Finite Group Databasebeta

← C2×#20243746 #26217621

Subgroups — C2×#20243746 #26217621

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×#10625335C211yesmaximal, characteristic
2C22×C4×Dic94C211yesmaximal, characteristic, complemented
3C22×#13093947C211yesmaximal, characteristic, complemented
4C22×#13093945C211yesmaximal, characteristic
5#20243746C218yesmaximal, direct product factor
6C2×#15688885C211yesmaximal, characteristic, complemented
7C2×#15688878C211yesmaximal, characteristic, complemented
8C2×C47×#390C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C22×#10625335—24no—
11C22×#10625335—24no—
12C22×#10625335C418yescomplemented
13C22×#10625335C2214yes—
14C23×C47⋊C8C2211yescharacteristic
15C23×C47⋊C8C2211yescharacteristic
16C2×C4×Dic94—24no—
17C2×C4×Dic94—24no—
18C2×C4×Dic94—22no—
19C2×C4×Dic94C2214yescomplemented
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4—21no—
22C22×C4×Dic47—21no—
23C2×#13093947—24no—
24C2×#13093947C2214yescomplemented
25C2×#13093945—24no—
26C2×#13093945C2214yes—
27#15688885C2214yescomplemented
28#15688878C2214yescomplemented
29C23×D4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47—21no—
32C22×C22⋊C4×C47—21no—
33C22×C42×C47C2211yescharacteristic
34C47×#390—22no—
35C47×#390C2214yes—
36C22×Dic94—21no—
37C22×Dic94—22no—
38C22×Dic94—22no—
39C22×Dic94—22no—
40C22×Dic94C2×C414yes—
41C22×Dic94D414yes—
42C22×Dic94D414yes—
43C22×Dic94C2312yes—
44C23×Dic47—21no—
45C23×Dic47—21no—
46C2×#10625335—232no—
47C2×#10625335—28no—
48C2×#10625335—28no—
49C2×#10625335C2×C4116yescomplemented
50C22×C47⋊C8—22no—
51C22×C47⋊C8—22no—
52C22×C47⋊C8—22no—
53C22×C47⋊C8—22no—
54C22×C47⋊C8—22no—
55C22×C47⋊C8C2×C414yes—
56C22×C47⋊C8C2312yes—
57C22×C47⋊C8C2312yes—
58C4×Dic94—216no—
59C4×Dic94—28no—
60C4×Dic94—28no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—22no—
63C2×C4⋊Dic47D414yes—
64C2×C4⋊Dic47C2312yes—
65C2×Dic47⋊C4—28no—
66C2×Dic47⋊C4—24no—
67C2×C4×Dic47—28no—
68C2×C4×Dic47—24no—
69#13093947—28no—
70#13093945—28no—
71C25×C47—21no—
72C22×D4×C47—22no—
73C22×D4×C47—22no—
74C22×D4×C47—22no—
75C22×D4×C47—21no—
76C22×D4×C47—22no—
77C22×D4×C47—22no—
78C22×D4×C47C2×C414yes—
79C22×D4×C47C2312yes—
80C23×C4×C47—21no—
81C23×C4×C47D411yescharacteristic
82C23×C4×C47D411yescharacteristic
83C23×C4×C47C2311yescharacteristic
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47—22no—
86C2×C4⋊C4×C47—21no—
87C2×C4⋊C4×C47C2312yes—
88C2×C22⋊C4×C47—24no—
89C2×C22⋊C4×C47—24no—
90C2×C22⋊C4×C47—44no—
91C2×C22⋊C4×C47—44no—
92C2×C42×C47—22no—
93C2×C42×C47—22no—
94C2×C42×C47D412yes—
95C2×C42×C47C2312yes—
96C24×C47—24no—
97C24×C47—22no—
98C24×C47—42no—
99C24×C47—42no—
100C24×C47—42no—
101C24×C47—42no—
102C24×C47—42no—
103C24×C47—42no—
104C24×C47C2×D411yescharacteristic
105C2×D4×C47—28no—
106C2×D4×C47—28no—
107C2×D4×C47—24no—
108C2×D4×C47—22no—
109C2×D4×C47—22no—
110C2×D4×C47—22no—
111C2×D4×C47—22no—
112C2×D4×C47—22no—
113C2×D4×C47—42no—
114C2×D4×C47—42no—
115C2×D4×C47—44no—
116C2×D4×C47—44no—
117C2×D4×C47C22×C414yes—
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—42no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47SD1614yescomplemented
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D412yes—
135C22×C4×C47#4112yes—
136C22×C4×C47#4112yes—
137C22×C4×C47C2411yescharacteristic
138C22×Dic47—24no—
139C22×Dic47—24no—
140C22×Dic47—24no—
141C22×Dic47—22no—
142C22×Dic47—24no—
143C22×Dic47—22no—
144C22×Dic47—42no—
145C22×Dic47—42no—
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—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—42no—
162C22⋊C4×C47—48no—
163C22⋊C4×C47—48no—
164C42×C47—24no—
165C42×C47—22no—
166C42×C47—22no—
167#10625335—232no—
168C4⋊Dic47—24no—
169C4⋊Dic47—22no—
170C4⋊Dic47C2×D414yes—
171Dic47⋊C4—216no—
172C4×Dic47—216no—
173C2×C47⋊C8—28no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—44no—
180C2×C47⋊C8C22×C414yes—
181C23×C47—24no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—48no—
189C23×C47—48no—
190C23×C47—48no—
191C23×C47—84no—
192C23×C47—84no—
193C23×C47—84no—
194C23×C47—84no—
195C23×C47C4×D412yes—
196C23×C47#12012yes—
197C23×C47#12112yes—
198C23×C47C2×SD1612yes—
199C23×C47#13512yes—
200C23×C47AGL(1,ℤ/8)12yes—
201C23×C47#13712yes—
202C23×C47C22×D411yescharacteristic
203D4×C47—24no—
204D4×C47—24no—
205D4×C47—42no—
206D4×C47—42no—
207D4×C47—48no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—28no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—44no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—84no—
219C2×C4×C47C2×C22⋊C414yes—
220C2×C4×C47C4×D414yes—
221C2×C4×C47C4×D412yes—
222C2×C4×C47#12012yes—
223C2×C4×C47#12112yes—
224C2×C4×C47#12114yes—
225C2×C4×C47#12314yes—
226C2×C4×C47C2×SD1614yescomplemented
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—216no—
237C2×Dic47—24no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—48no—
241Dic94—24no—
242Dic94—24no—
243Dic94—24no—
244Dic94—216no—
245Dic94C2×C22⋊C418yescomplemented
246C47⋊C8—24no—
247C47⋊C8—42no—
248C2×#1880—471nomaximal, Sylow
249C22×C47—44no—
250C22×C47—44no—
251C22×C47—44no—
252C22×C47—816no—
253C22×C47—82no—
254C22×C47—82no—
255C22×C47—82no—
256C22×C47—82no—
257C22×C47#38614yes—
258C22×C47C4×SD1614yes—
259C22×C47#44014yes—
260C22×C47#44814yes—
261C22×C47#44914yes—
262C22×C47#46014yes—
263C22×C47#46414yes—
264C22×C47C2×C4×D411yescharacteristic
265C22×C47C2×#12011yescharacteristic
266C22×C47C2×#12111yescharacteristic
267C22×C47C22×SD1611yescharacteristic
268C22×C47C2×#13511yescharacteristic
269C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C47C2×#13711yescharacteristic
271C4×C47—24no—
272C4×C47—42no—
273C4×C47—82no—
274C4×C47#38614yes—
275C4×C47C22×C22⋊C411yescharacteristic
276C4×C47C2×C4×D411yescharacteristic
277C4×C47C2×#12111yescharacteristic
278C4×C47C2×#12311yescharacteristic
279Dic47—28no—
280Dic47—44no—
281C23×SD16—471no—
282C22×C4×Q8—471no—
283C22×#103—471no—
284C22×#102—471no—
285C2×#390D4711yescharacteristic
286#1880—478no—
287C2×#340—471no—
288C2×#336—471no—
289C2×C47—84no—
290C2×C47#188018yescomplemented
291C2×C47C2×#38611yescharacteristic
292C2×C47C2×C4×SD1611yescharacteristic
293C2×C47C2×#44011yescharacteristic
294C2×C47C2×#44811yescharacteristic
295C2×C47C2×#44911yescharacteristic
296C2×C47C2×#46011yescharacteristic
297C2×C47C2×#46411yescharacteristic
298C23×Q8—471no—
299C23×D4C2×D4711yescharacteristic
300C22×SD16—478no—
301C22×SD16—474no—
302C22×SD16—944no—
303C22×SD16—944no—
304C23×C8—471no—
305C23×C8—471no—
306C2×C4×Q8—474no—
307C2×C4×Q8—944no—
308C2×C4×Q8—944no—
309C2×C4×Q8—942no—
310C22×C4⋊C4—471no—
311C22×C4⋊C4—941no—
312C22×C4⋊C4C2×D4711yescharacteristic
313C22×C22⋊C4—21no—
314C22×C22⋊C4—21no—
315C22×C42—941no—
316C22×C42C2×D4711yescharacteristic
317C2×#103—474no—
318C2×#103—944no—
319C2×#102—474no—
320C2×#102—944no—
321#390—22no—
322#390C2×D4714yes—
323#340—474no—
324#336—474no—
325C47C2×#188011yescharacteristic, complemented, Sylow
326C25—21no—
327C22×Q8—474no—
328C22×Q8—474no—
329C22×Q8—474no—
330C22×Q8—472no—
331C22×Q8—942no—
332C22×Q8—942no—
333C22×Q8—942no—
334C22×Q8—941no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4—22no—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—21no—
341C22×D4C4×D4714yes—
342C22×D4C22×D4712yes—
343C23×C4—21no—
344C23×C4—941no—
345C23×C4—941no—
346C23×C4C22⋊D4711yescharacteristic
347C23×C4C22⋊D4711yescharacteristic
348C23×C4C22×D4711yescharacteristic
349C2×SD16—4716no—
350C2×SD16—9432no—
351C2×SD16—948no—
352C2×SD16—948no—
353C22×C8—474no—
354C22×C8—472no—
355C22×C8—472no—
356C22×C8—942no—
357C22×C8—942no—
358C22×C8—942no—
359C22×C8—942no—
360C22×C8—942no—
361C4×Q8—948no—
362C4×Q8—948no—
363C4×Q8—9416no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—948no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—944no—
373C2×C4⋊C4C22×D4712yes—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—44no—
377C2×C22⋊C4—44no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—948no—
381C2×C42—944no—
382C2×C42C22⋊D4712yes—
383C2×C42C22×D4712yes—
384#103—948no—
385#102—948no—
386C24—24no—
387C24—22no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C2×C22⋊D4711yescharacteristic
395C2×Q8—4716no—
396C2×Q8—474no—
397C2×Q8—474no—
398C2×Q8—474no—
399C2×Q8—948no—
400C2×Q8—944no—
401C2×Q8—944no—
402C2×Q8—944no—
403C2×Q8—948no—
404C2×Q8—948no—
405C2×Q8—942no—
406C2×Q8—942no—
407C2×Q8—942no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4C2×C4×D4714yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—944no—
431C22×C4—944no—
432C22×C4—944no—
433C22×C4—942no—
434C22×C4—944no—
435C22×C4—942no—
436C22×C4—1882no—
437C22×C4—1882no—
438C22×C4#1062533514yescomplemented
439C22×C4C2×C4×D4712yes—
440C22×C4#1062535012yes—
441C22×C4D4×D4712yes—
442C22×C4D4×D4712yes—
443C22×C4D4×D4712yes—
444C22×C4#1062535212yes—
445C22×C4C2×C22⋊D4712yes—
446C22×C4C2×C22⋊D4712yes—
447C22×C4C2×C22⋊D4711yescharacteristic
448C22×C4C23×D4711yescharacteristic
449SD16—9432no—
450C2×C8—474no—
451C2×C8—948no—
452C2×C8—942no—
453C2×C8—942no—
454C2×C8—942no—
455C2×C8—942no—
456C2×C8—942no—
457C2×C8—1884no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—42no—
461C4⋊C4—474no—
462C4⋊C4—9416no—
463C4⋊C4—944no—
464C4⋊C4—942no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—9416no—
471C23—24no—
472C23—48no—
473C23—48no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—48no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23C4×C22⋊D4712yes—
486C23#1309399212yes—
487C23#1309399512yes—
488C23C2×#1062533512yes—
489C23#1309404412yes—
490C23#1309405012yes—
491C23#1309405412yes—
492C23C22×C22⋊D4711yescharacteristic
493Q8—478no—
494Q8—9416no—
495Q8—944no—
496Q8—944no—
497Q8—944no—
498D4—24no—
499D4—24no—
500D4—48no—
501D4—42no—
502D4—42no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—84no—
514C2×C4—9416no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—944no—
519C2×C4—1888no—
520C2×C4—1882no—
521C2×C4—1882no—
522C2×C4C22⋊C4×D4714yes—
523C2×C4(C22⋊D47)⋊C414yes—
524C2×C4C4×C22⋊D4712yes—
525C2×C4#1309399212yes—
526C2×C4#1309399512yes—
527C2×C4#1309399714yes—
528C2×C4#1309400614yes—
529C2×C4C2×#1062533514yescomplemented
530C2×C4C22×C4×D4711yescharacteristic
531C2×C4C2×D4×D4711yescharacteristic
532C2×C4C2×D4×D4711yescharacteristic
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C2×#1062535211yescharacteristic
535C2×C4C2×#1062535011yescharacteristic
536C8—944no—
537C8—1882no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—816no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22#1568899914yes—
547C22C4×#1062533514yes—
548C22#1568922314yes—
549C22#1568926114yes—
550C22#1568926614yes—
551C22#1568932014yes—
552C22#1568934214yes—
553C22C2×C4×C22⋊D4711yescharacteristic
554C22C2×#1309399211yescharacteristic
555C22C2×#1309399511yescharacteristic
556C22C22×#1062533511yescharacteristic
557C22C2×#1309404411yescharacteristic
558C22C2×#1309405011yescharacteristic
559C22C2×#1309405411yescharacteristic
560C4—24no—
561C4—42no—
562C4—82no—
563C4—948no—
564C4—1884no—
565C4#1568899914yes—
566C4C2×C22⋊C4×D4711yescharacteristic
567C4C2×(C22⋊D47)⋊C411yescharacteristic
568C4C2×#1309399711yescharacteristic
569C4C2×#1309400611yescharacteristic
570C2—84no—
571C2#2024374618yesdirect product factor
572C2C2×#1568899911yescharacteristic
573C2C2×C4×#1062533511yescharacteristic
574C2C2×#1568922311yescharacteristic
575C2C2×#1568926111yescharacteristic
576C2C2×#1568926611yescharacteristic
577C2C2×#1568932011yescharacteristic
578C2C2×#1568934211yescharacteristic
subgroupG/H
centerC24C2×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#390C2
socleC24×C47C2×D4
prime radicalC23×D4×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#390D47
hypocenterC47C2×#1880
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic94C2semidirect
3. C22×#13093947C2semidirect
5. #20243746C2direct (N × H)
6. C2×#15688885C2semidirect
7. C2×#15688878C2semidirect
12. C22×#10625335C4semidirect
19. C2×C4×Dic94C22semidirect
24. C2×#13093947C22semidirect
27. #15688885C22semidirect
28. #15688878C22semidirect
49. C2×#10625335C2×C4semidirect
127. C22×C4×C47SD16semidirect
155. C2×Dic94C22⋊C4semidirect
226. C2×C4×C47C2×SD16semidirect
245. Dic94C2×C22⋊C4semidirect
290. C2×C47#1880semidirect
325. C47C2×#1880semidirect
438. C22×C4#10625335semidirect
529. C2×C4C2×#10625335semidirect
571. C2#20243746direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253352
C22×C4×Dic942
C22×#130939472
C22×#130939452
#202437462
C2×#156888852
C2×#156888782
C2×C47×#3902
C2×#188047