Finite Group Databasebeta

← C2×#20244583 #26243733

Subgroups — C2×#20244583 #26243733

Order 12032.

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×#13094000C211yesmaximal, characteristic
2C22×#13093956C211yesmaximal, characteristic, complemented
3C22×#13093948C211yesmaximal, characteristic, complemented
4C22×(C2×D47)⋊C8C211yesmaximal, characteristic
5C2×#15688980C211yesmaximal, characteristic
6#20244583C218yesmaximal, direct product factor
7C2×#15688885C211yesmaximal, characteristic, complemented
8C22×#103×C47C211yesmaximal, characteristic, complemented
9C23×C4×D47C2211yescharacteristic
10C23×C47⋊C8C2211yescharacteristic
11C2×#13094000—22no—
12C2×#13094000—24no—
13C2×#13094000—24no—
14C2×#13094000C2214yes—
15C22×C4⋊Dic47C2211yescharacteristic
16C22×C4⋊Dic47C2211yescharacteristic
17C22×Dic47⋊C4—21no—
18C22×(C2×D47)⋊C4—21no—
19C22×(C2×D47)⋊C4—21no—
20C2×#13093956—24no—
21C2×#13093956C2214yescomplemented
22C2×#13093948—24no—
23C2×#13093948C2214yescomplemented
24C2×(C2×D47)⋊C8—24no—
25C2×(C2×D47)⋊C8C2214yes—
26#15688980—22no—
27#15688980C2214yes—
28C2×C47⋊#95—21no—
29#15688885C2214yescomplemented
30C23×Q8×C47C2211yescharacteristic
31C23×C8×C47C2211yescharacteristic
32C22×C4⋊C4×C47C2211yescharacteristic
33C2×#103×C47—24no—
34C2×#103×C47C2214yescomplemented
35C24×D47D411yescharacteristic
36C23×Dic47—21no—
37C23×Dic47—21no—
38C23×Dic47D411yescharacteristic
39C22×C4×D47—22no—
40C22×C4×D47—22no—
41C22×C4×D47—22no—
42C22×C4×D47—22no—
43C22×C4×D47—22no—
44C22×C4×D47—22no—
45C22×C4×D47D412yes—
46C22×C4×D47C2312yes—
47C22×C47⋊C8—22no—
48C22×C47⋊C8—22no—
49C22×C47⋊C8—22no—
50C22×C47⋊C8C2312yes—
51#13094000—216no—
52#13094000—28no—
53#13094000—28no—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47D414yes—
60C2×C4⋊Dic47C2312yes—
61C2×C4⋊Dic47C2312yes—
62C2×Dic47⋊C4—28no—
63C2×Dic47⋊C4—24no—
64C2×(C2×D47)⋊C4—28no—
65C2×(C2×D47)⋊C4—24no—
66C2×(C2×D47)⋊C4—24no—
67C2×(C2×D47)⋊C4—44no—
68#13093956—28no—
69#13093948—28no—
70(C2×D47)⋊C8—28no—
71C47⋊#95—24no—
72C22×Q8×C47—22no—
73C22×Q8×C47—21no—
74C22×Q8×C47—22no—
75C22×Q8×C47—22no—
76C22×Q8×C47—22no—
77C22×Q8×C47—22no—
78C22×Q8×C47D414yes—
79C22×Q8×C47C2312yes—
80C23×C4×C47—21no—
81C23×C4×C47—21no—
82C23×C4×C47C2311yescharacteristic
83C22×C8×C47—22no—
84C22×C8×C47—22no—
85C22×C8×C47D414yes—
86C22×C8×C47C2312yes—
87C2×C4⋊C4×C47—22no—
88C2×C4⋊C4×C47—22no—
89C2×C4⋊C4×C47—21no—
90C2×C4⋊C4×C47C2312yes—
91#103×C47—28no—
92C24×C47C2×D411yescharacteristic
93C23×D47—22no—
94C23×D47—22no—
95C23×D47—22no—
96C23×D47—22no—
97C23×D47—22no—
98C23×D47—22no—
99C23×D47SD1614yes—
100C23×D47C2×D412yes—
101C2×Q8×C47—24no—
102C2×Q8×C47—22no—
103C2×Q8×C47—28no—
104C2×Q8×C47—28no—
105C2×Q8×C47—22no—
106C2×Q8×C47—22no—
107C2×Q8×C47—22no—
108C2×Q8×C47—22no—
109C2×Q8×C47—42no—
110C2×Q8×C47—44no—
111C2×Q8×C47—42no—
112C2×Q8×C47—44no—
113C2×Q8×C47C2×D414yes—
114C22×C4×C47—24no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—42no—
118C22×C4×C47—42no—
119C22×C4×C47—42no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×D412yes—
126C22×C4×C47#4112yes—
127C22×C4×C47#4112yes—
128C22×C4×C47#4112yes—
129C22×C4×C47C2411yescharacteristic
130C22×Dic47—24no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C22×Dic47—42no—
140C22×Dic47—42no—
141C22×Dic47C2×D412yes—
142C2×C4×D47—24no—
143C2×C4×D47—24no—
144C2×C4×D47—24no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—44no—
152C2×C4×D47—44no—
153C2×C4×D47—44no—
154C2×C4×D47—44no—
155C2×C8×C47—28no—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47C2×D414yes—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—42no—
162(C2×D47)⋊C4—216no—
163(C2×D47)⋊C4—48no—
164C4⋊Dic47—24no—
165C4⋊Dic47—22no—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C4⋊Dic47—42no—
169C4⋊Dic47C2×D414yes—
170Dic47⋊C4—216no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—44no—
175C23×C47#12012yes—
176C23×C47#12112yes—
177C23×C47#12312yes—
178C23×C47C2×SD1612yes—
179C23×C47#13512yes—
180C23×C47#13712yes—
181C23×C47#13712yes—
182C23×C47C22×D411yescharacteristic
183C22×D47—22no—
184C22×D47—22no—
185C22×D47—22no—
186C22×D47—22no—
187C22×D47—22no—
188C22×D47—22no—
189C22×D47—28no—
190C22×D47—28no—
191C22×D47—28no—
192C22×D47—44no—
193C22×D47—44no—
194C22×D47—44no—
195C22×D47—44no—
196C22×D47C2×SD1614yes—
197Q8×C47—24no—
198Q8×C47—24no—
199Q8×C47—42no—
200Q8×C47—42no—
201Q8×C47—48no—
202C2×C4×C47—24no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—48no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—84no—
210C2×C4×C47#12012yes—
211C2×C4×C47#12112yes—
212C2×C4×C47#12114yes—
213C2×C4×C47#12114yes—
214C2×C4×C47#12312yes—
215C2×C4×C47#12314yes—
216C2×C4×C47#12414yes—
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×Dic47—22no—
224C2×Dic47—24no—
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—44no—
230C2×Dic47—48no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—84no—
235C4×D47—42no—
236C4×D47—42no—
237C4×D47—42no—
238C4×D47—42no—
239C4×D47—48no—
240C8×C47—24no—
241C47⋊C8—42no—
242C2×#2035—471nomaximal, Sylow
243C22×C47#39414yes—
244C22×C47#44814yes—
245C22×C47#45214yes—
246C22×C47#46514yes—
247C22×C47#47014yes—
248C22×C47#48314yes—
249C22×C47#48514yes—
250C22×C47C2×#12011yescharacteristic
251C22×C47C2×#12111yescharacteristic
252C22×C47C2×#12311yescharacteristic
253C22×C47C22×SD1611yescharacteristic
254C22×C47C2×#13511yescharacteristic
255C22×C47C2×#13711yescharacteristic
256C22×C47C2×#13711yescharacteristic
257C2×D47—24no—
258C2×D47—24no—
259C2×D47—24no—
260C2×D47—42no—
261C2×D47—42no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—416no—
265C4×C47—44no—
266C4×C47—82no—
267C4×C47#39414yes—
268C4×C47C2×#12111yescharacteristic
269C4×C47C2×#12111yescharacteristic
270C4×C47C2×#12311yescharacteristic
271C4×C47C2×#12411yescharacteristic
272Dic47—42no—
273Dic47—44no—
274Dic47—82no—
275C22×C22⋊Q8—471no—
276C22×#106—471no—
277C22×#103—471no—
278C22×#103D4711yescharacteristic, complemented
279C22×C22⋊C8—471no—
280C2×#380—471no—
281#2035—478no—
282C2×#340—471no—
283C2×C47#203518yescomplemented
284C2×C47C2×#39411yescharacteristic
285C2×C47C2×#44811yescharacteristic
286C2×C47C2×#45211yescharacteristic
287C2×C47C2×#46511yescharacteristic
288C2×C47C2×#47011yescharacteristic
289C2×C47C2×#48311yescharacteristic
290C2×C47C2×#48511yescharacteristic
291D47—44no—
292C23×Q8C2×D4711yescharacteristic
293C24×C4—471no—
294C23×C8—471no—
295C23×C8C2×D4711yescharacteristic
296C2×C22⋊Q8—474no—
297C2×C22⋊Q8—944no—
298C2×C22⋊Q8—942no—
299C2×C22⋊Q8—944no—
300C22×C4⋊C4—471no—
301C22×C4⋊C4—471no—
302C22×C4⋊C4—941no—
303C22×C4⋊C4C2×D4711yescharacteristic
304C22×C22⋊C4—941no—
305C22×C22⋊C4—941no—
306C2×#106—474no—
307C2×#106—944no—
308C2×#103—24no—
309C2×#103—474no—
310C2×#103—944no—
311C2×#103C2×D4714yescomplemented
312C2×C22⋊C8—474no—
313C2×C22⋊C8—944no—
314#380—474no—
315#380—942no—
316C2×#95—941no—
317#340—474no—
318C47C2×#203511yescharacteristic, complemented, Sylow
319C25—471no—
320C22×Q8—22no—
321C22×Q8—21no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8D18814yes—
327C22×Q8C22×D4712yes—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—471no—
331C23×C4—472no—
332C23×C4—472no—
333C23×C4—941no—
334C23×C4—942no—
335C23×C4—942no—
336C23×C4—942no—
337C23×C4—942no—
338C23×C4—942no—
339C23×C4—942no—
340C23×C4—941no—
341C23×C4C22×D4711yescharacteristic
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—472no—
345C22×C8—942no—
346C22×C8—942no—
347C22×C8—942no—
348C22×C8C22⋊D4714yes—
349C22×C8C22×D4712yes—
350C22⋊Q8—948no—
351C22⋊Q8—9416no—
352C22⋊Q8—948no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—474no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—472no—
359C2×C4⋊C4—944no—
360C2×C4⋊C4—948no—
361C2×C4⋊C4—942no—
362C2×C4⋊C4—941no—
363C2×C4⋊C4—942no—
364C2×C4⋊C4—942no—
365C2×C4⋊C4—941no—
366C2×C4⋊C4C22×D4712yes—
367C2×C22⋊C4—944no—
368C2×C22⋊C4—948no—
369C2×C22⋊C4—944no—
370C2×C22⋊C4—1884no—
371#106—948no—
372#103—28no—
373#103—948no—
374C22⋊C8—948no—
375#95—944no—
376C24—474no—
377C24—472no—
378C24—942no—
379C24—942no—
380C24—942no—
381C24—942no—
382C24—942no—
383C24—942no—
384C24D4×D4711yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×D18814yes—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—472no—
407C22×C4—944no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—944no—
413C22×C4—944no—
414C22×C4—944no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—1882no—
423C22×C4—1882no—
424C22×C4—1882no—
425C22×C4—1882no—
426C22×C4—1882no—
427C22×C4—1884no—
428C22×C4—1884no—
429C22×C4—1884no—
430C22×C4—1884no—
431C22×C4C2×D18812yes—
432C22×C4#1062535012yes—
433C22×C4D4×D4711yescharacteristic
434C22×C4D4×D4712yes—
435C22×C4#1062535212yes—
436C22×C4#1062535212yes—
437C22×C4C2×C22⋊D4712yes—
438C22×C4C23×D4711yescharacteristic
439C2×C8—28no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—942no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—1884no—
446C2×C8C2×C22⋊D4714yes—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—474no—
451C4⋊C4—9416no—
452C4⋊C4—944no—
453C4⋊C4—942no—
454C4⋊C4—942no—
455C4⋊C4—942no—
456C4⋊C4—1882no—
457C22⋊C4—9416no—
458C22⋊C4—1888no—
459C23—474no—
460C23—948no—
461C23—948no—
462C23—942no—
463C23—942no—
464C23—942no—
465C23—942no—
466C23—942no—
467C23—942no—
468C23—948no—
469C23—1884no—
470C23—1884no—
471C23—1884no—
472C23—1884no—
473C23#1309399412yes—
474C23#1309399512yes—
475C23#1309400612yes—
476C23SD16×D4712yes—
477C23#1309404512yes—
478C23#1309405612yes—
479C23#1309405612yes—
480C23C2×D4×D4711yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—944no—
495C2×C4—942no—
496C2×C4—942no—
497C2×C4—942no—
498C2×C4—3764no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1888no—
503C2×C4—1884no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1888no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1882no—
511C2×C4#1309399412yes—
512C2×C4#1309399512yes—
513C2×C4#1309399614yes—
514C2×C4#1309399814yes—
515C2×C4#1309400612yes—
516C2×C4#1309400814yes—
517C2×C4#1309401014yes—
518C2×C4C2×D4×D4711yescharacteristic
519C2×C4C22×C22⋊D4711yescharacteristic
520C2×C4C22×D18811yescharacteristic
521C2×C4C2×#1062535211yescharacteristic
522C2×C4C2×#1062535211yescharacteristic
523C2×C4C2×#1062535011yescharacteristic
524C8—24no—
525C8—1882no—
526C22—944no—
527C22—944no—
528C22—944no—
529C22—1882no—
530C22—1882no—
531C22—1882no—
532C22—18816no—
533C22—1882no—
534C22#1568903914yes—
535C22#1568926314yes—
536C22#1568928814yes—
537C22#1568934914yes—
538C22#1568937814yes—
539C22#1568945214yes—
540C22#1568946214yes—
541C22C2×#1309399411yescharacteristic
542C22C2×#1309399511yescharacteristic
543C22C2×#1309400611yescharacteristic
544C22C2×SD16×D4711yescharacteristic
545C22C2×#1309404511yescharacteristic
546C22C2×#1309405611yescharacteristic
547C22C2×#1309405611yescharacteristic
548C4—44no—
549C4—82no—
550C4—3762no—
551C4—1882no—
552C4—1884no—
553C4#1568903914yes—
554C4C2×#1309399611yescharacteristic
555C4C2×#1309399811yescharacteristic
556C4C2×#1309400811yescharacteristic
557C4C2×#1309401011yescharacteristic
558C2—1884no—
559C2#2024458318yesdirect product factor
560C2C2×#1568903911yescharacteristic
561C2C2×#1568926311yescharacteristic
562C2C2×#1568928811yescharacteristic
563C2C2×#1568934911yescharacteristic
564C2C2×#1568937811yescharacteristic
565C2C2×#1568945211yescharacteristic
566C2C2×#1568946211yescharacteristic
subgroupG/H
centerC24D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×#103×C47C2
socleC24×C47C2×D4
prime radicalC24×D47D4
hypercenterC22×#103D47
hypocenterC47C2×#2035
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C22×#13093956C2semidirect
3. C22×#13093948C2semidirect
6. #20244583C2direct (N × H)
7. C2×#15688885C2semidirect
8. C22×#103×C47C2semidirect
21. C2×#13093956C22semidirect
23. C2×#13093948C22semidirect
29. #15688885C22semidirect
34. C2×#103×C47C22semidirect
278. C22×#103D47semidirect
283. C2×C47#2035semidirect
311. C2×#103C2×D47semidirect
318. C47C2×#2035semidirect
559. C2#20244583direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940002
C22×#130939562
C22×#130939482
C22×(C2×D47)⋊C82
C2×#156889802
#202445832
C2×#156888852
C22×#103×C472
C2×#203547