×Dic83—22no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83C2×C22⋊C412yes—
231C2×Dic83C2×C22⋊C412yes—
232C4×D83—22no—
233C4×D83—22no—
234C4×D83—22no—
235C4×D83—22no—
236C4×D83—22no—
237C4×D83—22no—
238C4×D83—22no—
239C4×D83—22no—
240C4×D83—28no—
241C4×D83—28no—
242C4×D83—48no—
243C4×D83—48no—
244C8×C83—42no—
245C8×C83—42no—
246C83⋊C8—42no—
247C83⋊C8—42no—
248C22×C83—22no—
249C22×C83—22no—
250C22×C83—24no—
251C22×C83—24no—
252C22×C83#38114yes—
253C22×C83C22×C22⋊C411yescharacteristic
254C22×C83C2×#11711yescharacteristic
255C22×C83C2×#11711yescharacteristic
256C22×C83C2×#13112yes—
257C22×C83#56812yes—
258C2×D83—22no—
259C2×D83—24no—
260C2×D83—24no—
261C2×D83—22no—
262C2×D83—22no—
263C2×D83—22no—
264C2×D83—22no—
265C2×D83—416no—
266C2×D83—42no—
267C2×D83—42no—
268C2×D83#40814yes—
269C4×C83—42no—
270C4×C83—42no—
271C4×C83#38114yes—
272C4×C83C2×#12311yescharacteristic
273C4×C83C2×#12311yescharacteristic
274C4×C83C2×#12311yescharacteristic
275C4×C83C2×#12311yescharacteristic
276Dic83—22no—
277Dic83—22no—
278Dic83—42no—
279Dic83—42no—
280#20685—831nomaximal, Sylow
281C2×C83—24no—
282C2×C83C2×#38111yescharacteristic
283C2×C83C2×#40811yescharacteristic
284C2×C83#287911yescharacteristic
285C2×C83#292412yes—
286C2×C83#292412yes—
287D83—24no—
288D83—42no—
289C22×#117—831no—
290C2×#407—831no—
291C22×C22⋊C8—831no—
292#1818—834no—
293#1817—834no—
294#1729—831no—
295#1729—831no—
296#1729—831no—
297#1729D8311yescharacteristic, complemented
298C83#2068511yescharacteristic, complemented, Sylow
299C24×C4—831no—
300C22×M16—831no—
301C22×M16C2×D8311yescharacteristic
302C23×C8—831no—
303C23×C8C2×D8311yescharacteristic
304C2×#117—831no—
305C2×#117—831no—
306C2×#117—831no—
307C2×#117—832no—
308C2×#117—832no—
309C2×#117—1664no—
310C2×#117—1664no—
311C2×#117C2×D8311yescharacteristic
312C22×C4⋊C4—831no—
313C22×C4⋊C4—831no—
314C22×C22⋊C4—832no—
315C22×C42—832no—
316#407—1664no—
317C2×C22⋊C8—832no—
318C2×C22⋊C8—832no—
319C2×C22⋊C8—832no—
320C2×C22⋊C8—832no—
321C2×C22⋊C8—1664no—
322#336—834no—
323#336—834no—
324#336—834no—
325#336C2×D8314yescomplemented
326C25—831no—
327C23×C4—832no—
328C23×C4—831no—
329C23×C4—831no—
330C23×C4—831no—
331C23×C4—834no—
332C23×C4—834no—
333C23×C4—831no—
334C23×C4—831no—
335C23×C4—832no—
336C23×C4—1662no—
337C23×C4—1662no—
338C23×C4—1662no—
339C23×C4—1662no—
340C23×C4C22×D8311yescharacteristic
341C2×M16—22no—
342C2×M16—22no—
343C2×M16—22no—
344C2×M16—1662no—
345C2×M16—1662no—
346C2×M16—1662no—
347C22×C8—22no—
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—832no—
351C22×C8—832no—
352C22×C8—1662no—
353C22×C8—1662no—
354C22×C8—1662no—
355C22×C8C22×D8312yes—
356C22×C8C22×D8312yes—
357#117—22no—
358#117—22no—
359#117—1664no—
360#117—1664no—
361#117—1668no—
362#117—1664no—
363#117—1664no—
364#117—1662no—
365#117—1662no—
366C2×C4⋊C4—832no—
367C2×C4⋊C4—832no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—1664no—
371C2×C4⋊C4—1664no—
372C2×C4⋊C4C4×D8311yescharacteristic
373C2×C4⋊C4C4×D8311yescharacteristic
374C2×C22⋊C4—832no—
375C2×C22⋊C4—832no—
376C2×C22⋊C4—832no—
377C2×C22⋊C4—1668no—
378C2×C22⋊C4C4×D8312yes—
379C2×C42—832no—
380C2×C42—832no—
381C2×C42—832no—
382C2×C42—1664no—
383C2×C42—1664no—
384C2×C42C22×D8312yes—
385C22⋊C8—1664no—
386C22⋊C8—1664no—
387C22⋊C8—1664no—
388C22⋊C8—1664no—
389C24—3322no—
390C24—834no—
391C24—831no—
392C24—831no—
393C24—1664no—
394C24—1662no—
395C24—1662no—
396C24C2×C4×D8311yescharacteristic
397C22×C4—3324no—
398C22×C4—831no—
399C22×C4—831no—
400C22×C4—834no—
401C22×C4—832no—
402C22×C4—832no—
403C22×C4—832no—
404C22×C4—832no—
405C22×C4—834no—
406C22×C4—834no—
407C22×C4—832no—
408C22×C4—832no—
409C22×C4—1664no—
410C22×C4—1664no—
411C22×C4—1668no—
412C22×C4—1662no—
413C22×C4—1662no—
414C22×C4—1668no—
415C22×C4—1668no—
416C22×C4—1668no—
417C22×C4—1668no—
418C22×C4—1661no—
419C22×C4—1661no—
420C22×C4—1661no—
421C22×C4—1661no—
422C22×C4—1661no—
423C22×C4—1661no—
424C22×C4—1661no—
425C22×C4—1661no—
426C22×C4—1662no—
427C22×C4—1662no—
428C22×C4—1662no—
429C22×C4—1662no—
430C22×C4C2×C4×D8312yes—
431C22×C4C2×C4×D8312yes—
432C22×C4C2×C4×D8311yescharacteristic
433C22×C4D4×D8312yes—
434C22×C4D4×D8312yes—
435C22×C4#1305058912yes—
436C22×C4#1305059012yes—
437C22×C4#1305059012yes—
438C22×C4#1305059112yes—
439C22×C4C23×D8311yescharacteristic
440M16—44no—
441M16—3324no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—44no—
449C2×C8—3324no—
450C2×C8—1662no—
451C2×C8—1662no—
452C2×C8—1662no—
453C2×C8—1662no—
454C2×C8—1662no—
455C2×C8—1662no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—1664no—
459C4⋊C4—1664no—
460C4⋊C4—1662no—
461C4⋊C4—1662no—
462C22⋊C4—24no—
463C22⋊C4—1664no—
464C22⋊C4—1664no—
465C22⋊C4—1664no—
466C42—22no—
467C42—22no—
468C42—1664no—
469C42—1664no—
470C42—1662no—
471C42—1662no—
472C23—22no—
473C23—22no—
474C23—3328no—
475C23—3328no—
476C23—3324no—
477C23—832no—
478C23—834no—
479C23—832no—
480C23—1668no—
481C23—1668no—
482C23—1661no—
483C23—1661no—
484C23—1662no—
485C23—1662no—
486C23—1661no—
487C23—1661no—
488C23—1662no—
489C23C22⋊C4×D8312yes—
490C23#1458174912yes—
491C23#1458175012yes—
492C23C22×C4×D8311yescharacteristic
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—3328no—
498C2×C4—3328no—
499C2×C4—832no—
500C2×C4—832no—
501C2×C4—1662no—
502C2×C4—1662no—
503C2×C4—1664no—
504C2×C4—1664no—
505C2×C4—1668no—
506C2×C4—1661no—
507C2×C4—1661no—
508C2×C4—1661no—
509C2×C4—1661no—
510C2×C4—1668no—
511C2×C4—1668no—
512C2×C4—1662no—
513C2×C4—1662no—
514C2×C4—1662no—
515C2×C4—1662no—
516C2×C4—1662no—
517C2×C4—1662no—
518C2×C4—1662no—
519C2×C4—1662no—
520C2×C4C22⋊C4×D8312yes—
521C2×C4#1458174912yes—
522C2×C4#1458175012yes—
523C2×C4#1458177414yes—
524C2×C4#1458177414yes—
525C2×C4#1458177714yes—
526C2×C4#1458177714yes—
527C2×C4C2×D4×D8311yescharacteristic
528C2×C4C2×D4×D8311yescharacteristic
529C2×C4C2×#1305058911yescharacteristic
530C2×C4C2×#1305059011yescharacteristic
531C2×C4C2×#1305059011yescharacteristic
532C2×C4C2×#1305059111yescharacteristic
533C8—42no—
534C8—42no—
535C8—3322no—
536C8—3322no—
537C22—24no—
538C22—24no—
539C22—22no—
540C22—22no—
541C22—33216no—
542C22—3322no—
543C22—3322no—
544C22—834no—
545C22—1662no—
546C22—1662no—
547C22—1662no—
548C22—1662no—
549C22—1664no—
550C22—1664no—
551C22—1662no—
552C22#1912756314yes—
553C22C2×C22⋊C4×D8311yescharacteristic
554C22C2×#1458174911yescharacteristic
555C22C2×#1458175011yescharacteristic
556C22#131×D8312yes—
557C22#1912846612yes—
558C4—42no—
559C4—42no—
560C4—3322no—
561C4—3322no—
562C4—1662no—
563C4—1662no—
564C4#1912756314yes—
565C4C2×#1458177411yescharacteristic
566C4C2×#1458177411yescharacteristic
567C4C2×#1458177711yescharacteristic
568C4C2×#1458177711yescharacteristic
569C2—24no—
570C2—3322no—
571C2—1664no—
572C2C2×#1912756311yescharacteristic
573C2#408×D8311yescharacteristic
574C2#2491590511yescharacteristic
575C2#2491611812yes—
576C2#2491612012yes—
subgroupG/H
center, FrattiniC22×C4C23×D83
commutator, socleC23×C83C23×C4
FittingC83×#1729C2
prime radicalC24×D83C2×C4
cosocleC22×C4×C83C24
hypercenter#1729D83
hypocenterC83#20685
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
4. #24907533C2semidirect
5. #24907530C2semidirect
6. #24907161C2semidirect
7. #24907160C2semidirect
8. #24907159C2semidirect
9. C83×#1729C2semidirect
30. #19127456C22semidirect
31. #19127455C22semidirect
32. #19127455C22semidirect
36. #336×C83C22semidirect
297. #1729D83semidirect
298. C83#20685semidirect
325. #336C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D832
C2×#191276642
C22×(C2×D83)⋊C82
#249075332
#249075302
#249071612
#249071602
#249071592
C83×#17292
#2068583