Finite Group Databasebeta

← C2×#24908659 #35431860

Subgroups — C2×#24908659 #35431860

Order 21248.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581771C211yesmaximal, characteristic
2C22×#14581723C211yesmaximal, characteristic, complemented
3C22×#14581715C211yesmaximal, characteristic, complemented
4C22×#14581715C211yesmaximal, characteristic, complemented
5C22×C22⋊(C83⋊C8)C211yesmaximal, characteristic
6#24908659C218yesmaximal, direct product factor
7C2×#19127463C211yesmaximal, characteristic, complemented
8C2×#380×C83C211yesmaximal, characteristic
9C23×Dic166C2211yescharacteristic
10C23×C83⋊C8C2211yescharacteristic
11C23×C83⋊C8C2211yescharacteristic
12C2×#14581771—24no—
13C2×#14581771—22no—
14C2×#14581771—24no—
15C2×#14581771C2214yes—
16C22×C4⋊Dic83C2211yescharacteristic
17C22×Dic83⋊C4—21no—
18C22×C22⋊Dic83—21no—
19C2×#14581723—24no—
20C2×#14581723C2214yescomplemented
21C2×#14581715—24no—
22C2×#14581715—24no—
23C2×#14581715C2214yescomplemented
24C2×#14581715C2214yescomplemented
25C2×C22⋊(C83⋊C8)—24no—
26C2×C22⋊(C83⋊C8)C2214yes—
27#19127463C2214yescomplemented
28C24×C4×C83C2211yescharacteristic
29C22×C4⋊C4×C83C2211yescharacteristic
30C22×C4⋊C4×C83C2211yescharacteristic
31C22×C22⋊C4×C83—21no—
32#380×C83—22no—
33#380×C83C2214yes—
34C2×#95×C83—21no—
35C22×Dic166—22no—
36C22×Dic166—21no—
37C22×Dic166—22no—
38C22×Dic166—22no—
39C22×Dic166—22no—
40C22×Dic166—22no—
41C22×Dic166D414yes—
42C22×Dic166C2312yes—
43C23×Dic83—21no—
44C23×Dic83—21no—
45C22×C83⋊C8—22no—
46C22×C83⋊C8—22no—
47C22×C83⋊C8—22no—
48C22×C83⋊C8—22no—
49C22×C83⋊C8—22no—
50C22×C83⋊C8D414yes—
51C22×C83⋊C8C2312yes—
52C22×C83⋊C8C2312yes—
53#14581771—216no—
54#14581771—28no—
55#14581771—28no—
56C2×C4⋊Dic83—22no—
57C2×C4⋊Dic83—21no—
58C2×C4⋊Dic83D414yes—
59C2×C4⋊Dic83C2312yes—
60C2×Dic83⋊C4—28no—
61C2×Dic83⋊C4—24no—
62C2×C22⋊Dic83—28no—
63C2×C22⋊Dic83—24no—
64#14581723—28no—
65#14581715—28no—
66#14581715—28no—
67C22⋊(C83⋊C8)—28no—
68C25×C83D411yescharacteristic
69C23×C4×C83—22no—
70C23×C4×C83—22no—
71C23×C4×C83—22no—
72C23×C4×C83—22no—
73C23×C4×C83—22no—
74C23×C4×C83—22no—
75C23×C4×C83—21no—
76C23×C4×C83—21no—
77C23×C4×C83D411yescharacteristic
78C23×C4×C83D412yes—
79C23×C4×C83C2312yes—
80C23×C4×C83C2311yescharacteristic
81C2×C4⋊C4×C83—22no—
82C2×C4⋊C4×C83—22no—
83C2×C4⋊C4×C83—22no—
84C2×C4⋊C4×C83—22no—
85C2×C4⋊C4×C83—21no—
86C2×C4⋊C4×C83—21no—
87C2×C4⋊C4×C83C2312yes—
88C2×C4⋊C4×C83C2312yes—
89C2×C22⋊C4×C83—24no—
90C2×C22⋊C4×C83—44no—
91#95×C83—24no—
92C2×Dic166—28no—
93C2×Dic166—28no—
94C2×Dic166—22no—
95C2×Dic166—22no—
96C2×Dic166—22no—
97C2×Dic166—22no—
98C2×Dic166—24no—
99C2×Dic166—22no—
100C2×Dic166—42no—
101C2×Dic166—44no—
102C2×Dic166—42no—
103C2×Dic166—44no—
104C2×Dic166C2×D414yes—
105C22×Dic83—24no—
106C22×Dic83—24no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—42no—
110C22×Dic83—42no—
111C22×Dic83—42no—
112C22×Dic83—42no—
113C2×C83⋊C8—28no—
114C2×C83⋊C8—22no—
115C2×C83⋊C8—22no—
116C2×C83⋊C8—22no—
117C2×C83⋊C8—22no—
118C2×C83⋊C8—22no—
119C2×C83⋊C8—44no—
120C2×C83⋊C8C2×D414yes—
121C4⋊Dic83—24no—
122C4⋊Dic83—22no—
123C4⋊Dic83C2×D414yes—
124Dic83⋊C4—216no—
125C22⋊Dic83—216no—
126C24×C83—22no—
127C24×C83—22no—
128C24×C83—22no—
129C24×C83—22no—
130C24×C83—22no—
131C24×C83—22no—
132C24×C83SD1614yes—
133C24×C83C2×D412yes—
134C24×C83C2×D411yescharacteristic
135C22×C4×C83—22no—
136C22×C4×C83—22no—
137C22×C4×C83—22no—
138C22×C4×C83—24no—
139C22×C4×C83—24no—
140C22×C4×C83—24no—
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—44no—
150C22×C4×C83—44no—
151C22×C4×C83—44no—
152C22×C4×C83—44no—
153C22×C4×C83—42no—
154C22×C4×C83—42no—
155C22×C4×C83—42no—
156C22×C4×C83—42no—
157C22×C4×C83—42no—
158C22×C4×C83—42no—
159C22×C4×C83C2×D412yes—
160C22×C4×C83C2×D412yes—
161C22×C4×C83C2×D412yes—
162C22×C4×C83C2×D412yes—
163C22×C4×C83C2×D411yescharacteristic
164C22×C4×C83#4112yes—
165C22×C4×C83#4112yes—
166C22×C4×C83#4112yes—
167C22×C4×C83C2411yescharacteristic
168C4⋊C4×C83—22no—
169C4⋊C4×C83—22no—
170C4⋊C4×C83—22no—
171C4⋊C4×C83—22no—
172C4⋊C4×C83—42no—
173C4⋊C4×C83—42no—
174C22⋊C4×C83—48no—
175C23×C83—22no—
176C23×C83—22no—
177C23×C83—22no—
178C23×C83—22no—
179C23×C83—22no—
180C23×C83—22no—
181C23×C83—28no—
182C23×C83—28no—
183C23×C83—28no—
184C23×C83—44no—
185C23×C83—44no—
186C23×C83—44no—
187C23×C83—44no—
188C23×C83#12012yes—
189C23×C83#12112yes—
190C23×C83#12312yes—
191C23×C83C2×SD1614yes—
192C23×C83C2×SD1612yes—
193C23×C83#13512yes—
194C23×C83#13712yes—
195C23×C83#13712yes—
196C23×C83C22×D411yescharacteristic
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—22no—
200C2×C4×C83—44no—
201C2×C4×C83—42no—
202C2×C4×C83—42no—
203C2×C4×C83—42no—
204C2×C4×C83—42no—
205C2×C4×C83—48no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83—42no—
212C2×C4×C83—84no—
213C2×C4×C83—84no—
214C2×C4×C83#12012yes—
215C2×C4×C83#12112yes—
216C2×C4×C83#12114yes—
217C2×C4×C83#12114yes—
218C2×C4×C83#12314yes—
219C2×C4×C83#12312yes—
220C2×C4×C83#12414yes—
221C2×C4×C83C22×D411yescharacteristic
222C2×C4×C83C22×D411yescharacteristic
223C2×C4×C83C22×D411yescharacteristic
224C2×C4×C83C2×#4111yescharacteristic
225C2×C4×C83C2×#4111yescharacteristic
226C2×C4×C83C2×#4111yescharacteristic
227C2×Dic83—24no—
228C2×Dic83—24no—
229C2×Dic83—42no—
230C2×Dic83—42no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—48no—
234C2×Dic83—48no—
235Dic166—24no—
236Dic166—24no—
237Dic166—42no—
238Dic166—42no—
239Dic166—48no—
240C83⋊C8—24no—
241C83⋊C8—42no—
242C22×C83—24no—
243C22×C83—24no—
244C22×C83—24no—
245C22×C83—42no—
246C22×C83—42no—
247C22×C83—42no—
248C22×C83—42no—
249C22×C83—416no—
250C22×C83#39414yes—
251C22×C83#44814yes—
252C22×C83#45214yes—
253C22×C83#46514yes—
254C22×C83#47014yes—
255C22×C83#48314yes—
256C22×C83#48514yes—
257C22×C83C2×#12011yescharacteristic
258C22×C83C2×#12111yescharacteristic
259C22×C83C2×#12311yescharacteristic
260C22×C83C22×SD1611yescharacteristic
261C22×C83C2×#13511yescharacteristic
262C22×C83C2×#13711yescharacteristic
263C22×C83C2×#13711yescharacteristic
264C4×C83—42no—
265C4×C83—82no—
266C4×C83—82no—
267C4×C83#39414yes—
268C4×C83C2×#12111yescharacteristic
269C4×C83C2×#12111yescharacteristic
270C4×C83C2×#12311yescharacteristic
271C4×C83C2×#12411yescharacteristic
272Dic83—44no—
273Dic83—44no—
274C2×#2035—831nomaximal, Sylow
275C2×C83—44no—
276C2×C83#203518yescomplemented
277C2×C83C2×#39411yescharacteristic
278C2×C83C2×#44811yescharacteristic
279C2×C83C2×#45211yescharacteristic
280C2×C83C2×#46511yescharacteristic
281C2×C83C2×#47011yescharacteristic
282C2×C83C2×#48311yescharacteristic
283C2×C83C2×#48511yescharacteristic
284C22×C22⋊Q8—831no—
285C22×#106—831no—
286C22×#103—831no—
287C22×#103—831no—
288C22×C22⋊C8—831no—
289C2×#380D8311yescharacteristic
290#2035—838no—
291C2×#340—831no—
292C83C2×#203511yescharacteristic, complemented, Sylow
293C23×Q8—831no—
294C24×C4C2×D8311yescharacteristic
295C23×C8—831no—
296C23×C8—831no—
297C2×C22⋊Q8—834no—
298C2×C22⋊Q8—1664no—
299C2×C22⋊Q8—1664no—
300C2×C22⋊Q8—1662no—
301C22×C4⋊C4—831no—
302C22×C4⋊C4—1661no—
303C22×C4⋊C4C2×D8311yescharacteristic
304C22×C4⋊C4C2×D8311yescharacteristic
305C22×C22⋊C4—21no—
306C22×C22⋊C4—1661no—
307C2×#106—834no—
308C2×#106—1664no—
309C2×#103—834no—
310C2×#103—834no—
311C2×#103—1664no—
312C2×#103—1664no—
313C2×C22⋊C8—834no—
314C2×C22⋊C8—1664no—
315#380—22no—
316#380C2×D8314yes—
317C2×#95—21no—
318#340—834no—
319C25C22⋊D8311yescharacteristic
320C22×Q8—834no—
321C22×Q8—832no—
322C22×Q8—1662no—
323C22×Q8—1662no—
324C22×Q8—1662no—
325C22×Q8—1662no—
326C22×Q8—1662no—
327C22×Q8—1661no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—1661no—
337C23×C4—1661no—
338C23×C4C22⋊D8312yes—
339C23×C4C22⋊D8311yescharacteristic
340C23×C4C22×D8312yes—
341C23×C4C22×D8311yescharacteristic
342C22×C8—834no—
343C22×C8—832no—
344C22×C8—832no—
345C22×C8—1662no—
346C22×C8—1662no—
347C22×C8—1662no—
348C22×C8—1662no—
349C22×C8—1662no—
350C22⋊Q8—1668no—
351C22⋊Q8—1668no—
352C22⋊Q8—16616no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—834no—
360C2×C4⋊C4—832no—
361C2×C4⋊C4—1668no—
362C2×C4⋊C4—1662no—
363C2×C4⋊C4—1661no—
364C2×C4⋊C4—1664no—
365C2×C4⋊C4C22×D8312yes—
366C2×C4⋊C4C22×D8312yes—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4—1668no—
370C2×C22⋊C4—1664no—
371#106—1668no—
372#103—1668no—
373#103—1668no—
374C22⋊C8—1668no—
375#95—24no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24#1305057814yes—
383C24C2×C22⋊D8312yes—
384C24C2×C22⋊D8311yescharacteristic
385C2×Q8—3322no—
386C2×Q8—3324no—
387C2×Q8—3322no—
388C2×Q8—3324no—
389C2×Q8—834no—
390C2×Q8—1668no—
391C2×Q8—1668no—
392C2×Q8—1662no—
393C2×Q8—1662no—
394C2×Q8—1662no—
395C2×Q8—1662no—
396C2×Q8—1664no—
397C2×Q8—1662no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—3322no—
423C22×C4—3322no—
424C22×C4—3322no—
425C22×C4—3322no—
426C22×C4—1664no—
427C22×C4—1664no—
428C22×C4—1662no—
429C22×C4—1662no—
430C22×C4D4×D8312yes—
431C22×C4D4×D8312yes—
432C22×C4D4×D8312yes—
433C22×C4C2×C22⋊D8312yes—
434C22×C4C2×C22⋊D8311yescharacteristic
435C22×C4#1305058912yes—
436C22×C4#1305059012yes—
437C22×C4#1305059012yes—
438C22×C4C23×D8311yescharacteristic
439C2×C8—3324no—
440C2×C8—834no—
441C2×C8—1662no—
442C2×C8—1662no—
443C2×C8—1662no—
444C2×C8—1668no—
445C2×C8—1662no—
446C2×C8—1662no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—834no—
454C4⋊C4—16616no—
455C4⋊C4—1664no—
456C4⋊C4—1662no—
457C22⋊C4—48no—
458C22⋊C4—16616no—
459C23—28no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—28no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23#1458176012yes—
473C23#1458176412yes—
474C23#1458177512yes—
475C23C2×#1305057814yes—
476C23C2×#1305057812yes—
477C23#1458181212yes—
478C23#1458182212yes—
479C23#1458182312yes—
480C23C22×C22⋊D8311yescharacteristic
481Q8—3328no—
482Q8—3322no—
483Q8—3322no—
484Q8—1664no—
485Q8—1664no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—3328no—
504C2×C4—3328no—
505C2×C4—3322no—
506C2×C4—3322no—
507C2×C4—3322no—
508C2×C4—3322no—
509C2×C4—1664no—
510C2×C4—1664no—
511C2×C4#1458176012yes—
512C2×C4#1458176412yes—
513C2×C4#1458176514yes—
514C2×C4#1458176514yes—
515C2×C4#1458177414yes—
516C2×C4#1458177512yes—
517C2×C4#1458177814yes—
518C2×C4C2×D4×D8311yescharacteristic
519C2×C4C2×D4×D8311yescharacteristic
520C2×C4C2×D4×D8311yescharacteristic
521C2×C4C2×#1305058911yescharacteristic
522C2×C4C2×#1305059011yescharacteristic
523C2×C4C2×#1305059011yescharacteristic
524C8—3322no—
525C8—1664no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—416no—
533C22—42no—
534C22#1912761614yes—
535C22#1912783914yes—
536C22#1912786114yes—
537C22#1912792814yes—
538C22#1912795714yes—
539C22#1912803114yes—
540C22#1912804214yes—
541C22C2×#1458176011yescharacteristic
542C22C2×#1458176411yescharacteristic
543C22C2×#1458177511yescharacteristic
544C22C22×#1305057811yescharacteristic
545C22C2×#1458181211yescharacteristic
546C22C2×#1458182211yescharacteristic
547C22C2×#1458182311yescharacteristic
548C4—42no—
549C4—82no—
550C4—82no—
551C4—3324no—
552C4—3324no—
553C4#1912761614yes—
554C4C2×#1458176511yescharacteristic
555C4C2×#1458176511yescharacteristic
556C4C2×#1458177411yescharacteristic
557C4C2×#1458177811yescharacteristic
558C2—44no—
559C2#2490865918yesdirect product factor
560C2C2×#1912761611yescharacteristic
561C2C2×#1912783911yescharacteristic
562C2C2×#1912786111yescharacteristic
563C2C2×#1912792811yescharacteristic
564C2C2×#1912795711yescharacteristic
565C2C2×#1912803111yescharacteristic
566C2C2×#1912804211yescharacteristic
subgroupG/H
centerC24C2×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC2×#380×C83C2
socleC24×C83C2×D4
prime radicalC25×C83D4
hypercenterC2×#380D83
hypocenterC83C2×#2035
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C24 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#14581723C2semidirect
3. C22×#14581715C2semidirect
4. C22×#14581715C2semidirect
6. #24908659C2direct (N × H)
7. C2×#19127463C2semidirect
20. C2×#14581723C22semidirect
23. C2×#14581715C22semidirect
24. C2×#14581715C22semidirect
27. #19127463C22semidirect
276. C2×C83#2035semidirect
292. C83C2×#2035semidirect
559. C2#24908659direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817712
C22×#145817232
C22×#145817152
C22×C22⋊(C83⋊C8)2
#249086592
C2×#191274632
C2×#380×C832
C2×#203583