Finite Group Databasebeta

← C2×#20244581 #26243731

Subgroups — C2×#20244581 #26243731

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×#13093955C211yesmaximal, characteristic, complemented
2C22×#13093948C211yesmaximal, characteristic, complemented
3C22×#13093948C211yesmaximal, characteristic, complemented
4C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic
5C2×#15688983C211yesmaximal, characteristic
6#20244581C218yesmaximal, direct product factor
7C2×#15688885C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8×C47C211yesmaximal, characteristic
9C23×C47⋊C8C2211yescharacteristic
10C23×C47⋊C8C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C22⋊Dic47—21no—
14C2×#13093955—24no—
15C2×#13093955C2214yescomplemented
16C2×#13093948—24no—
17C2×#13093948—24no—
18C2×#13093948C2214yescomplemented
19C2×#13093948C2214yescomplemented
20C2×C22⋊(C47⋊C8)—24no—
21C2×C22⋊(C47⋊C8)C2214yes—
22#15688983—22no—
23#15688983C2214yes—
24C2×C47⋊#95—21no—
25#15688885C2214yescomplemented
26C23×Q8×C47C2211yescharacteristic
27C24×C4×C47C2211yescharacteristic
28C2×C22⋊Q8×C47—24no—
29C2×C22⋊Q8×C47—22no—
30C2×C22⋊Q8×C47—24no—
31C2×C22⋊Q8×C47C2214yes—
32C22×C4⋊C4×C47—21no—
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C22⋊C4×C47—21no—
35C23×Dic47—21no—
36C23×Dic47—21no—
37C22×C47⋊C8—22no—
38C22×C47⋊C8—22no—
39C22×C47⋊C8—22no—
40C22×C47⋊C8—22no—
41C22×C47⋊C8—22no—
42C22×C47⋊C8D414yes—
43C22×C47⋊C8C2312yes—
44C22×C47⋊C8C2312yes—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47C2312yes—
52C2×C4⋊Dic47C2312yes—
53C2×C22⋊Dic47—24no—
54C2×C22⋊Dic47—44no—
55#13093955—28no—
56#13093948—28no—
57#13093948—28no—
58C22⋊(C47⋊C8)—28no—
59C47⋊#95—24no—
60C25×C47D411yescharacteristic
61C22×Q8×C47—22no—
62C22×Q8×C47—21no—
63C22×Q8×C47—22no—
64C22×Q8×C47—22no—
65C22×Q8×C47—22no—
66C22×Q8×C47—22no—
67C22×Q8×C47D414yes—
68C22×Q8×C47C2312yes—
69C23×C4×C47—21no—
70C23×C4×C47—21no—
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×C47D411yescharacteristic
78C23×C4×C47D412yes—
79C23×C4×C47C2312yes—
80C23×C4×C47C2311yescharacteristic
81C22⋊Q8×C47—216no—
82C22⋊Q8×C47—28no—
83C22⋊Q8×C47—28no—
84C2×C4⋊C4×C47—28no—
85C2×C4⋊C4×C47—24no—
86C2×C4⋊C4×C47—21no—
87C2×C4⋊C4×C47—22no—
88C2×C4⋊C4×C47D414yes—
89C2×C4⋊C4×C47C2312yes—
90C2×C22⋊C4×C47—28no—
91C2×C22⋊C4×C47—24no—
92C24×C47—22no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—22no—
96C24×C47—22no—
97C24×C47—22no—
98C24×C47SD1614yes—
99C24×C47C2×D412yes—
100C24×C47C2×D411yescharacteristic
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—24no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—24no—
122C22×C4×C47—24no—
123C22×C4×C47—24no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—44no—
135C22×C4×C47—44no—
136C22×C4×C47—44no—
137C22×C4×C47—44no—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D412yes—
143C22×C4×C47#4112yes—
144C22×C4×C47#4112yes—
145C22×C4×C47#4112yes—
146C22×C4×C47C2411yescharacteristic
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—42no—
150C22×Dic47—42no—
151C22×Dic47—42no—
152C22×Dic47—42no—
153C22×Dic47—42no—
154C22×Dic47—42no—
155C4⋊C4×C47—216no—
156C4⋊C4×C47—24no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47C2×D414yes—
159C22⋊C4×C47—216no—
160C22⋊Dic47—48no—
161C4⋊Dic47—22no—
162C4⋊Dic47—22no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—42no—
166C4⋊Dic47—42no—
167C2×C47⋊C8—28no—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—22no—
173C2×C47⋊C8—44no—
174C2×C47⋊C8C2×D414yes—
175C23×C47—22no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—28no—
182C23×C47—28no—
183C23×C47—28no—
184C23×C47—44no—
185C23×C47—44no—
186C23×C47—44no—
187C23×C47—44no—
188C23×C47#12012yes—
189C23×C47#12112yes—
190C23×C47#12312yes—
191C23×C47C2×SD1614yes—
192C23×C47C2×SD1612yes—
193C23×C47#13512yes—
194C23×C47#13712yes—
195C23×C47#13712yes—
196C23×C47C22×D411yescharacteristic
197Q8×C47—24no—
198Q8×C47—24no—
199Q8×C47—42no—
200Q8×C47—42no—
201Q8×C47—48no—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—44no—
212C2×C4×C47—48no—
213C2×C4×C47—48no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—48no—
219C2×C4×C47#12012yes—
220C2×C4×C47#12114yes—
221C2×C4×C47#12114yes—
222C2×C4×C47#12112yes—
223C2×C4×C47#12314yes—
224C2×C4×C47#12312yes—
225C2×C4×C47#12414yes—
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—84no—
239C2×Dic47—84no—
240C47⋊C8—24no—
241C47⋊C8—42no—
242C2×#2035—471nomaximal, Sylow
243C22×C47—24no—
244C22×C47—24no—
245C22×C47—24no—
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—42no—
249C22×C47—42no—
250C22×C47—416no—
251C22×C47#39414yes—
252C22×C47#44814yes—
253C22×C47#45214yes—
254C22×C47#46514yes—
255C22×C47#47014yes—
256C22×C47#48314yes—
257C22×C47#48514yes—
258C22×C47C2×#12011yescharacteristic
259C22×C47C2×#12111yescharacteristic
260C22×C47C2×#12311yescharacteristic
261C22×C47C22×SD1611yescharacteristic
262C22×C47C2×#13511yescharacteristic
263C22×C47C2×#13711yescharacteristic
264C22×C47C2×#13711yescharacteristic
265C4×C47—42no—
266C4×C47—44no—
267C4×C47—44no—
268C4×C47#39414yes—
269C4×C47C2×#12111yescharacteristic
270C4×C47C2×#12111yescharacteristic
271C4×C47C2×#12311yescharacteristic
272C4×C47C2×#12411yescharacteristic
273Dic47—82no—
274Dic47—82no—
275C22×C22⋊Q8D4711yescharacteristic
276C22×#106—471no—
277C22×#103—471no—
278C22×#103—471no—
279C22×C22⋊C8—471no—
280C2×#380—471no—
281#2035—478no—
282C2×#340—471no—
283C2×C47—44no—
284C2×C47#203518yescomplemented
285C2×C47C2×#39411yescharacteristic
286C2×C47C2×#44811yescharacteristic
287C2×C47C2×#45211yescharacteristic
288C2×C47C2×#46511yescharacteristic
289C2×C47C2×#47011yescharacteristic
290C2×C47C2×#48311yescharacteristic
291C2×C47C2×#48511yescharacteristic
292C23×Q8C2×D4711yescharacteristic
293C24×C4C2×D4711yescharacteristic
294C23×C8—471no—
295C23×C8—471no—
296C2×C22⋊Q8—24no—
297C2×C22⋊Q8—22no—
298C2×C22⋊Q8—24no—
299C2×C22⋊Q8C2×D4714yes—
300C22×C4⋊C4—21no—
301C22×C4⋊C4—471no—
302C22×C4⋊C4—471no—
303C22×C4⋊C4C2×D4711yescharacteristic
304C22×C22⋊C4—21no—
305C22×C22⋊C4—941no—
306C2×#106—474no—
307C2×#106—944no—
308C2×#103—474no—
309C2×#103—474no—
310C2×#103—944no—
311C2×#103—944no—
312C2×C22⋊C8—474no—
313C2×C22⋊C8—944no—
314#380—474no—
315#380—942no—
316C2×#95—941no—
317#340—474no—
318C47C2×#203511yescharacteristic, complemented, Sylow
319C25C22⋊D4711yescharacteristic
320C22×Q8—22no—
321C22×Q8—21no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8C22⋊D4714yes—
327C22×Q8C22×D4712yes—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—941no—
337C23×C4—941no—
338C23×C4C22⋊D4712yes—
339C23×C4C22⋊D4711yescharacteristic
340C23×C4C22×D4712yes—
341C23×C4C22×D4711yescharacteristic
342C22×C8—474no—
343C22×C8—472no—
344C22×C8—472no—
345C22×C8—942no—
346C22×C8—942no—
347C22×C8—942no—
348C22×C8—942no—
349C22×C8—942no—
350C22⋊Q8—216no—
351C22⋊Q8—28no—
352C22⋊Q8—28no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—472no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—942no—
361C2×C4⋊C4—942no—
362C2×C4⋊C4—942no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4—941no—
365C2×C4⋊C4C22⋊D4714yes—
366C2×C4⋊C4C22×D4712yes—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—944no—
370C2×C22⋊C4—1884no—
371#106—948no—
372#103—948no—
373#103—948no—
374C22⋊C8—948no—
375#95—944no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24#1062533614yes—
383C24C2×C22⋊D4712yes—
384C24C2×C22⋊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—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×C22⋊D4714yes—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—1882no—
425C22×C4—1882no—
426C22×C4—1882no—
427C22×C4—1882no—
428C22×C4—1882no—
429C22×C4—1882no—
430C22×C4D4×D4712yes—
431C22×C4#1062535212yes—
432C22×C4#1062535212yes—
433C22×C4#1062535212yes—
434C22×C4C2×C22⋊D4712yes—
435C22×C4C2×C22⋊D4711yescharacteristic
436C22×C4C2×C22⋊D4712yes—
437C22×C4C2×C22⋊D4712yes—
438C22×C4C23×D4711yescharacteristic
439C2×C8—474no—
440C2×C8—948no—
441C2×C8—942no—
442C2×C8—942no—
443C2×C8—942no—
444C2×C8—942no—
445C2×C8—942no—
446C2×C8—1884no—
447C4⋊C4—216no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—942no—
451C4⋊C4—942no—
452C4⋊C4—942no—
453C4⋊C4—942no—
454C4⋊C4—1882no—
455C4⋊C4—1882no—
456C4⋊C4C2×C22⋊D4714yes—
457C22⋊C4—216no—
458C22⋊C4—1888no—
459C23—28no—
460C23—22no—
461C23—28no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—28no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23#1309399312yes—
473C23#1309399612yes—
474C23#1309400512yes—
475C23C2×#1062533614yes—
476C23C2×#1062533612yes—
477C23#1309404412yes—
478C23#1309405412yes—
479C23#1309405512yes—
480C23C22×C22⋊D4711yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—3764no—
504C2×C4—3764no—
505C2×C4—1882no—
506C2×C4—1882no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1882no—
511C2×C4#1309399312yes—
512C2×C4#1309399514yes—
513C2×C4#1309399612yes—
514C2×C4#1309399614yes—
515C2×C4#1309400512yes—
516C2×C4#1309400514yes—
517C2×C4#1309401114yes—
518C2×C4C2×D4×D4711yescharacteristic
519C2×C4C22×C22⋊D4711yescharacteristic
520C2×C4C22×C22⋊D4711yescharacteristic
521C2×C4C2×#1062535211yescharacteristic
522C2×C4C2×#1062535211yescharacteristic
523C2×C4C2×#1062535211yescharacteristic
524C8—944no—
525C8—1882no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—416no—
533C22—42no—
534C22#1568903714yes—
535C22#1568926214yes—
536C22#1568928514yes—
537C22#1568935414yes—
538C22#1568937914yes—
539C22#1568945114yes—
540C22#1568946114yes—
541C22C2×#1309399311yescharacteristic
542C22C2×#1309399611yescharacteristic
543C22C2×#1309400511yescharacteristic
544C22C22×#1062533611yescharacteristic
545C22C2×#1309404411yescharacteristic
546C22C2×#1309405411yescharacteristic
547C22C2×#1309405511yescharacteristic
548C4—42no—
549C4—44no—
550C4—44no—
551C4—3762no—
552C4—3762no—
553C4#1568903714yes—
554C4C2×#1309399511yescharacteristic
555C4C2×#1309399611yescharacteristic
556C4C2×#1309400511yescharacteristic
557C4C2×#1309401111yescharacteristic
558C2—44no—
559C2#2024458118yesdirect product factor
560C2C2×#1568903711yescharacteristic
561C2C2×#1568926211yescharacteristic
562C2C2×#1568928511yescharacteristic
563C2C2×#1568935411yescharacteristic
564C2C2×#1568937911yescharacteristic
565C2C2×#1568945111yescharacteristic
566C2C2×#1568946111yescharacteristic
subgroupG/H
centerC24C2×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×C22⋊Q8×C47C2
socleC24×C47C2×D4
prime radicalC25×C47D4
hypercenterC22×C22⋊Q8D47
hypocenterC47C2×#2035
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13093955C2semidirect
2. C22×#13093948C2semidirect
3. C22×#13093948C2semidirect
6. #20244581C2direct (N × H)
7. C2×#15688885C2semidirect
15. C2×#13093955C22semidirect
18. C2×#13093948C22semidirect
19. C2×#13093948C22semidirect
25. #15688885C22semidirect
284. C2×C47#2035semidirect
318. C47C2×#2035semidirect
559. C2#20244581direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939552
C22×#130939482
C22×C22⋊(C47⋊C8)2
C2×#156889832
#202445812
C2×#156888852
C22×C22⋊Q8×C472
C2×#203547