Finite Group Databasebeta

← C2×#20243699 #26217483

Subgroups — C2×#20243699 #26217483

Order 12032.

1966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094004C211yesmaximal, characteristic
2C22×#13093982C211yesmaximal, characteristic
3C22×#13093949C211yesmaximal, characteristic, complemented
4C22×#13093947C211yesmaximal, characteristic, complemented
5C22×(C2×D47)⋊C8C211yesmaximal, characteristic
6#20243699C218yesmaximal, direct product factor
7C2×#15688885C211yesmaximal, characteristic, complemented
8C2×C47×#340C211yesmaximal, characteristic, complemented
9C23×Dic94C2211yescharacteristic
10C23×C4×D47C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12C2×#13094004—24no—
13C2×#13094004—22no—
14C2×#13094004C418yescomplemented
15C2×#13094004C2214yes—
16C2×#13093982—22no—
17C2×#13093982—24no—
18C2×#13093982—24no—
19C2×#13093982C2214yes—
20C22×C4⋊Dic47—21no—
21C22×C4⋊Dic47C2211yescharacteristic
22C22×(C2×D47)⋊C4—21no—
23C22×(C2×D47)⋊C4—21no—
24C22×C4×Dic47—21no—
25C2×#13093949—24no—
26C2×#13093949C2214yescomplemented
27C2×#13093947—24no—
28C2×#13093947C2214yescomplemented
29C2×(C2×D47)⋊C8—24no—
30C2×(C2×D47)⋊C8C2214yes—
31#15688885C2214yescomplemented
32C23×C8×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C47×#340C2214yescomplemented
36C24×D47D411yescharacteristic
37C22×Dic94—22no—
38C22×Dic94—22no—
39C22×Dic94—22no—
40C22×Dic94—22no—
41C22×Dic94—21no—
42C22×Dic94—22no—
43C22×Dic94C2×C414yes—
44C22×Dic94C2312yes—
45C23×Dic47—21no—
46C23×Dic47—21no—
47C23×Dic47D411yescharacteristic
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47C2×C414yes—
53C22×C4×D47D412yes—
54C22×C4×D47D414yes—
55C22×C4×D47C2312yes—
56C22×C47⋊C8—22no—
57C22×C47⋊C8—22no—
58C22×C47⋊C8—22no—
59C22×C47⋊C8C2312yes—
60#13094004—216no—
61#13094004—28no—
62#13094004C2×C4116yescomplemented
63#13093982—216no—
64#13093982—28no—
65#13093982—28no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—22no—
68C2×C4⋊Dic47—28no—
69C2×C4⋊Dic47—24no—
70C2×C4⋊Dic47D414yes—
71C2×C4⋊Dic47C2312yes—
72C2×(C2×D47)⋊C4—28no—
73C2×(C2×D47)⋊C4—28no—
74C2×(C2×D47)⋊C4—24no—
75C2×(C2×D47)⋊C4—24no—
76C2×C4×Dic47—28no—
77C2×C4×Dic47—24no—
78#13093949—28no—
79#13093947—28no—
80(C2×D47)⋊C8—28no—
81C23×C4×C47—21no—
82C23×C4×C47—21no—
83C23×C4×C47C2311yescharacteristic
84C22×C8×C47—22no—
85C22×C8×C47—22no—
86C22×C8×C47—22no—
87C22×C8×C47C2312yes—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47C2×C414yes—
93C2×C4⋊C4×C47D414yes—
94C2×C4⋊C4×C47C2312yes—
95C2×C4⋊C4×C47C2312yes—
96C24×C47C2×D411yescharacteristic
97C23×D47—22no—
98C23×D47—22no—
99C23×D47—22no—
100C23×D47—22no—
101C23×D47—22no—
102C23×D47—22no—
103C23×D47C22⋊C414yes—
104C23×D47C2×D412yes—
105C22×C4×C47—24no—
106C22×C4×C47—24no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47—42no—
112C22×C4×C47—42no—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D412yes—
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47#4112yes—
119C22×C4×C47#4112yes—
120C22×C4×C47C2411yescharacteristic
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—24no—
124C22×Dic47—24no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—42no—
128C22×Dic47—42no—
129C22×Dic47—42no—
130C22×Dic47—42no—
131C22×Dic47C22⋊C414yes—
132C22×Dic47C2×D412yes—
133C2×C4×D47—28no—
134C2×C4×D47—28no—
135C2×C4×D47—24no—
136C2×C4×D47—28no—
137C2×C4×D47—28no—
138C2×C4×D47—24no—
139C2×C4×D47—22no—
140C2×C4×D47—22no—
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47C22⋊C418yes—
144C2×C4×D47C22×C414yes—
145C2×C4×D47C2×D414yes—
146C2×Dic94—28no—
147C2×Dic94—28no—
148C2×Dic94—22no—
149C2×Dic94—22no—
150C2×Dic94—22no—
151C2×Dic94—22no—
152C2×Dic94—24no—
153C2×Dic94—22no—
154C2×Dic94—42no—
155C2×Dic94—44no—
156C2×Dic94—44no—
157C2×Dic94—42no—
158C2×Dic94C22×C414yes—
159C2×C8×C47—22no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—44no—
163C4⋊C4×C47—24no—
164C4⋊C4×C47—24no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47C22×C414yes—
168C4⋊C4×C47C2×D414yes—
169(C2×D47)⋊C4—216no—
170(C2×D47)⋊C4—216no—
171C4⋊Dic47—24no—
172C4⋊Dic47—22no—
173C4⋊Dic47—216no—
174C4⋊Dic47C2×D414yes—
175C4×Dic47—216no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—44no—
180C23×C47C2×C22⋊C412yes—
181C23×C47#12012yes—
182C23×C47#12312yes—
183C23×C47#13512yes—
184C23×C47#13512yes—
185C23×C47#13712yes—
186C23×C47#13712yes—
187C23×C47C22×D411yescharacteristic
188C22×D47—22no—
189C22×D47—22no—
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—28no—
195C22×D47—28no—
196C22×D47—28no—
197C22×D47—44no—
198C22×D47—44no—
199C22×D47—44no—
200C22×D47—44no—
201C22×D47C2×C22⋊C414yes—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—48no—
209C2×C4×C47—48no—
210C2×C4×C47C2×C22⋊C412yes—
211C2×C4×C47C4×D414yes—
212C2×C4×C47C4×D414yes—
213C2×C4×C47#12012yes—
214C2×C4×C47#12114yes—
215C2×C4×C47#12114yes—
216C2×C4×C47#12312yes—
217C2×C4×C47C23×C411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—28no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—48no—
233C2×Dic47—48no—
234C2×Dic47C2×C22⋊C414yes—
235C4×D47—24no—
236C4×D47—24no—
237C4×D47—24no—
238C4×D47—24no—
239C4×D47—216no—
240Dic94—24no—
241Dic94—24no—
242Dic94—42no—
243Dic94—42no—
244Dic94—48no—
245C8×C47—42no—
246C47⋊C8—42no—
247C2×#1873—471nomaximal, Sylow
248C22×C47#38614yes—
249C22×C47#41614yes—
250C22×C47#41914yes—
251C22×C47#45214yes—
252C22×C47#45214yes—
253C22×C47#48514yes—
254C22×C47#48514yes—
255C22×C47C22×C22⋊C411yescharacteristic
256C22×C47C2×#12011yescharacteristic
257C22×C47C2×#12311yescharacteristic
258C22×C47C2×#13511yescharacteristic
259C22×C47C2×#13511yescharacteristic
260C22×C47C2×#13711yescharacteristic
261C22×C47C2×#13711yescharacteristic
262C2×D47—24no—
263C2×D47—24no—
264C2×D47—24no—
265C2×D47—42no—
266C2×D47—42no—
267C2×D47—416no—
268C2×D47—42no—
269C2×D47—42no—
270C4×C47—44no—
271C4×C47—44no—
272C4×C47#38614yes—
273C4×C47C2×C4×D411yescharacteristic
274C4×C47C2×C4×D411yescharacteristic
275C4×C47C2×#12111yescharacteristic
276C4×C47C2×#12111yescharacteristic
277Dic47—24no—
278Dic47—44no—
279Dic47—44no—
280C22×C22⋊Q8—471no—
281C22×#117—471no—
282C22×#103—471no—
283C22×#103—471no—
284C22×C22⋊C8—471no—
285#1873—478no—
286C2×#340—471no—
287C2×#340D4711yescharacteristic, complemented
288C2×C47#187318yescomplemented
289C2×C47C2×#38611yescharacteristic
290C2×C47C2×#41611yescharacteristic
291C2×C47C2×#41911yescharacteristic
292C2×C47C2×#45211yescharacteristic
293C2×C47C2×#45211yescharacteristic
294C2×C47C2×#48511yescharacteristic
295C2×C47C2×#48511yescharacteristic
296D47—44no—
297C23×Q8—471no—
298C24×C4—471no—
299C23×C8—471no—
300C23×C8C2×D4711yescharacteristic
301C2×C22⋊Q8—478no—
302C2×C22⋊Q8—474no—
303C2×C22⋊Q8—942no—
304C2×C22⋊Q8—944no—
305C2×#117—474no—
306C2×#117—944no—
307C2×#117—942no—
308C2×#117—944no—
309C22×C4⋊C4—471no—
310C22×C4⋊C4—941no—
311C22×C4⋊C4C2×D4711yescharacteristic
312C22×C4⋊C4C2×D4711yescharacteristic
313C22×C22⋊C4—941no—
314C22×C22⋊C4—941no—
315C22×C42—941no—
316C2×#103—474no—
317C2×#103—474no—
318C2×#103—944no—
319C2×#103—944no—
320C2×C22⋊C8—474no—
321C2×C22⋊C8—944no—
322#340—474no—
323#340C2×D4714yescomplemented
324C47C2×#187311yescharacteristic, complemented, Sylow
325C25—471no—
326C22×Q8—474no—
327C22×Q8—472no—
328C22×Q8—942no—
329C22×Q8—941no—
330C22×Q8—942no—
331C22×Q8—942no—
332C22×Q8—942no—
333C22×Q8—942no—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—471no—
337C23×C4—474no—
338C23×C4—472no—
339C23×C4—474no—
340C23×C4—472no—
341C23×C4—942no—
342C23×C4—942no—
343C23×C4—942no—
344C23×C4—942no—
345C23×C4—941no—
346C23×C4—941no—
347C23×C4C22×D4711yescharacteristic
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—472no—
352C22×C8—942no—
353C22×C8—942no—
354C22×C8—942no—
355C22×C8C22×D4712yes—
356C22⋊Q8—4716no—
357C22⋊Q8—9416no—
358C22⋊Q8—948no—
359#117—948no—
360#117—9416no—
361#117—948no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—474no—
367C2×C4⋊C4—472no—
368C2×C4⋊C4—948no—
369C2×C4⋊C4—944no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4C4×D4714yes—
373C2×C4⋊C4D18814yes—
374C2×C4⋊C4C22×D4712yes—
375C2×C4⋊C4C22×D4712yes—
376C2×C22⋊C4—944no—
377C2×C22⋊C4—948no—
378C2×C22⋊C4—944no—
379C2×C22⋊C4—948no—
380C2×C42—944no—
381C2×C42—948no—
382#103—948no—
383#103—948no—
384C22⋊C8—948no—
385C24—474no—
386C24—472no—
387C24—942no—
388C24—942no—
389C24—942no—
390C24—942no—
391C24—942no—
392C24—942no—
393C24D4×D4711yescharacteristic
394C2×Q8—474no—
395C2×Q8—944no—
396C2×Q8—942no—
397C2×Q8—948no—
398C2×Q8—948no—
399C2×Q8—942no—
400C2×Q8—942no—
401C2×Q8—942no—
402C2×Q8—942no—
403C2×Q8—1882no—
404C2×Q8—1884no—
405C2×Q8—1884no—
406C2×Q8—1882no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—474no—
416C22×C4—472no—
417C22×C4—478no—
418C22×C4—474no—
419C22×C4—474no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—948no—
423C22×C4—948no—
424C22×C4—944no—
425C22×C4—948no—
426C22×C4—948no—
427C22×C4—944no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—944no—
433C22×C4—944no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—1882no—
437C22×C4—1882no—
438C22×C4—1882no—
439C22×C4—1882no—
440C22×C4C2×C4×D4712yes—
441C22×C4C2×D18812yes—
442C22×C4#1062535012yes—
443C22×C4D4×D4711yescharacteristic
444C22×C4D4×D4712yes—
445C22×C4D4×D4712yes—
446C22×C4#1062535412yes—
447C22×C4C23×D4711yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—942no—
453C2×C8—942no—
454C2×C8—942no—
455C2×C8—1884no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—474no—
461C4⋊C4—9416no—
462C4⋊C4—944no—
463C4⋊C4—942no—
464C4⋊C4C2×C4×D4714yes—
465C4⋊C4C2×D18814yes—
466C22⋊C4—9416no—
467C22⋊C4—9416no—
468C42—9416no—
469C23—474no—
470C23—948no—
471C23—942no—
472C23—948no—
473C23—942no—
474C23—948no—
475C23—942no—
476C23—942no—
477C23—942no—
478C23—942no—
479C23—1884no—
480C23—1884no—
481C23—1884no—
482C23—1884no—
483C23C22⋊C4×D4712yes—
484C23#1309399412yes—
485C23#1309400912yes—
486C23#1309404312yes—
487C23#1309404512yes—
488C23#1309405312yes—
489C23#1309405612yes—
490C23C2×D4×D4711yescharacteristic
491Q8—944no—
492Q8—944no—
493Q8—1888no—
494Q8—1882no—
495Q8—1882no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—474no—
505C2×C4—948no—
506C2×C4—942no—
507C2×C4—942no—
508C2×C4—9416no—
509C2×C4—944no—
510C2×C4—944no—
511C2×C4—944no—
512C2×C4—944no—
513C2×C4—944no—
514C2×C4—944no—
515C2×C4—1888no—
516C2×C4—1888no—
517C2×C4—1882no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4C22⋊C4×D4712yes—
522C2×C4#1309398614yes—
523C2×C4C4×D18814yes—
524C2×C4#1309399412yes—
525C2×C4#1309399714yes—
526C2×C4#1309399914yes—
527C2×C4#1309400912yes—
528C2×C4C22×C4×D4711yescharacteristic
529C2×C4C2×D4×D4711yescharacteristic
530C2×C4C2×D4×D4711yescharacteristic
531C2×C4C22×D18811yescharacteristic
532C2×C4C2×#1062535011yescharacteristic
533C2×C4C2×#1062535411yescharacteristic
534C8—42no—
535C8—1882no—
536C22—944no—
537C22—944no—
538C22—944no—
539C22—1882no—
540C22—1882no—
541C22—1882no—
542C22—1882no—
543C22—18816no—
544C22#1568900114yes—
545C22#1568911614yes—
546C22#1568913014yes—
547C22#1568928614yes—
548C22#1568928814yes—
549C22#1568946614yes—
550C22#1568946714yes—
551C22C2×C22⋊C4×D4711yescharacteristic
552C22C2×#1309399411yescharacteristic
553C22C2×#1309400911yescharacteristic
554C22C2×#1309404311yescharacteristic
555C22C2×#1309404511yescharacteristic
556C22C2×#1309405311yescharacteristic
557C22C2×#1309405611yescharacteristic
558C4—44no—
559C4—44no—
560C4—944no—
561C4—1884no—
562C4—1884no—
563C4#1568900114yes—
564C4C2×#1309398611yescharacteristic
565C4C2×C4×D18811yescharacteristic
566C4C2×#1309399711yescharacteristic
567C4C2×#1309399911yescharacteristic
568C2—1884no—
569C2#2024369918yesdirect product factor
570C2C2×#1568900111yescharacteristic
571C2C2×#1568911611yescharacteristic
572C2C2×#1568913011yescharacteristic
573C2C2×#1568928611yescharacteristic
574C2C2×#1568928811yescharacteristic
575C2C2×#1568946611yescharacteristic
576C2C2×#1568946711yescharacteristic
subgroupG/H
centerC24D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#340C2
socleC24×C47C2×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenterC2×#340D47
hypocenterC47C2×#1873
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
3. C22×#13093949C2semidirect
4. C22×#13093947C2semidirect
6. #20243699C2direct (N × H)
7. C2×#15688885C2semidirect
8. C2×C47×#340C2semidirect
14. C2×#13094004C4semidirect
26. C2×#13093949C22semidirect
28. C2×#13093947C22semidirect
31. #15688885C22semidirect
35. C47×#340C22semidirect
62. #13094004C2×C4semidirect
287. C2×#340D47semidirect
288. C2×C47#1873semidirect
323. #340C2×D47semidirect
324. C47C2×#1873semidirect
569. C2#20243699direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940042
C22×#130939822
C22×#130939492
C22×#130939472
C22×(C2×D47)⋊C82
#202436992
C2×#156888852
C2×C47×#3402
C2×#187347