Finite Group Databasebeta

← C2×#501×D47 #27085884

Subgroups — C2×#501×D47 #27085884

Order 12032.

2142 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D47C211yesmaximal, characteristic
2C2×#125×D47C211yesmaximal, characteristic
3C2×#15689533C211yesmaximal, characteristic, complemented
4C2×#15689532C211yesmaximal, characteristic, complemented
5C2×#15689531C212yesmaximal, complemented
6C2×#15689530C212yesmaximal, complemented
7C2×#15689529C211yesmaximal, characteristic, complemented
8#501×D47C2116yesmaximal, direct product factor
9C2×#107×D47C212yesmaximal
10C2×#106×D47C212yesmaximal
11C2×#97×D47C211yesmaximal, characteristic
12C2×C47×#501C211yesmaximal, characteristic, complemented
13C22×Q8×D47—21no—
14C22×C8×D47C2212yes—
15C2×#13094024C2211yescharacteristic
16C2×#13094023—21no—
17C2×#13094023C2212yes—
18C2×#13094022—21no—
19C4⋊Q8×D47—24no—
20C4⋊Q8×D47C2218yes—
21C2×#13094016C2211yescharacteristic
22C2×#13094015C2211yescharacteristic
23C2×#13094015C2211yescharacteristic
24C2×#13094014—21no—
25C2×#13094014—21no—
26#125×D47—24no—
27#125×D47C2218yes—
28C2×C4⋊C4×D47—21no—
29C2×C4⋊C4×D47—21no—
30C2×C4⋊C4×D47—21no—
31C2×C4⋊C4×D47C2211yescharacteristic
32C2×C4⋊C4×D47C2212yes—
33C2×C4⋊C4×D47C2211yescharacteristic
34C2×C42×D47C2211yescharacteristic
35#15689533C2218yescomplemented
36#15689532C2218yescomplemented
37#15689531C22116yescomplemented
38#15689530C22116yescomplemented
39#15689529C2218yescomplemented
40C2×#13093958C2212yes—
41C2×#13093957C2212yes—
42C2×#13093957C2212yes—
43#107×D47C22116yes—
44C2×#13093956C2212yes—
45C2×#13093955C2212yes—
46C2×#13093955C2212yes—
47#106×D47C22116yes—
48C2×#13093935C2211yescharacteristic
49C2×#13093934C2212yes—
50#97×D47C2218yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×#125×C47C2211yescharacteristic
53C47×#501C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×#106×C47C2212yes—
56C2×#97×C47C2211yescharacteristic
57C22×Dic94—21no—
58C22×Dic94—22no—
59C2×Q8×D47—28no—
60C2×Q8×D47—48no—
61C22×C4×D47—21no—
62C22×C4×D47—21no—
63C22×C4×D47—22no—
64C22×C4×D47D411yescharacteristic
65C22×C4×D47D411yescharacteristic
66C22×C4×D47C2311yescharacteristic
67C22×C47⋊C8C2312yes—
68C2×C8×D47Q8116yescomplemented
69C2×C8×D47C2318yes—
70#13094024—22no—
71#13094024C2314yes—
72#13094023—24no—
73#13094023—28no—
74#13094023C2318yes—
75#13094022—28no—
76#13094016—22no—
77#13094016C2314yes—
78#13094015—22no—
79#13094015—22no—
80#13094015C2314yes—
81#13094015C2314yes—
82#13094014—28no—
83#13094014—28no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47—21no—
86C2×C4⋊Dic47—21no—
87C2×C4⋊Dic47C2311yescharacteristic
88C2×C4⋊Dic47C2311yescharacteristic
89C2×C4⋊Dic47C2312yes—
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—22no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—22no—
98C4⋊C4×D47—28no—
99C4⋊C4×D47—28no—
100C4⋊C4×D47—28no—
101C4⋊C4×D47—22no—
102C4⋊C4×D47—22no—
103C4⋊C4×D47—24no—
104C4⋊C4×D47C2314yes—
105C4⋊C4×D47C2314yes—
106C4⋊C4×D47C2318yes—
107C2×C4×Dic47D411yescharacteristic
108C2×C4×Dic47D411yescharacteristic
109C2×C4×Dic47C2311yescharacteristic
110C42×D47D414yes—
111C42×D47C2314yes—
112#13093958C2318yes—
113#13093957C2318yes—
114#13093957C2318yes—
115#13093956C2318yes—
116#13093955C2318yes—
117#13093955C2318yes—
118#13093935C2314yes—
119#13093934C2318yes—
120C22×Q8×C47—21no—
121C22×C8×C47C2312yes—
122C4⋊Q8×C47—22no—
123C4⋊Q8×C47C2314yes—
124#125×C47—22no—
125#125×C47C2314yes—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C4⋊C4×C47C2312yes—
132C2×C42×C47C2311yescharacteristic
133#107×C47C2318yes—
134#106×C47C2318yes—
135#97×C47C2314yes—
136C23×D47C2×D411yescharacteristic
137C2×Q8×C47—24no—
138C2×Q8×C47—44no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—22no—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2411yescharacteristic
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—22no—
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151Q8×D47—432no—
152C2×C4×D47—24no—
153C2×C4×D47—24no—
154C2×C4×D47—24no—
155C2×C4×D47—24no—
156C2×C4×D47—28no—
157C2×C4×D47—44no—
158C2×C4×D47—48no—
159C2×C4×D47—44no—
160C2×C4×D47C2×D414yes—
161C2×C4×D47C2×D414yes—
162C2×C4×D47C2×D412yes—
163C2×C4×D47C2×Q818yes—
164C2×C4×D47C2412yes—
165C2×Dic94—24no—
166C2×Dic94—28no—
167C2×Dic94—48no—
168C2×Dic94—44no—
169C2×C8×C47C2×Q818yescomplemented
170C2×C8×C47C2414yes—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47C2412yes—
178C4⋊C4×C47C2414yes—
179C4⋊C4×C47C2412yes—
180C42×C47C2×D412yes—
181C42×C47C2412yes—
182C4⋊Dic47—24no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—22no—
186C4⋊Dic47—24no—
187C4⋊Dic47—24no—
188C4⋊Dic47C2412yes—
189C4⋊Dic47C2414yes—
190C4⋊Dic47C2412yes—
191Dic47⋊C4—24no—
192Dic47⋊C4—24no—
193Dic47⋊C4—24no—
194Dic47⋊C4—24no—
195Dic47⋊C4—28no—
196Dic47⋊C4—24no—
197Dic47⋊C4—24no—
198Dic47⋊C4—28no—
199C4×Dic47C2×D412yes—
200C4×Dic47C2×D414yes—
201C4×Dic47C2×D414yes—
202C4×Dic47C2412yes—
203C2×C47⋊C8C2×Q818yescomplemented
204C2×C47⋊C8C2414yes—
205C8×D47C2×Q8132yescomplemented
206C23×C47C22×D411yescharacteristic
207C22×D47C4⋊Q814yes—
208C22×D47AGL(1,ℤ/8)14yes—
209C22×D47#13714yes—
210C22×D47C22×D412yes—
211Q8×C47—48no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—44no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47C22×D412yes—
221C2×C4×C47C22×D412yes—
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C22×Q814yes—
224C2×C4×C47C2511yescharacteristic
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—24no—
230C2×Dic47—44no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47C4⋊Q814yes—
234C2×Dic47C22×D412yes—
235C2×Dic47C22×D412yes—
236C2×Dic47C22×D412yes—
237C4×D47—28no—
238C4×D47—28no—
239C4×D47—48no—
240C4×D47—416no—
241C4×D47—48no—
242C4×D47C4⋊Q818yes—
243C4×D47C22×Q818yes—
244Dic94—416no—
245Dic94—48no—
246C8×C47C22×Q818yescomplemented
247C47⋊C8C22×Q818yescomplemented
248C22×#501—471nomaximal, Sylow
249C22×C47C2×C4⋊Q812yes—
250C22×C47C2×AGL(1,ℤ/8)12yes—
251C22×C47C2×#13712yes—
252C22×C47C23×D411yescharacteristic
253C2×D47#501116yesdirect product factor
254C2×D47C2×C4⋊Q814yes—
255C2×D47C2×AGL(1,ℤ/8)14yes—
256C2×D47C2×#13714yes—
257C4×C47—22no—
258C4×C47—22no—
259C4×C47—44no—
260C4×C47—42no—
261C4×C47—42no—
262C4×C47C2×C4⋊Q812yes—
263C4×C47C23×Q812yes—
264Dic47—22no—
265Dic47—22no—
266Dic47—44no—
267Dic47—42no—
268Dic47—42no—
269Dic47C2×C4⋊Q814yes—
270C22×C4⋊Q8—471no—
271C22×#125—471no—
272C2×#501—4716no—
273C2×#501—472no—
274C2×#501—472no—
275C2×#501—471no—
276C2×#501—471no—
277C2×#501—471no—
278C2×#501D4711yescharacteristic, direct product factor
279C22×#107—472no—
280C22×#106—472no—
281C22×#97—471no—
282C2×C47C2×#50114yescomplemented
283C2×C47C22×C4⋊Q811yescharacteristic
284C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
285C2×C47C22×#13711yescharacteristic
286D47C2×#50118yesdirect product factor
287C23×Q8—941no—
288C23×C8—472no—
289C2×C4⋊Q8—478no—
290C2×C4⋊Q8—472no—
291C2×C4⋊Q8—471no—
292C2×C4⋊Q8—941no—
293C2×C4⋊Q8—944no—
294C2×C4⋊Q8—941no—
295C2×C4⋊Q8C2×D4711yescharacteristic
296C2×#125—478no—
297C2×#125—471no—
298C2×#125—471no—
299C2×#125—471no—
300C2×#125—941no—
301C2×#125—941no—
302C2×#125—944no—
303C2×#125C2×D4711yescharacteristic
304C22×C4⋊C4—471no—
305C22×C4⋊C4—472no—
306C22×C4⋊C4—471no—
307C22×C4⋊C4—941no—
308C22×C4⋊C4—941no—
309C22×C4⋊C4—941no—
310C22×C42—471no—
311#501—478no—
312#501—478no—
313#501—4716no—
314#501—4716no—
315#501—478no—
316#501C2×D4718yesdirect product factor
317C2×#107—472no—
318C2×#107—472no—
319C2×#107—472no—
320C2×#107—4716no—
321C2×#107C2×D4712yes—
322C2×#106—472no—
323C2×#106—472no—
324C2×#106—472no—
325C2×#106—4716no—
326C2×#106C2×D4712yes—
327C2×#97—472no—
328C2×#97—471no—
329C2×#97—478no—
330C2×#97C2×D4711yescharacteristic
331C47C22×#50111yescharacteristic, complemented, Sylow
332C22×Q8—21no—
333C22×Q8—942no—
334C22×Q8—948no—
335C22×Q8—941no—
336C22×Q8—1888no—
337C23×C4—471no—
338C23×C4—471no—
339C23×C4—471no—
340C23×C4—941no—
341C23×C4—941no—
342C23×C4—942no—
343C22×C8—472no—
344C22×C8—4716no—
345C22×C8—478no—
346C22×C8C22×D4712yes—
347C4⋊Q8—22no—
348C4⋊Q8—478no—
349C4⋊Q8—474no—
350C4⋊Q8—948no—
351C4⋊Q8—944no—
352C4⋊Q8—948no—
353C4⋊Q8—942no—
354C4⋊Q8C22×D4714yes—
355#125—22no—
356#125—474no—
357#125—474no—
358#125—474no—
359#125—948no—
360#125—948no—
361#125—942no—
362#125—942no—
363#125—942no—
364#125C22×D4714yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—472no—
371C2×C4⋊C4—474no—
372C2×C4⋊C4—474no—
373C2×C4⋊C4—478no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—944no—
377C2×C4⋊C4—948no—
378C2×C4⋊C4—948no—
379C2×C4⋊C4—948no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C4⋊C4C22×D4712yes—
394C2×C42—471no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—474no—
398C2×C42—474no—
399C2×C42C22×D4711yescharacteristic
400#107—478no—
401#107—478no—
402#107—478no—
403#107C22×D4718yes—
404#106—478no—
405#106—478no—
406#106—478no—
407#106C22×D4718yes—
408#97—478no—
409#97—474no—
410#97C22×D4714yes—
411C24—471no—
412C2×Q8—24no—
413C2×Q8—44no—
414C2×Q8—948no—
415C2×Q8—944no—
416C2×Q8—1884no—
417C2×Q8—1888no—
418C2×Q8—18832no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—471no—
423C22×C4—472no—
424C22×C4—478no—
425C22×C4—472no—
426C22×C4—471no—
427C22×C4—471no—
428C22×C4—474no—
429C22×C4—474no—
430C22×C4—944no—
431C22×C4—944no—
432C22×C4—944no—
433C22×C4—944no—
434C22×C4—948no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4—942no—
438C22×C4—1884no—
439C22×C4—1884no—
440C22×C4—1888no—
441C22×C4D4×D4711yescharacteristic
442C22×C4D4×D4711yescharacteristic
443C22×C4C23×D4711yescharacteristic
444C2×C8—478no—
445C2×C8—474no—
446C2×C8—4732no—
447C2×C8Q8×D4718yescomplemented
448C2×C8C23×D4714yes—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—472no—
456C4⋊C4—474no—
457C4⋊C4—472no—
458C4⋊C4—944no—
459C4⋊C4—944no—
460C4⋊C4—948no—
461C4⋊C4—944no—
462C4⋊C4—944no—
463C4⋊C4—948no—
464C4⋊C4—944no—
465C4⋊C4—944no—
466C4⋊C4—944no—
467C4⋊C4—944no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—942no—
471C4⋊C4—944no—
472C4⋊C4C23×D4712yes—
473C4⋊C4C23×D4714yes—
474C4⋊C4C23×D4712yes—
475C42—474no—
476C42—474no—
477C42—472no—
478C42—472no—
479C42D4×D4712yes—
480C42C23×D4712yes—
481C23—474no—
482C23—474no—
483C23—474no—
484C23—472no—
485C23C2×D4×D4711yescharacteristic
486Q8—48no—
487Q8—1888no—
488Q8—18816no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—474no—
498C2×C4—472no—
499C2×C4—478no—
500C2×C4—478no—
501C2×C4—472no—
502C2×C4—472no—
503C2×C4—948no—
504C2×C4—948no—
505C2×C4—942no—
506C2×C4—942no—
507C2×C4—942no—
508C2×C4—942no—
509C2×C4—944no—
510C2×C4—1888no—
511C2×C4—1888no—
512C2×C4—18816no—
513C2×C4—1882no—
514C2×C4—1884no—
515C2×C4—1882no—
516C2×C4C2×D4×D4712yes—
517C2×C4C2×D4×D4712yes—
518C2×C4C2×D4×D4711yescharacteristic
519C2×C4C2×Q8×D4714yes—
520C2×C4C24×D4711yescharacteristic
521C8—478no—
522C8C2×Q8×D4718yescomplemented
523C22—474no—
524C22—474no—
525C22—474no—
526C22—4716no—
527C22C4⋊Q8×D4712yes—
528C22AGL(1,ℤ/8)×D4712yes—
529C22#137×D4712yes—
530C22C22×D4×D4711yescharacteristic
531C4—22no—
532C4—22no—
533C4—44no—
534C4—42no—
535C4—42no—
536C4—474no—
537C4—942no—
538C4—942no—
539C4—1884no—
540C4—1882no—
541C4—1882no—
542C4C4⋊Q8×D4712yes—
543C4C22×Q8×D4712yes—
544C2—478no—
545C2#501×D4714yesdirect product factor
546C2C2×C4⋊Q8×D4711yescharacteristic
547C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
548C2C2×#137×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#501C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenterC2×#501D47
hypocenterC47C22×#501
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. C2×#15689533C2semidirect
4. C2×#15689532C2semidirect
5. C2×#15689531C2semidirect
6. C2×#15689530C2semidirect
7. C2×#15689529C2semidirect
8. #501×D47C2direct (N × H)
12. C2×C47×#501C2semidirect
35. #15689533C22semidirect
36. #15689532C22semidirect
37. #15689531C22semidirect
38. #15689530C22semidirect
39. #15689529C22semidirect
53. C47×#501C22semidirect
68. C2×C8×D47Q8semidirect
169. C2×C8×C47C2×Q8semidirect
203. C2×C47⋊C8C2×Q8semidirect
205. C8×D47C2×Q8semidirect
246. C8×C47C22×Q8semidirect
247. C47⋊C8C22×Q8semidirect
253. C2×D47#501direct (N × H)
278. C2×#501D47direct (N × H)
282. C2×C47C2×#501semidirect
286. D47C2×#501direct (N × H)
316. #501C2×D47direct (N × H)
331. C47C22×#501semidirect
447. C2×C8Q8×D47semidirect
522. C8C2×Q8×D47semidirect
545. C2#501×D47direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D472
C2×#125×D472
C2×#156895332
C2×#156895322
C2×#156895312
C2×#156895302
C2×#156895292
#501×D472
C2×#107×D472
C2×#106×D472
C2×#97×D472
C2×C47×#5012
C22×#50147