Finite Group Databasebeta

← C2×#20243614 #26217132

Subgroups — C2×#20243614 #26217132

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×#15689166C211yesmaximal, characteristic, complemented
2C22×#13093951C211yesmaximal, characteristic
3C2×#15689124C211yesmaximal, characteristic
4C2×#15688992C211yesmaximal, characteristic, complemented
5#20243614C218yesmaximal, direct product factor
6C2×#15688892C211yesmaximal, characteristic, complemented
7C2×#15688882C211yesmaximal, characteristic, complemented
8C2×C47×#520C211yesmaximal, characteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×Dic47⋊C4—22no—
13C22×C4×Dic47C2211yescharacteristic
14#15689166—22no—
15#15689166C2214yescomplemented
16C2×#13093957—21no—
17C2×#13093955—21no—
18C2×#13093951C418yescomplemented
19C2×#13093951C2214yes—
20#15689124—22no—
21#15689124C2214yes—
22C2×#13093948—21no—
23C2×#13093945—21no—
24#15688992—22no—
25#15688992C2214yescomplemented
26#15688892C2214yescomplemented
27#15688882C2214yescomplemented
28C22×#41×C47C2211yescharacteristic
29C47×#520—22no—
30C47×#520—22no—
31C47×#520—22no—
32C47×#520C2214yes—
33C2×C4×Q8×C47—21no—
34C2×C4×D4×C47—21no—
35C2×C4×D4×C47—22no—
36C2×#117×C47—21no—
37C2×#117×C47C2211yescharacteristic
38C22×C4⋊C4×C47—21no—
39C22×C4⋊C4×C47C2211yescharacteristic
40C23×Dic47—21no—
41C23×Dic47—21no—
42C2×#10625329—21no—
43C2×#10625329—22no—
44C2×#10625329—21no—
45C2×#10625329—22no—
46C2×#10625329—22no—
47C2×#10625329C2×C414yescomplemented
48C2×#10625329C2312yes—
49C2×#10625329C2312yes—
50C22×C47⋊C8—21no—
51C22×C47⋊C8—21no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47—21no—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47C2312yes—
58C2×Dic47⋊C4—28no—
59C2×Dic47⋊C4—48no—
60C2×C4×Dic47—21no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47C2×C414yescomplemented
63C2×C4×Dic47C2312yes—
64#13093957—24no—
65#13093955—24no—
66#13093951C2×C4116yescomplemented
67#13093948—24no—
68#13093945—24no—
69C2×#41×C47—21no—
70C2×#41×C47—21no—
71C2×#41×C47—22no—
72C2×#41×C47—22no—
73C2×#41×C47C2×C414yes—
74C2×#41×C47D412yes—
75C2×#41×C47Q814yes—
76C2×#41×C47C2312yes—
77C22×Q8×C47D411yescharacteristic
78C22×D4×C47—21no—
79C22×D4×C47D411yescharacteristic
80C23×C4×C47—21no—
81C23×C4×C47—22no—
82C23×C4×C47D412yes—
83C23×C4×C47C2311yescharacteristic
84C4×Q8×C47—24no—
85C4×Q8×C47—24no—
86C4×D4×C47—28no—
87C4×D4×C47—24no—
88C4×D4×C47—24no—
89C4×D4×C47—28no—
90#117×C47—24no—
91#117×C47—21no—
92#117×C47—21no—
93#117×C47—24no—
94#117×C47D412yes—
95#117×C47C2312yes—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47—24no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47D412yes—
104C2×C4⋊C4×C47D411yescharacteristic
105C2×C4⋊C4×C47D411yescharacteristic
106C2×C4⋊C4×C47C2312yes—
107C2×C22⋊C4×C47—22no—
108C2×C22⋊C4×C47—21no—
109C2×C42×C47—22no—
110C2×C42×C47—21no—
111C24×C47—21no—
112C24×C47C2×D411yescharacteristic
113#41×C47—24no—
114#41×C47—24no—
115#41×C47—24no—
116#41×C47—24no—
117#41×C47—24no—
118#41×C47—22no—
119#41×C47—22no—
120#41×C47C4⋊C418yescomplemented
121#41×C47C22×C414yes—
122#41×C47C2×Q814yes—
123C2×Q8×C47—21no—
124C2×Q8×C47—22no—
125C2×Q8×C47C4⋊C414yescomplemented
126C2×Q8×C47C2×D412yes—
127C2×D4×C47—24no—
128C2×D4×C47—22no—
129C2×D4×C47—24no—
130C2×D4×C47—22no—
131C2×D4×C47—21no—
132C2×D4×C47—22no—
133C2×D4×C47C4⋊C414yescomplemented
134C2×D4×C47C2×D412yes—
135C22×C4×C47—21no—
136C22×C4×C47—24no—
137C22×C4×C47—22no—
138C22×C4×C47—24no—
139C22×C4×C47—22no—
140C22×C4×C47—21no—
141C22×C4×C47—22no—
142C22×C4×C47—21no—
143C22×C4×C47—28no—
144C22×C4×C47—22no—
145C22×C4×C47—44no—
146C22×C4×C47—84no—
147C22×C4×C47C22×C412yes—
148C22×C4×C47C2×D414yes—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×Q812yes—
155C22×C4×C47#4112yes—
156C22×C4×C47#4112yes—
157C22×C4×C47C2411yescharacteristic
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—24no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—44no—
164C22×Dic47—41no—
165C22×Dic47—41no—
166C4⋊C4×C47—28no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47C2×D412yes—
174C4⋊C4×C47C2×D412yes—
175C22⋊C4×C47—28no—
176C22⋊C4×C47—22no—
177C22⋊C4×C47—22no—
178C42×C47—28no—
179C42×C47—22no—
180C42×C47—22no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4⋊Dic47—41no—
184C4⋊Dic47—41no—
185Dic47⋊C4—416no—
186C4×Dic47—24no—
187C4×Dic47—22no—
188C4×Dic47C22×C414yescomplemented
189#10625329—24no—
190#10625329—22no—
191#10625329—22no—
192#10625329—22no—
193#10625329—42no—
194#10625329C22×C414yescomplemented
195C2×C47⋊C8—24no—
196C2×C47⋊C8—22no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—42no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—22no—
202C23×C47—24no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47C4×D412yes—
206C23×C47#12011yescharacteristic
207C23×C47#12011yescharacteristic
208C23×C47C22⋊Q812yes—
209C23×C47C22×D411yescharacteristic
210Q8×C47—22no—
211Q8×C47—24no—
212Q8×C47C2×C4⋊C414yescomplemented
213D4×C47—24no—
214D4×C47—24no—
215D4×C47—28no—
216D4×C47—22no—
217D4×C47—24no—
218D4×C47C2×C4⋊C414yescomplemented
219C2×C4×C47—22no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—28no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—44no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—44no—
231C2×C4×C47—82no—
232C2×C4×C47—88no—
233C2×C4×C47C2×C4⋊C412yes—
234C2×C4×C47C2×C4⋊C412yes—
235C2×C4×C47C4×D412yes—
236C2×C4×C47C4×D412yes—
237C2×C4×C47C4×D412yes—
238C2×C4×C47#12011yescharacteristic
239C2×C4×C47#12011yescharacteristic
240C2×C4×C47C22⋊Q812yes—
241C2×C4×C47C22⋊Q812yes—
242C2×C4×C47C22⋊Q812yes—
243C2×C4×C47#12312yes—
244C2×C4×C47#12312yes—
245C2×C4×C47C23×C411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C22×Q811yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×Dic47—24no—
252C2×Dic47—216no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—44no—
256C2×Dic47—42no—
257C2×Dic47—82no—
258C2×Dic47—82no—
259C47⋊C8—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—24no—
269C22×C47—48no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47#38512yes—
273C22×C47#38512yes—
274C22×C47#45312yes—
275C22×C47#45612yes—
276C22×C47C2×C4×D411yescharacteristic
277C22×C47C2×#12011yescharacteristic
278C22×C47C2×C22⋊Q811yescharacteristic
279C4×C47—24no—
280C4×C47—44no—
281C4×C47—42no—
282C4×C47—42no—
283C4×C47#38512yes—
284C4×C47#38512yes—
285C4×C47C22×C4⋊C411yescharacteristic
286C4×C47C2×C4×D411yescharacteristic
287C4×C47C2×C22⋊Q811yescharacteristic
288C4×C47C2×#12311yescharacteristic
289Dic47—28no—
290Dic47—82no—
291C2×#520D4711yescharacteristic
292C2×#428—471no—
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—44no—
303C2×C47#185914yescomplemented
304C2×C47C2×#38511yescharacteristic
305C2×C47C2×#45311yescharacteristic
306C2×C47C2×#45611yescharacteristic
307C22×#41C2×D4711yescharacteristic
308C22×M16—471no—
309C22×M16—471no—
310#520—22no—
311#520—22no—
312#520—22no—
313#520C2×D4714yes—
314C2×C4×Q8—21no—
315C2×C4×D4—21no—
316C2×C4×D4—22no—
317C2×#117—21no—
318C2×#117C2×D4711yescharacteristic
319C22×C4⋊C4—21no—
320C22×C4⋊C4—471no—
321C22×C4⋊C4—942no—
322C22×C4⋊C4C2×D4711yescharacteristic
323C22×C42—471no—
324#428—474no—
325#428—942no—
326C2×#107—941no—
327C2×#106—941no—
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—22no—
340C2×#41—22no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41Dic9414yes—
344C2×#41C4×D4714yes—
345C2×#41C22⋊D4712yes—
346C2×#41C22×D4712yes—
347C22×Q8C22⋊D4711yescharacteristic
348C22×D4—21no—
349C22×D4C22⋊D4711yescharacteristic
350C23×C4—21no—
351C23×C4—22no—
352C23×C4—941no—
353C23×C4—941no—
354C23×C4C22⋊D4712yes—
355C23×C4C22×D4711yescharacteristic
356C2×M16—474no—
357C2×M16—472no—
358C2×M16—472no—
359C2×M16—942no—
360C2×M16—941no—
361C2×M16—941no—
362C2×M16—942no—
363C2×M16—942no—
364C22×C8—941no—
365C22×C8—941no—
366C4×Q8—24no—
367C4×Q8—24no—
368C4×D4—28no—
369C4×D4—24no—
370C4×D4—24no—
371C4×D4—28no—
372#117—24no—
373#117—21no—
374#117—21no—
375#117—24no—
376#117C22⋊D4712yes—
377#117C22×D4712yes—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—472no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—948no—
392C2×C4⋊C4—1888no—
393C2×C4⋊C4C22⋊D4712yes—
394C2×C4⋊C4C22⋊D4711yescharacteristic
395C2×C4⋊C4C22⋊D4711yescharacteristic
396C2×C4⋊C4C22×D4712yes—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—21no—
399C2×C42—22no—
400C2×C42—21no—
401C2×C42—474no—
402C2×C42—472no—
403C2×C42—942no—
404C2×C42—941no—
405#107—944no—
406#106—944no—
407#104—4716no—
408#103—944no—
409#102—944no—
410C24—21no—
411C24C2×C22⋊D4711yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—22no—
419#41Dic47⋊C418yescomplemented
420#41C2×Dic9414yes—
421#41C2×C4×D4714yes—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8Dic47⋊C414yescomplemented
425C2×Q8C2×C22⋊D4712yes—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4Dic47⋊C414yescomplemented
433C2×D4C2×C22⋊D4712yes—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—84no—
446C22×C4—944no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—1884no—
452C22×C4—1881no—
453C22×C4—1881no—
454C22×C4C2×Dic9412yes—
455C22×C4C2×C4×D4712yes—
456C22×C4#1062535012yes—
457C22×C4#1062535012yes—
458C22×C4C2×C22⋊D4714yes—
459C22×C4C2×C22⋊D4711yescharacteristic
460C22×C4C2×C22⋊D4711yescharacteristic
461C22×C4C2×C22⋊D4711yescharacteristic
462C22×C4C2×C22⋊D4711yescharacteristic
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C23×D4711yescharacteristic
465M16—474no—
466M16—944no—
467M16—942no—
468M16—942no—
469M16—942no—
470M16—1882no—
471C2×C8—944no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—1882no—
475C4⋊C4—28no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—942no—
483C4⋊C4—942no—
484C4⋊C4—18816no—
485C4⋊C4—1881no—
486C4⋊C4—1881no—
487C4⋊C4C2×C22⋊D4712yes—
488C4⋊C4C2×C22⋊D4712yes—
489C22⋊C4—28no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C42—28no—
493C42—22no—
494C42—22no—
495C42—474no—
496C42—944no—
497C42—942no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23C4×C22⋊D4712yes—
505C23#1309399311yescharacteristic
506C23#1309399311yescharacteristic
507C23#1309400312yes—
508C23C22×C22⋊D4711yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8C2×Dic47⋊C414yescomplemented
512D4—28no—
513D4—24no—
514D4—24no—
515D4—24no—
516D4—22no—
517D4C2×Dic47⋊C414yescomplemented
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—82no—
531C2×C4—88no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—9416no—
535C2×C4—944no—
536C2×C4—3762no—
537C2×C4—3762no—
538C2×C4—1884no—
539C2×C4—1882no—
540C2×C4C2×Dic47⋊C412yes—
541C2×C4C2×Dic47⋊C412yes—
542C2×C4C4×C22⋊D4712yes—
543C2×C4C4×C22⋊D4712yes—
544C2×C4C4×C22⋊D4712yes—
545C2×C4#1309399311yescharacteristic
546C2×C4#1309399311yescharacteristic
547C2×C4#1309400312yes—
548C2×C4#1309400312yes—
549C2×C4#1309400312yes—
550C2×C4#1309400712yes—
551C2×C4#1309400712yes—
552C2×C4C22×C4×D4711yescharacteristic
553C2×C4C22×C22⋊D4711yescharacteristic
554C2×C4C22×C22⋊D4711yescharacteristic
555C2×C4C22×Dic9411yescharacteristic
556C2×C4C2×#1062535011yescharacteristic
557C2×C4C2×#1062535011yescharacteristic
558C8—944no—
559C8—1882no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22#1568899612yes—
571C22#1568899612yes—
572C22#1568929112yes—
573C22#1568930612yes—
574C22C2×C4×C22⋊D4711yescharacteristic
575C22C2×#1309399311yescharacteristic
576C22C2×#1309400311yescharacteristic
577C4—24no—
578C4—44no—
579C4—42no—
580C4—42no—
581C4—948no—
582C4—3762no—
583C4#1568899612yes—
584C4#1568899612yes—
585C4C22×Dic47⋊C411yescharacteristic
586C4C2×C4×C22⋊D4711yescharacteristic
587C4C2×#1309400311yescharacteristic
588C4C2×#1309400711yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—44no—
593C2#2024361414yesdirect product factor
594C2C2×#1568899611yescharacteristic
595C2C2×#1568929111yescharacteristic
596C2C2×#1568930611yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#520C2
socleC23×C47#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#520D47
hypocenterC47C2×#1859
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#15689166C2semidirect
4. C2×#15688992C2semidirect
5. #20243614C2direct (N × H)
6. C2×#15688892C2semidirect
7. C2×#15688882C2semidirect
15. #15689166C22semidirect
18. C2×#13093951C4semidirect
25. #15688992C22semidirect
26. #15688892C22semidirect
27. #15688882C22semidirect
47. C2×#10625329C2×C4semidirect
62. C2×C4×Dic47C2×C4semidirect
66. #13093951C2×C4semidirect
120. #41×C47C4⋊C4semidirect
125. C2×Q8×C47C4⋊C4semidirect
133. C2×D4×C47C4⋊C4semidirect
188. C4×Dic47C22×C4semidirect
194. #10625329C22×C4semidirect
212. Q8×C47C2×C4⋊C4semidirect
218. D4×C47C2×C4⋊C4semidirect
303. C2×C47#1859semidirect
338. C47C2×#1859semidirect
419. #41Dic47⋊C4semidirect
424. C2×Q8Dic47⋊C4semidirect
432. C2×D4Dic47⋊C4semidirect
511. Q8C2×Dic47⋊C4semidirect
517. D4C2×Dic47⋊C4semidirect
593. C2#20243614direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156891662
C22×#130939512
C2×#156891242
C2×#156889922
#202436142
C2×#156888922
C2×#156888822
C2×C47×#5202
C2×#185947