Finite Group Databasebeta

← C2×#20243744 #26217619

Subgroups — C2×#20243744 #26217619

Order 12032.

2038 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625325C211yesmaximal, characteristic
2C22×#13093989C211yesmaximal, characteristic, complemented
3C22×#13093947C211yesmaximal, characteristic, complemented
4C22×#13093944C211yesmaximal, characteristic
5C2×#15689024C211yesmaximal, characteristic
6#20243744C218yesmaximal, direct product factor
7C2×#15688877C211yesmaximal, characteristic, complemented
8C2×C47×#340C211yesmaximal, characteristic, complemented
9C23×Dic94C2211yescharacteristic
10C23×D188C2211yescharacteristic
11C22×#10625325—24no—
12C22×#10625325—24no—
13C22×#10625325C418yescomplemented
14C22×#10625325C2214yes—
15C23×C47⋊C8C2211yescharacteristic
16C2×#13093989—24no—
17C2×#13093989—22no—
18C2×#13093989—24no—
19C2×#13093989C2214yescomplemented
20C22×Dic47⋊C4—21no—
21C22×(C2×D47)⋊C4—21no—
22C22×(C2×D47)⋊C4—21no—
23C22×C4×Dic47—21no—
24C22×C4×Dic47C2211yescharacteristic
25C2×#13093947—24no—
26C2×#13093947C2214yescomplemented
27C2×#13093944—24no—
28C2×#13093944C2214yes—
29#15689024—22no—
30#15689024C2214yes—
31#15688877C2214yescomplemented
32C23×C8×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C47×#340C2214yescomplemented
36C24×D47—21no—
37C22×Dic94—21no—
38C22×Dic94—22no—
39C22×Dic94—22no—
40C22×Dic94—22no—
41C22×Dic94C2×C414yes—
42C22×Dic94D414yes—
43C22×Dic94D414yes—
44C22×Dic94C2312yes—
45C22×D188—22no—
46C22×D188—22no—
47C22×D188—22no—
48C22×D188—22no—
49C22×D188—21no—
50C22×D188—22no—
51C22×D188C2×C414yes—
52C22×D188C2312yes—
53C23×Dic47—21no—
54C23×Dic47D411yescharacteristic
55C23×Dic47D411yescharacteristic
56C2×#10625325—232no—
57C2×#10625325—28no—
58C2×#10625325—28no—
59C2×#10625325C2×C4116yescomplemented
60C22×C47⋊C8—22no—
61C22×C47⋊C8—22no—
62C22×C47⋊C8—22no—
63C22×C47⋊C8C2312yes—
64#13093989—216no—
65#13093989—28no—
66#13093989—28no—
67C2×Dic47⋊C4—28no—
68C2×Dic47⋊C4—24no—
69C2×(C2×D47)⋊C4—24no—
70C2×(C2×D47)⋊C4—24no—
71C2×(C2×D47)⋊C4—44no—
72C2×(C2×D47)⋊C4—44no—
73C2×C4×Dic47—28no—
74C2×C4×Dic47—22no—
75C2×C4×Dic47—22no—
76C2×C4×Dic47—24no—
77C2×C4×Dic47D412yes—
78C2×C4×Dic47C2312yes—
79#13093947—28no—
80#13093944—28no—
81C23×C4×C47—21no—
82C23×C4×C47—21no—
83C23×C4×C47C2311yescharacteristic
84C22×C8×C47—22no—
85C22×C8×C47—22no—
86C22×C8×C47C2×C414yes—
87C22×C8×C47C2312yes—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—22no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47D414yes—
94C2×C4⋊C4×C47C2312yes—
95C2×C4⋊C4×C47C2312yes—
96C24×C47C2×D411yescharacteristic
97C23×D47—24no—
98C23×D47—22no—
99C23×D47—42no—
100C23×D47—42no—
101C23×D47—42no—
102C23×D47—42no—
103C23×D47—42no—
104C23×D47—42no—
105C22×C4×C47—24no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—42no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47—42no—
112C22×C4×C47—42no—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C2×D412yes—
115C22×C4×C47C2×D412yes—
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D412yes—
118C22×C4×C47#4112yes—
119C22×C4×C47#4112yes—
120C22×C4×C47C2411yescharacteristic
121C22×Dic47—24no—
122C22×Dic47—22no—
123C22×Dic47—24no—
124C22×Dic47—24no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47SD1614yescomplemented
131C22×Dic47C2×D412yes—
132C22×Dic47C2×D412yes—
133C2×D188—28no—
134C2×D188—28no—
135C2×D188—22no—
136C2×D188—22no—
137C2×D188—22no—
138C2×D188—22no—
139C2×D188—24no—
140C2×D188—22no—
141C2×D188—42no—
142C2×D188—44no—
143C2×D188—42no—
144C2×D188—44no—
145C2×D188C22×C414yes—
146C2×Dic94—28no—
147C2×Dic94—24no—
148C2×Dic94—24no—
149C2×Dic94—28no—
150C2×Dic94—24no—
151C2×Dic94—28no—
152C2×Dic94—22no—
153C2×Dic94—22no—
154C2×Dic94—22no—
155C2×Dic94C22⋊C4116yescomplemented
156C2×Dic94C22×C414yes—
157C2×Dic94C2×D414yes—
158C2×Dic94C2×D414yes—
159C2×C8×C47—28no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47C22×C414yes—
163C4⋊C4×C47—24no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—42no—
168C4⋊C4×C47C2×D414yes—
169(C2×D47)⋊C4—48no—
170(C2×D47)⋊C4—48no—
171Dic47⋊C4—216no—
172C4×Dic47—216no—
173C4×Dic47—24no—
174C4×Dic47—22no—
175C4×Dic47—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—44no—
180#10625325—232no—
181C23×C47C4×D412yes—
182C23×C47#12012yes—
183C23×C47#12112yes—
184C23×C47C2×SD1612yes—
185C23×C47#13512yes—
186C23×C47AGL(1,ℤ/8)12yes—
187C23×C47#13712yes—
188C23×C47C22×D411yescharacteristic
189C22×D47—24no—
190C22×D47—42no—
191C22×D47—42no—
192C22×D47—42no—
193C22×D47—42no—
194C22×D47—42no—
195C22×D47—42no—
196C22×D47—48no—
197C22×D47—48no—
198C22×D47—48no—
199C22×D47—84no—
200C22×D47—84no—
201C22×D47—84no—
202C22×D47—84no—
203C2×C4×C47—24no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—48no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—84no—
211C2×C4×C47C2×C22⋊C414yes—
212C2×C4×C47C4×D412yes—
213C2×C4×C47C4×D414yes—
214C2×C4×C47#12012yes—
215C2×C4×C47#12112yes—
216C2×C4×C47#12114yes—
217C2×C4×C47#12314yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×Dic47—24no—
225C2×Dic47—22no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—22no—
229C2×Dic47—216no—
230C2×Dic47—28no—
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—44no—
235C2×Dic47C2×SD1614yescomplemented
236D188—24no—
237D188—24no—
238D188—42no—
239D188—42no—
240D188—48no—
241Dic94—24no—
242Dic94—24no—
243Dic94—24no—
244Dic94—216no—
245Dic94C2×C22⋊C418yescomplemented
246C8×C47—24no—
247C47⋊C8—42no—
248C2×#1880—471nomaximal, Sylow
249C22×C47#38614yes—
250C22×C47C4×SD1614yes—
251C22×C47#44014yes—
252C22×C47#44814yes—
253C22×C47#44914yes—
254C22×C47#46014yes—
255C22×C47#46414yes—
256C22×C47C2×C4×D411yescharacteristic
257C22×C47C2×#12011yescharacteristic
258C22×C47C2×#12111yescharacteristic
259C22×C47C22×SD1611yescharacteristic
260C22×C47C2×#13511yescharacteristic
261C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C47C2×#13711yescharacteristic
263C2×D47—44no—
264C2×D47—44no—
265C2×D47—44no—
266C2×D47—816no—
267C2×D47—82no—
268C2×D47—82no—
269C2×D47—82no—
270C2×D47—82no—
271C4×C47—44no—
272C4×C47—82no—
273C4×C47#38614yes—
274C4×C47C22×C22⋊C411yescharacteristic
275C4×C47C2×C4×D411yescharacteristic
276C4×C47C2×#12111yescharacteristic
277C4×C47C2×#12311yescharacteristic
278Dic47—24no—
279Dic47—28no—
280Dic47—42no—
281C23×SD16—471no—
282C22×C4×Q8—471no—
283C22×#103—471no—
284C22×#102—471no—
285C2×#390—471no—
286#1880—478no—
287C2×#340D4711yescharacteristic, complemented
288C2×#336—471no—
289C2×C47#188018yescomplemented
290C2×C47C2×#38611yescharacteristic
291C2×C47C2×C4×SD1611yescharacteristic
292C2×C47C2×#44011yescharacteristic
293C2×C47C2×#44811yescharacteristic
294C2×C47C2×#44911yescharacteristic
295C2×C47C2×#46011yescharacteristic
296C2×C47C2×#46411yescharacteristic
297D47—84no—
298C23×Q8—471no—
299C23×D4—471no—
300C22×SD16—478no—
301C22×SD16—474no—
302C22×SD16—944no—
303C22×SD16—944no—
304C23×C8—471no—
305C23×C8C2×D4711yescharacteristic
306C2×C4×Q8—474no—
307C2×C4×Q8—942no—
308C2×C4×Q8—944no—
309C2×C4×Q8—944no—
310C22×C4⋊C4—941no—
311C22×C4⋊C4C2×D4711yescharacteristic
312C22×C4⋊C4C2×D4711yescharacteristic
313C22×C22⋊C4—941no—
314C22×C22⋊C4—941no—
315C22×C42—471no—
316C22×C42—941no—
317C2×#103—474no—
318C2×#103—944no—
319C2×#102—474no—
320C2×#102—944no—
321#390—474no—
322#390—942no—
323#340C2×D4714yescomplemented
324#336—474no—
325C47C2×#188011yescharacteristic, complemented, Sylow
326C25—941no—
327C22×Q8—474no—
328C22×Q8—474no—
329C22×Q8—474no—
330C22×Q8—472no—
331C22×Q8—942no—
332C22×Q8—942no—
333C22×Q8—941no—
334C22×Q8—942no—
335C22×D4—474no—
336C22×D4—472no—
337C22×D4—942no—
338C22×D4—942no—
339C22×D4—942no—
340C22×D4—942no—
341C22×D4—941no—
342C22×D4—942no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—471no—
346C23×C4—471no—
347C23×C4—941no—
348C23×C4C22×D4711yescharacteristic
349C2×SD16—4716no—
350C2×SD16—9432no—
351C2×SD16—948no—
352C2×SD16—948no—
353C22×C8—22no—
354C22×C8—22no—
355C22×C8—472no—
356C22×C8—942no—
357C22×C8—942no—
358C22×C8—942no—
359C22×C8C4×D4714yes—
360C22×C8C22×D4712yes—
361C4×Q8—948no—
362C4×Q8—9416no—
363C4×Q8—948no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—944no—
370C2×C4⋊C4—948no—
371C2×C4⋊C4D18814yes—
372C2×C4⋊C4C22×D4712yes—
373C2×C4⋊C4C22×D4712yes—
374C2×C22⋊C4—944no—
375C2×C22⋊C4—944no—
376C2×C22⋊C4—1884no—
377C2×C22⋊C4—1884no—
378C2×C42—472no—
379C2×C42—472no—
380C2×C42—944no—
381C2×C42—948no—
382C2×C42—942no—
383C2×C42—942no—
384#103—948no—
385#102—948no—
386C24—944no—
387C24—942no—
388C24—1882no—
389C24—1882no—
390C24—1882no—
391C24—1882no—
392C24—1882no—
393C24—1882no—
394C24D4×D4711yescharacteristic
395C2×Q8—4716no—
396C2×Q8—474no—
397C2×Q8—474no—
398C2×Q8—474no—
399C2×Q8—944no—
400C2×Q8—948no—
401C2×Q8—944no—
402C2×Q8—944no—
403C2×Q8—948no—
404C2×Q8—948no—
405C2×Q8—942no—
406C2×Q8—942no—
407C2×Q8—942no—
408C2×D4—474no—
409C2×D4—948no—
410C2×D4—944no—
411C2×D4—948no—
412C2×D4—942no—
413C2×D4—942no—
414C2×D4—942no—
415C2×D4—942no—
416C2×D4—942no—
417C2×D4—1882no—
418C2×D4—1882no—
419C2×D4—1884no—
420C2×D4—1884no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—474no—
430C22×C4—472no—
431C22×C4—472no—
432C22×C4—944no—
433C22×C4—942no—
434C22×C4—944no—
435C22×C4—944no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4C2×C4×D4712yes—
442C22×C4C2×D18812yes—
443C22×C4#1062535012yes—
444C22×C4D4×D4712yes—
445C22×C4D4×D4712yes—
446C22×C4D4×D4711yescharacteristic
447C22×C4#1062535412yes—
448C22×C4C23×D4711yescharacteristic
449SD16—9432no—
450C2×C8—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—942no—
454C2×C8—942no—
455C2×C8—942no—
456C2×C8—1884no—
457C2×C8C2×C4×D4714yes—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—9416no—
464C4⋊C4C2×D18814yes—
465C22⋊C4—1888no—
466C22⋊C4—1888no—
467C42—9416no—
468C42—944no—
469C42—942no—
470C42—942no—
471C23—944no—
472C23—3764no—
473C23—3764no—
474C23—3764no—
475C23—3764no—
476C23—1888no—
477C23—1888no—
478C23—1882no—
479C23—1882no—
480C23—1882no—
481C23—1882no—
482C23—1882no—
483C23—1882no—
484C23—1888no—
485C23#1309398612yes—
486C23#1309399412yes—
487C23#1309399712yes—
488C23SD16×D4712yes—
489C23#1309404512yes—
490C23#1309404812yes—
491C23#1309405612yes—
492C23C2×D4×D4711yescharacteristic
493Q8—478no—
494Q8—9416no—
495Q8—944no—
496Q8—944no—
497Q8—944no—
498D4—944no—
499D4—944no—
500D4—1888no—
501D4—1882no—
502D4—1882no—
503C2×C4—24no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—474no—
512C2×C4—9416no—
513C2×C4—948no—
514C2×C4—944no—
515C2×C4—942no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—1884no—
523C2×C4C22⋊C4×D4714yes—
524C2×C4#1309398612yes—
525C2×C4C4×D18814yes—
526C2×C4#1309399412yes—
527C2×C4#1309399712yes—
528C2×C4#1309399914yes—
529C2×C4#1309400914yes—
530C2×C4C22×C4×D4711yescharacteristic
531C2×C4C2×D4×D4711yescharacteristic
532C2×C4C2×D4×D4711yescharacteristic
533C2×C4C22×D18811yescharacteristic
534C2×C4C2×#1062535011yescharacteristic
535C2×C4C2×#1062535411yescharacteristic
536C8—24no—
537C8—1882no—
538C22—37616no—
539C22—3762no—
540C22—3762no—
541C22—3762no—
542C22—3762no—
543C22—1884no—
544C22—1884no—
545C22—1884no—
546C22#1568900114yes—
547C22#1568921014yes—
548C22#1568922214yes—
549C22#1568926314yes—
550C22#1568926914yes—
551C22#1568932214yes—
552C22#1568934814yes—
553C22C2×#1309398611yescharacteristic
554C22C2×#1309399411yescharacteristic
555C22C2×#1309399711yescharacteristic
556C22C2×SD16×D4711yescharacteristic
557C22C2×#1309404511yescharacteristic
558C22C2×#1309404811yescharacteristic
559C22C2×#1309405611yescharacteristic
560C4—44no—
561C4—82no—
562C4—948no—
563C4—944no—
564C4—1882no—
565C4#1568900114yes—
566C4C2×C22⋊C4×D4711yescharacteristic
567C4C2×C4×D18811yescharacteristic
568C4C2×#1309399911yescharacteristic
569C4C2×#1309400911yescharacteristic
570C2—3764no—
571C2#2024374418yesdirect product factor
572C2C2×#1568900111yescharacteristic
573C2C2×#1568921011yescharacteristic
574C2C2×#1568922211yescharacteristic
575C2C2×#1568926311yescharacteristic
576C2C2×#1568926911yescharacteristic
577C2C2×#1568932211yescharacteristic
578C2C2×#1568934811yescharacteristic
subgroupG/H
centerC24D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#340C2
socleC24×C47C2×D4
prime radicalC23×D188C22
cosocleC22×C4×C47C24
hypercenterC2×#340D47
hypocenterC47C2×#1880
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C22×#13093989C2semidirect
3. C22×#13093947C2semidirect
6. #20243744C2direct (N × H)
7. C2×#15688877C2semidirect
8. C2×C47×#340C2semidirect
13. C22×#10625325C4semidirect
19. C2×#13093989C22semidirect
26. C2×#13093947C22semidirect
31. #15688877C22semidirect
35. C47×#340C22semidirect
59. C2×#10625325C2×C4semidirect
130. C22×Dic47SD16semidirect
155. C2×Dic94C22⋊C4semidirect
235. C2×Dic47C2×SD16semidirect
245. Dic94C2×C22⋊C4semidirect
287. C2×#340D47semidirect
289. C2×C47#1880semidirect
323. #340C2×D47semidirect
325. C47C2×#1880semidirect
571. C2#20243744direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253252
C22×#130939892
C22×#130939472
C22×#130939442
C2×#156890242
#202437442
C2×#156888772
C2×C47×#3402
C2×#188047