Finite Group Databasebeta

← C2×#20244577 #26243742

Subgroups — C2×#20244577 #26243742

Order 12032.

2006 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093998C211yesmaximal, characteristic
2C22×#13093955C211yesmaximal, characteristic, complemented
3C22×#13093944C211yesmaximal, characteristic, complemented
4C22×#13093944C211yesmaximal, characteristic, complemented
5C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
6#20244577C218yesmaximal, direct product factor
7C2×#15688885C211yesmaximal, characteristic, complemented
8C2×C47×#380C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C23×C22⋊D47—21no—
11C23×C47⋊C8C2211yescharacteristic
12C23×C47⋊C8C2211yescharacteristic
13C2×#13093998—24no—
14C2×#13093998—24no—
15C2×#13093998—22no—
16C2×#13093998C2214yes—
17C22×C4⋊Dic47C2211yescharacteristic
18C22×(C2×D47)⋊C4—21no—
19C2×#13093955—24no—
20C2×#13093955C2214yescomplemented
21C2×#13093944—24no—
22C2×#13093944—24no—
23C2×#13093944C2214yescomplemented
24C2×#13093944C2214yescomplemented
25C2×C22⋊(C47⋊C8)—24no—
26C2×C22⋊(C47⋊C8)C2214yescomplemented
27#15688885C2214yescomplemented
28C24×C4×C47C2211yescharacteristic
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47—21no—
32C47×#380—22no—
33C47×#380C2214yescomplemented
34C2×#95×C47—21no—
35C24×D47—21no—
36C22×D188—22no—
37C22×D188—22no—
38C22×D188—22no—
39C22×D188—22no—
40C22×D188—21no—
41C22×D188—22no—
42C22×D188D414yes—
43C22×D188C2312yes—
44C22×C22⋊D47—28no—
45C22×C22⋊D47—24no—
46C22×C22⋊D47—44no—
47C22×C22⋊D47—44no—
48C23×Dic47—21no—
49C22×C47⋊C8—22no—
50C22×C47⋊C8—22no—
51C22×C47⋊C8—22no—
52C22×C47⋊C8—22no—
53C22×C47⋊C8—22no—
54C22×C47⋊C8D414yescomplemented
55C22×C47⋊C8C2312yes—
56C22×C47⋊C8C2312yes—
57#13093998—216no—
58#13093998—28no—
59#13093998—28no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47—22no—
62C2×C4⋊Dic47D414yes—
63C2×C4⋊Dic47C2312yes—
64C2×(C2×D47)⋊C4—28no—
65C2×(C2×D47)⋊C4—24no—
66#13093955—28no—
67#13093944—28no—
68#13093944—28no—
69C22⋊(C47⋊C8)—28no—
70C25×C47D411yescharacteristic
71C23×C4×C47—22no—
72C23×C4×C47—22no—
73C23×C4×C47—22no—
74C23×C4×C47—22no—
75C23×C4×C47—22no—
76C23×C4×C47—22no—
77C23×C4×C47—21no—
78C23×C4×C47—21no—
79C23×C4×C47D411yescharacteristic
80C23×C4×C47D412yes—
81C23×C4×C47C2312yes—
82C23×C4×C47C2311yescharacteristic
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47—22no—
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47—21no—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47C2312yes—
90C2×C4⋊C4×C47C2312yes—
91C2×C22⋊C4×C47—24no—
92C2×C22⋊C4×C47—44no—
93#95×C47—24no—
94C24×C47—22no—
95C24×C47—22no—
96C24×C47—22no—
97C24×C47—22no—
98C24×C47—22no—
99C24×C47—22no—
100C24×C47SD1614yes—
101C24×C47C2×D412yes—
102C24×C47C2×D411yescharacteristic
103C23×D47—24no—
104C23×D47—22no—
105C23×D47—42no—
106C23×D47—42no—
107C23×D47—42no—
108C23×D47—42no—
109C23×D47—42no—
110C23×D47—42no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—24no—
115C22×C4×C47—24no—
116C22×C4×C47—24no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—44no—
126C22×C4×C47—44no—
127C22×C4×C47—44no—
128C22×C4×C47—44no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47#4112yes—
141C22×C4×C47#4112yes—
142C22×C4×C47#4112yes—
143C22×C4×C47C2411yescharacteristic
144C2×C22⋊D47—216no—
145C2×C22⋊D47—48no—
146C2×C22⋊D47—432no—
147C2×C22⋊D47—48no—
148C22×Dic47—24no—
149C22×Dic47—22no—
150C22×Dic47—42no—
151C22×Dic47—42no—
152C2×D188—28no—
153C2×D188—28no—
154C2×D188—22no—
155C2×D188—22no—
156C2×D188—22no—
157C2×D188—22no—
158C2×D188—24no—
159C2×D188—22no—
160C2×D188—44no—
161C2×D188—42no—
162C2×D188—42no—
163C2×D188—44no—
164C2×D188C2×D414yes—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—42no—
170C4⋊C4×C47—42no—
171C22⋊C4×C47—48no—
172(C2×D47)⋊C4—216no—
173C4⋊Dic47—24no—
174C4⋊Dic47—22no—
175C4⋊Dic47C2×D414yes—
176C2×C47⋊C8—28no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—44no—
183C2×C47⋊C8C2×D414yescomplemented
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—28no—
191C23×C47—28no—
192C23×C47—28no—
193C23×C47—44no—
194C23×C47—44no—
195C23×C47—44no—
196C23×C47—44no—
197C23×C47#12012yes—
198C23×C47#12112yes—
199C23×C47#12312yes—
200C23×C47C2×SD1614yes—
201C23×C47C2×SD1612yes—
202C23×C47#13512yes—
203C23×C47AGL(1,ℤ/8)12yes—
204C23×C47AGL(1,ℤ/8)12yes—
205C23×C47C22×D411yescharacteristic
206C22×D47—24no—
207C22×D47—42no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47—48no—
214C22×D47—48no—
215C22×D47—48no—
216C22×D47—84no—
217C22×D47—84no—
218C22×D47—84no—
219C22×D47—84no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—44no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—48no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—84no—
236C2×C4×C47—84no—
237C2×C4×C47#12012yes—
238C2×C4×C47#12112yes—
239C2×C4×C47#12114yes—
240C2×C4×C47#12114yes—
241C2×C4×C47#12314yes—
242C2×C4×C47#12312yes—
243C2×C4×C47#12414yes—
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C22⋊D47—432no—
251C2×Dic47—24no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—48no—
255D188—24no—
256D188—24no—
257D188—42no—
258D188—42no—
259D188—48no—
260C47⋊C8—24no—
261C47⋊C8—42no—
262C2×#2034—471nomaximal, Sylow
263C22×C47—24no—
264C22×C47—24no—
265C22×C47—24no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—416no—
271C22×C47#39414yes—
272C22×C47#44914yes—
273C22×C47#45014yes—
274C22×C47#46514yes—
275C22×C47#46914yes—
276C22×C47#48114yes—
277C22×C47#48214yes—
278C22×C47C2×#12011yescharacteristic
279C22×C47C2×#12111yescharacteristic
280C22×C47C2×#12311yescharacteristic
281C22×C47C22×SD1611yescharacteristic
282C22×C47C2×#13511yescharacteristic
283C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
285C2×D47—44no—
286C2×D47—44no—
287C2×D47—44no—
288C2×D47—82no—
289C2×D47—82no—
290C2×D47—82no—
291C2×D47—82no—
292C2×D47—816no—
293C4×C47—42no—
294C4×C47—82no—
295C4×C47—82no—
296C4×C47#39414yes—
297C4×C47C2×#12111yescharacteristic
298C4×C47C2×#12111yescharacteristic
299C4×C47C2×#12311yescharacteristic
300C4×C47C2×#12411yescharacteristic
301Dic47—44no—
302C22×#121—471no—
303C22×#106—471no—
304C22×#102—471no—
305C22×#102—471no—
306C22×C22⋊C8—471no—
307C2×#380D4711yescharacteristic, complemented
308#2034—478no—
309C2×#340—471no—
310C2×C47—44no—
311C2×C47#203418yescomplemented
312C2×C47C2×#39411yescharacteristic
313C2×C47C2×#45011yescharacteristic
314C2×C47C2×#44911yescharacteristic
315C2×C47C2×#46511yescharacteristic
316C2×C47C2×#46911yescharacteristic
317C2×C47C2×#48211yescharacteristic
318C2×C47C2×#48111yescharacteristic
319D47—84no—
320C23×D4—471no—
321C23×D4—941no—
322C24×C4C2×D4711yescharacteristic
323C23×C8—471no—
324C23×C8—471no—
325C2×#121—474no—
326C2×#121—944no—
327C2×#121—944no—
328C2×#121—942no—
329C22×C4⋊C4—471no—
330C22×C4⋊C4C2×D4711yescharacteristic
331C22×C4⋊C4C2×D4711yescharacteristic
332C22×C22⋊C4—21no—
333C22×C22⋊C4—941no—
334C2×#106—474no—
335C2×#106—944no—
336C2×#102—474no—
337C2×#102—474no—
338C2×#102—944no—
339C2×#102—944no—
340C2×C22⋊C8—474no—
341C2×C22⋊C8—944no—
342#380—22no—
343#380C2×D4714yescomplemented
344C2×#95—21no—
345#340—474no—
346C47C2×#203411yescharacteristic, complemented, Sylow
347C25—941no—
348C25C22⋊D4711yescharacteristic
349C22×D4—474no—
350C22×D4—472no—
351C22×D4—948no—
352C22×D4—944no—
353C22×D4—942no—
354C22×D4—942no—
355C22×D4—942no—
356C22×D4—942no—
357C22×D4—942no—
358C22×D4—941no—
359C22×D4—1884no—
360C22×D4—1884no—
361C23×C4—22no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—941no—
370C23×C4C22⋊D4712yes—
371C23×C4C22⋊D4711yescharacteristic
372C23×C4C22×D4712yes—
373C23×C4C22×D4711yescharacteristic
374C22×C8—474no—
375C22×C8—472no—
376C22×C8—472no—
377C22×C8—942no—
378C22×C8—942no—
379C22×C8—942no—
380C22×C8—942no—
381C22×C8—942no—
382#121—948no—
383#121—948no—
384#121—9416no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—474no—
392C2×C4⋊C4—472no—
393C2×C4⋊C4—942no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4C22×D4712yes—
396C2×C4⋊C4C22×D4712yes—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—44no—
399C2×C22⋊C4—948no—
400C2×C22⋊C4—944no—
401#106—948no—
402#102—948no—
403#102—948no—
404C22⋊C8—948no—
405#95—24no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—944no—
413C24—942no—
414C24—1882no—
415C24—1882no—
416C24—1882no—
417C24—1882no—
418C24—1882no—
419C24—1882no—
420C24#1062533614yes—
421C24C2×C22⋊D4712yes—
422C24C2×C22⋊D4711yescharacteristic
423C2×D4—474no—
424C2×D4—9416no—
425C2×D4—944no—
426C2×D4—948no—
427C2×D4—948no—
428C2×D4—942no—
429C2×D4—942no—
430C2×D4—942no—
431C2×D4—942no—
432C2×D4—942no—
433C2×D4—18832no—
434C2×D4—1888no—
435C2×D4—1888no—
436C2×D4—1882no—
437C2×D4—1884no—
438C2×D4—1882no—
439C2×D4—1884no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—944no—
465C22×C4—942no—
466C22×C4—1882no—
467C22×C4—1882no—
468C22×C4#1062535012yes—
469C22×C4#1062535012yes—
470C22×C4D4×D4712yes—
471C22×C4D4×D4712yes—
472C22×C4D4×D4712yes—
473C22×C4#1062535212yes—
474C22×C4C2×C22⋊D4712yes—
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C23×D4711yescharacteristic
477C2×C8—474no—
478C2×C8—948no—
479C2×C8—942no—
480C2×C8—942no—
481C2×C8—942no—
482C2×C8—942no—
483C2×C8—942no—
484C2×C8—1884no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—474no—
492C4⋊C4—944no—
493C4⋊C4—942no—
494C22⋊C4—48no—
495C22⋊C4—9416no—
496C23—28no—
497C23—22no—
498C23—28no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—944no—
510C23—3764no—
511C23—3764no—
512C23—3764no—
513C23—3764no—
514C23—1888no—
515C23—1888no—
516C23—1888no—
517C23—1882no—
518C23—1882no—
519C23—1882no—
520C23—1882no—
521C23—1882no—
522C23—1882no—
523C23#1309399212yes—
524C23#1309399612yes—
525C23#1309400712yes—
526C23C2×#1062533614yes—
527C23C2×#1062533612yes—
528C23#1309404412yes—
529C23#1309404912yes—
530C23#1309405012yes—
531C23C22×C22⋊D4711yescharacteristic
532D4—944no—
533D4—944no—
534D4—18832no—
535D4—1888no—
536D4—1882no—
537D4—1882no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—944no—
556C2×C4—1888no—
557C2×C4—1882no—
558C2×C4—1882no—
559C2×C4#1309399212yes—
560C2×C4#1309399612yes—
561C2×C4#1309399714yes—
562C2×C4#1309399714yes—
563C2×C4#1309400614yes—
564C2×C4#1309400712yes—
565C2×C4#1309401014yes—
566C2×C4C2×D4×D4711yescharacteristic
567C2×C4C2×D4×D4711yescharacteristic
568C2×C4C2×D4×D4711yescharacteristic
569C2×C4C2×#1062535211yescharacteristic
570C2×C4C2×#1062535011yescharacteristic
571C2×C4C2×#1062535011yescharacteristic
572C8—944no—
573C8—1882no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—416no—
581C22—42no—
582C22—3762no—
583C22—3762no—
584C22—3762no—
585C22—37616no—
586C22—3762no—
587C22—1884no—
588C22—1884no—
589C22—1884no—
590C22#1568903814yes—
591C22#1568926614yes—
592C22#1568927314yes—
593C22#1568935414yes—
594C22#1568937414yes—
595C22#1568944114yes—
596C22#1568944714yes—
597C22C2×#1309399211yescharacteristic
598C22C2×#1309399611yescharacteristic
599C22C2×#1309400711yescharacteristic
600C22C22×#1062533611yescharacteristic
601C22C2×#1309404411yescharacteristic
602C22C2×#1309404911yescharacteristic
603C22C2×#1309405011yescharacteristic
604C4—42no—
605C4—82no—
606C4—82no—
607C4—1884no—
608C4#1568903814yes—
609C4C2×#1309399711yescharacteristic
610C4C2×#1309399711yescharacteristic
611C4C2×#1309400611yescharacteristic
612C4C2×#1309401011yescharacteristic
613C2—44no—
614C2—3764no—
615C2#2024457718yesdirect product factor
616C2C2×#1568903811yescharacteristic
617C2C2×#1568927311yescharacteristic
618C2C2×#1568926611yescharacteristic
619C2C2×#1568935411yescharacteristic
620C2C2×#1568937411yescharacteristic
621C2C2×#1568944711yescharacteristic
622C2C2×#1568944111yescharacteristic
subgroupG/H
centerC24C2×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#380C2
socleC24×C47C2×D4
prime radicalC22×#13093998C2
hypercenterC2×#380D47
hypocenterC47C2×#2034
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#13093955C2semidirect
3. C22×#13093944C2semidirect
4. C22×#13093944C2semidirect
5. C22×C22⋊(C47⋊C8)C2semidirect
6. #20244577C2direct (N × H)
7. C2×#15688885C2semidirect
8. C2×C47×#380C2semidirect
20. C2×#13093955C22semidirect
23. C2×#13093944C22semidirect
24. C2×#13093944C22semidirect
26. C2×C22⋊(C47⋊C8)C22semidirect
27. #15688885C22semidirect
33. C47×#380C22semidirect
54. C22×C47⋊C8D4semidirect
183. C2×C47⋊C8C2×D4semidirect
307. C2×#380D47semidirect
311. C2×C47#2034semidirect
343. #380C2×D47semidirect
346. C47C2×#2034semidirect
615. C2#20244577direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939982
C22×#130939552
C22×#130939442
C22×C22⋊(C47⋊C8)2
#202445772
C2×#156888852
C2×C47×#3802
C2×#203447