Finite Group Databasebeta

← C2×#20243611 #26217129

Subgroups — C2×#20243611 #26217129

Order 12032.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689603C211yesmaximal, characteristic
2C22×#13093950C211yesmaximal, characteristic
3C2×#15689122C211yesmaximal, characteristic
4C2×#15688992C211yesmaximal, characteristic, complemented
5#20243611C218yesmaximal, direct product factor
6C2×#15688891C211yesmaximal, characteristic, complemented
7C2×#15688882C211yesmaximal, characteristic, complemented
8C2×C47×#428C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11#15689603—22no—
12#15689603—22no—
13#15689603—22no—
14#15689603C2214yes—
15C2×C4×Dic94—21no—
16C2×C4×D188—21no—
17C2×(C22⋊D47)⋊C4—22no—
18C2×#13093982—21no—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×Dic47⋊C4—22no—
21C2×C4⋊C4×D47—21no—
22C22×C4×Dic47C2211yescharacteristic
23C2×#13093950C418yescomplemented
24C2×#13093950C2214yes—
25#15689122—22no—
26#15689122C2214yes—
27C2×#13093947—21no—
28C2×#13093944—21no—
29#15688992—22no—
30#15688992C2214yescomplemented
31#15688891C2214yescomplemented
32#15688882C2214yescomplemented
33C22×M16×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C22×C4⋊C4×C47C2211yescharacteristic
36C47×#428—22no—
37C47×#428C2214yescomplemented
38C2×#107×C47—21no—
39C2×#106×C47—21no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350C2×C414yes—
45C2×#10625350D412yes—
46C2×#10625350Q814yes—
47C2×#10625350C2312yes—
48C22×Dic94D411yescharacteristic
49C22×D188D411yescharacteristic
50C22×C22⋊D47—21no—
51C23×Dic47—21no—
52C23×Dic47D412yes—
53C22×C4×D47—21no—
54C22×C4×D47—22no—
55C2×#10625329—21no—
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329C2312yes—
59C22×C47⋊C8—21no—
60C4×Dic94—24no—
61C4×Dic94—24no—
62C4×D188—24no—
63C4×D188—24no—
64(C22⋊D47)⋊C4—28no—
65(C22⋊D47)⋊C4—28no—
66#13093982—24no—
67#13093982—24no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47D412yes—
73C2×C4⋊Dic47C2312yes—
74C2×Dic47⋊C4—22no—
75C2×Dic47⋊C4—28no—
76C2×Dic47⋊C4—48no—
77C4⋊C4×D47—24no—
78C4⋊C4×D47—24no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—22no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47C2×C414yescomplemented
84C2×C4×Dic47C2312yes—
85#13093950C2×C4116yescomplemented
86#13093947—24no—
87#13093944—24no—
88C23×C4×C47—21no—
89C23×C4×C47C2311yescharacteristic
90C2×M16×C47—21no—
91C2×M16×C47—22no—
92C2×M16×C47C2×C414yescomplemented
93C2×M16×C47C2312yes—
94C22×C8×C47—21no—
95#117×C47—21no—
96#117×C47—21no—
97#117×C47D412yes—
98#117×C47C2312yes—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47D411yescharacteristic
105C2×C4⋊C4×C47D411yescharacteristic
106C2×C4⋊C4×C47C2312yes—
107C2×C22⋊C4×C47—21no—
108C2×C42×C47—21no—
109#107×C47—24no—
110#106×C47—24no—
111C24×C47C2×D411yescharacteristic
112C23×D47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—22no—
116C22×C4×C47—44no—
117C22×C4×C47—41no—
118C22×C4×C47—41no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×Q812yes—
126C22×C4×C47#4112yes—
127C22×C4×C47#4112yes—
128C22×C4×C47C2411yescharacteristic
129C2×C22⋊D47—24no—
130C2×C22⋊D47—22no—
131C2×C22⋊D47—24no—
132C2×C22⋊D47—22no—
133C22×Dic47—24no—
134C22×Dic47—24no—
135C22×Dic47—24no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—44no—
141C22×Dic47C2×D414yes—
142#10625350—24no—
143#10625350—24no—
144#10625350—24no—
145#10625350—24no—
146#10625350—24no—
147#10625350—22no—
148#10625350—22no—
149#10625350C4⋊C418yescomplemented
150#10625350C22×C414yes—
151#10625350C2×Q814yes—
152C2×D188—21no—
153C2×D188—22no—
154C2×D188C4⋊C414yescomplemented
155C2×D188C2×D412yes—
156C2×C4×D47—24no—
157C2×C4×D47—22no—
158C2×C4×D47—24no—
159C2×C4×D47—22no—
160C2×C4×D47—28no—
161C2×C4×D47—84no—
162C2×Dic94—21no—
163C2×Dic94—22no—
164C2×Dic94C4⋊C414yescomplemented
165C2×Dic94C2×D412yes—
166M16×C47—24no—
167M16×C47—22no—
168M16×C47C22×C414yescomplemented
169C2×C8×C47—24no—
170C2×C8×C47—22no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—41no—
176C4⋊C4×C47—41no—
177C4⋊C4×C47C2×D412yes—
178C4⋊C4×C47C2×D412yes—
179C22⋊C4×C47—22no—
180C22⋊C4×C47—22no—
181C42×C47—22no—
182C42×C47—22no—
183(C2×D47)⋊C4—28no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47—22no—
187C4⋊Dic47—22no—
188Dic47⋊C4—28no—
189Dic47⋊C4—416no—
190C4×Dic47—28no—
191C4×Dic47—24no—
192C4×Dic47—22no—
193C4×Dic47C22×C414yescomplemented
194#10625329—22no—
195#10625329—22no—
196#10625329—42no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—42no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47C4×D412yes—
202C23×C47#12011yescharacteristic
203C23×C47#12011yescharacteristic
204C23×C47C22⋊Q812yes—
205C23×C47C22×D411yescharacteristic
206C22×D47—22no—
207C22×D47—24no—
208C22×D47—42no—
209C22×D47—42no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—44no—
215C2×C4×C47—42no—
216C2×C4×C47—82no—
217C2×C4×C47—82no—
218C2×C4×C47C2×C4⋊C412yes—
219C2×C4×C47C2×C4⋊C412yes—
220C2×C4×C47C4×D412yes—
221C2×C4×C47C4×D412yes—
222C2×C4×C47C4×D412yes—
223C2×C4×C47#12011yescharacteristic
224C2×C4×C47#12011yescharacteristic
225C2×C4×C47C22⋊Q812yes—
226C2×C4×C47C22⋊Q812yes—
227C2×C4×C47C22⋊Q812yes—
228C2×C4×C47#12312yes—
229C2×C4×C47#12312yes—
230C2×C4×C47C23×C411yescharacteristic
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×Q811yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C22⋊D47—24no—
237C22⋊D47—24no—
238C22⋊D47—28no—
239C2×Dic47—24no—
240C2×Dic47—216no—
241C2×Dic47—24no—
242C2×Dic47—24no—
243C2×Dic47—22no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—44no—
247C2×Dic47—44no—
248C2×Dic47—82no—
249D188—22no—
250D188—24no—
251D188C2×C4⋊C414yescomplemented
252C4×D47—24no—
253C4×D47—24no—
254C4×D47—28no—
255C4×D47—88no—
256Dic94—22no—
257Dic94—24no—
258Dic94C2×C4⋊C414yescomplemented
259C8×C47—24no—
260C47⋊C8—42no—
261C2×#1859—471nomaximal, Sylow
262C22×C47—24no—
263C22×C47—24no—
264C22×C47—21no—
265C22×C47—21no—
266C22×C47—21no—
267C22×C47—21no—
268C22×C47#38512yes—
269C22×C47#38512yes—
270C22×C47#45312yes—
271C22×C47#45612yes—
272C22×C47C2×C4×D411yescharacteristic
273C22×C47C2×#12011yescharacteristic
274C22×C47C2×C22⋊Q811yescharacteristic
275C2×D47—24no—
276C2×D47—48no—
277C2×D47—42no—
278C2×D47—42no—
279C4×C47—42no—
280C4×C47—42no—
281C4×C47—82no—
282C4×C47#38512yes—
283C4×C47#38512yes—
284C4×C47C22×C4⋊C411yescharacteristic
285C4×C47C2×C4×D411yescharacteristic
286C4×C47C2×C22⋊Q811yescharacteristic
287C4×C47C2×#12311yescharacteristic
288Dic47—28no—
289Dic47—24no—
290Dic47—44no—
291C2×#520—471no—
292C2×#428D4711yescharacteristic, complemented
293C22×#104—471no—
294C2×#417—471no—
295C2×#384—471no—
296#1859—478no—
297C2×#343—471no—
298C2×#339—471no—
299C2×C47—21no—
300C2×C47—21no—
301C2×C47—22no—
302C2×C47#185914yescomplemented
303C2×C47C2×#38511yescharacteristic
304C2×C47C2×#45311yescharacteristic
305C2×C47C2×#45611yescharacteristic
306D47—44no—
307C22×#41—471no—
308C22×M16—471no—
309C22×M16C2×D4711yescharacteristic
310#520—474no—
311#520—942no—
312#520—942no—
313#520—942no—
314C2×C4×Q8—941no—
315C2×C4×D4—941no—
316C2×C4×D4—942no—
317C2×#117—941no—
318C2×#117C2×D4711yescharacteristic
319C22×C4⋊C4—471no—
320C22×C4⋊C4—941no—
321C22×C4⋊C4—942no—
322C22×C4⋊C4C2×D4711yescharacteristic
323C22×C42—471no—
324#428—22no—
325#428C2×D4714yescomplemented
326C2×#107—21no—
327C2×#106—21no—
328C2×#104—478no—
329C2×#104—474no—
330#417—474no—
331#417—942no—
332C2×#103—941no—
333C2×#102—941no—
334#384—474no—
335#384—942no—
336#343—474no—
337#339—474no—
338C47C2×#185911yescharacteristic, complemented, Sylow
339C2×#41—474no—
340C2×#41—472no—
341C2×#41—474no—
342C2×#41—472no—
343C2×#41—942no—
344C2×#41—941no—
345C2×#41—942no—
346C2×#41—941no—
347C22×Q8—471no—
348C22×D4—471no—
349C22×D4—941no—
350C23×C4—21no—
351C23×C4—472no—
352C23×C4—942no—
353C23×C4—941no—
354C23×C4—941no—
355C23×C4C22×D4711yescharacteristic
356C2×M16—22no—
357C2×M16—21no—
358C2×M16—472no—
359C2×M16—941no—
360C2×M16—942no—
361C2×M16—942no—
362C2×M16C4×D4714yescomplemented
363C2×M16C22×D4712yes—
364C22×C8—21no—
365C22×C8—941no—
366C4×Q8—944no—
367C4×Q8—944no—
368C4×D4—944no—
369C4×D4—944no—
370C4×D4—948no—
371C4×D4—948no—
372#117—21no—
373#117—21no—
374#117—944no—
375#117—944no—
376#117C22⋊D4712yes—
377#117C22×D4712yes—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—472no—
384C2×C4⋊C4—472no—
385C2×C4⋊C4—944no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—948no—
393C2×C4⋊C4—1888no—
394C2×C4⋊C4C22⋊D4711yescharacteristic
395C2×C4⋊C4C22⋊D4711yescharacteristic
396C2×C4⋊C4C22×D4712yes—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—942no—
399C2×C42—21no—
400C2×C42—474no—
401C2×C42—472no—
402C2×C42—942no—
403C2×C42—941no—
404C2×C42—942no—
405#107—24no—
406#106—24no—
407#104—4716no—
408#103—944no—
409#102—944no—
410C24—941no—
411C24D4×D4711yescharacteristic
412#41—478no—
413#41—474no—
414#41—474no—
415#41—944no—
416#41—944no—
417#41—944no—
418#41—944no—
419#41—944no—
420#41—942no—
421#41—942no—
422C2×Q8—474no—
423C2×Q8—472no—
424C2×Q8—941no—
425C2×Q8—942no—
426C2×D4—474no—
427C2×D4—472no—
428C2×D4—944no—
429C2×D4—942no—
430C2×D4—944no—
431C2×D4—942no—
432C2×D4—941no—
433C2×D4—942no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—474no—
441C22×C4—948no—
442C22×C4—944no—
443C22×C4—944no—
444C22×C4—944no—
445C22×C4—941no—
446C22×C4—942no—
447C22×C4—944no—
448C22×C4—942no—
449C22×C4—944no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—3764no—
454C22×C4—1884no—
455C22×C4C2×C4×D4712yes—
456C22×C4#1062535012yes—
457C22×C4D4×D4711yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4D4×D4711yescharacteristic
460C22×C4Q8×D4712yes—
461C22×C4#1062535412yes—
462C22×C4C2×C22⋊D4711yescharacteristic
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C23×D4711yescharacteristic
465M16—24no—
466M16—22no—
467M16—942no—
468M16—942no—
469M16—1882no—
470M16C2×C4×D4714yescomplemented
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—942no—
474C2×C8—1882no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—948no—
482C4⋊C4—942no—
483C4⋊C4—942no—
484C4⋊C4—942no—
485C4⋊C4—942no—
486C4⋊C4—18816no—
487C4⋊C4C2×C22⋊D4712yes—
488C4⋊C4C2×C22⋊D4712yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—948no—
492C42—22no—
493C42—22no—
494C42—474no—
495C42—948no—
496C42—944no—
497C42—942no—
498C23—22no—
499C23—22no—
500C23—944no—
501C23—942no—
502C23—1882no—
503C23—1882no—
504C23#1309398612yes—
505C23#1309399211yescharacteristic
506C23#1309399211yescharacteristic
507C23#1309400212yes—
508C23C2×D4×D4711yescharacteristic
509Q8—474no—
510Q8—944no—
511Q8—942no—
512D4—474no—
513D4—948no—
514D4—944no—
515D4—944no—
516D4—944no—
517D4—942no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—944no—
527C2×C4—944no—
528C2×C4—9416no—
529C2×C4—944no—
530C2×C4—944no—
531C2×C4—942no—
532C2×C4—948no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—3768no—
537C2×C4—3762no—
538C2×C4—1884no—
539C2×C4—1884no—
540C2×C4C4⋊C4×D4712yes—
541C2×C4C4⋊C4×D4712yes—
542C2×C4C4×C22⋊D4712yes—
543C2×C4C4×C22⋊D4712yes—
544C2×C4#1309398612yes—
545C2×C4#1309399211yescharacteristic
546C2×C4#1309399211yescharacteristic
547C2×C4#1309400012yes—
548C2×C4#1309400012yes—
549C2×C4#1309400212yes—
550C2×C4#1309400912yes—
551C2×C4#1309400912yes—
552C2×C4C22×C4×D4711yescharacteristic
553C2×C4C2×D4×D4711yescharacteristic
554C2×C4C22×C22⋊D4711yescharacteristic
555C2×C4C2×Q8×D4711yescharacteristic
556C2×C4C2×#1062535011yescharacteristic
557C2×C4C2×#1062535411yescharacteristic
558C8—24no—
559C8—1882no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—944no—
567C22—1882no—
568C22—1882no—
569C22—1888no—
570C22#1568899412yes—
571C22#1568899412yes—
572C22#1568928912yes—
573C22#1568930412yes—
574C22C2×#1309398611yescharacteristic
575C22C2×#1309399211yescharacteristic
576C22C2×#1309400211yescharacteristic
577C4—42no—
578C4—42no—
579C4—82no—
580C4—948no—
581C4—944no—
582C4—1884no—
583C4#1568899412yes—
584C4#1568899412yes—
585C4C2×C4⋊C4×D4711yescharacteristic
586C4C2×C4×C22⋊D4711yescharacteristic
587C4C2×#1309400011yescharacteristic
588C4C2×#1309400911yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—1884no—
593C2#2024361114yesdirect product factor
594C2C2×#1568899411yescharacteristic
595C2C2×#1568928911yescharacteristic
596C2C2×#1568930411yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#428C2
socleC23×C47#120
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenterC2×#428D47
hypocenterC47C2×#1859
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. C2×#15688992C2semidirect
5. #20243611C2direct (N × H)
6. C2×#15688891C2semidirect
7. C2×#15688882C2semidirect
8. C2×C47×#428C2semidirect
23. C2×#13093950C4semidirect
30. #15688992C22semidirect
31. #15688891C22semidirect
32. #15688882C22semidirect
37. C47×#428C22semidirect
83. C2×C4×Dic47C2×C4semidirect
85. #13093950C2×C4semidirect
92. C2×M16×C47C2×C4semidirect
149. #10625350C4⋊C4semidirect
154. C2×D188C4⋊C4semidirect
164. C2×Dic94C4⋊C4semidirect
168. M16×C47C22×C4semidirect
193. C4×Dic47C22×C4semidirect
251. D188C2×C4⋊C4semidirect
258. Dic94C2×C4⋊C4semidirect
292. C2×#428D47semidirect
302. C2×C47#1859semidirect
325. #428C2×D47semidirect
338. C47C2×#1859semidirect
362. C2×M16C4×D47semidirect
470. M16C2×C4×D47semidirect
593. C2#20243611direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896032
C22×#130939502
C2×#156891222
C2×#156889922
#202436112
C2×#156888912
C2×#156888822
C2×C47×#4282
C2×#185947