Finite Group Databasebeta

← C2×#20254565 #27088354

Subgroups — C2×#20254565 #27088354

Order 12032.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic, complemented
2C2×#15689655C211yesmaximal, characteristic
3#20254565C2116yesmaximal, direct product factor
4C2×#15689524C212yesmaximal, complemented
5C2×#15689476C212yesmaximal, complemented
6C2×#15689466C212yesmaximal, complemented
7C2×#15689445C212yesmaximal
8C2×#15689419C211yesmaximal, characteristic, complemented
9C2×#15689405C211yesmaximal, characteristic, complemented
10C2×#15689413C211yesmaximal, characteristic, complemented
11C2×#15689175C211yesmaximal, characteristic, complemented
12C2×C47×#474C211yesmaximal, characteristic, complemented
13#15689678—24no—
14#15689678C2218yescomplemented
15C2×#13094024C2211yescharacteristic
16#15689655—24no—
17#15689655C2218yes—
18C2×#13094017—21no—
19C2×#13094015C2212yes—
20C2×#13094015C2211yescharacteristic
21C2×#13094011C2211yescharacteristic
22C2×#13094006—21no—
23C2×#13094004—21no—
24C2×#13094004C2211yescharacteristic
25C2×#13094000—21no—
26C2×#13093995C2211yescharacteristic
27C2×Q8×Dic47—21no—
28C2×C4×Dic94C2211yescharacteristic
29C2×#13093989—21no—
30C2×C4×C22⋊D47—21no—
31C2×#13093983—21no—
32C2×#13093982C2212yes—
33C2×C42×D47C2211yescharacteristic
34#15689524C22116yescomplemented
35#15689476C22116yescomplemented
36#15689466C22116yescomplemented
37#15689445C22116yes—
38#15689419C2218yescomplemented
39#15689413C2218yescomplemented
40#15689405C2218yescomplemented
41#15689175C2218yescomplemented
42C2×#13093958C2212yes—
43C2×#13093955C2212yes—
44C2×#13093955C2212yes—
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46C2×#13093949C2212yes—
47C2×#13093947C2212yes—
48C2×#13093945C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×C8×Dic47C2212yes—
51C2×C4⋊Q8×C47C2211yescharacteristic
52C2×C4×D4×C47C2211yescharacteristic
53C47×#474C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#102×C47C2212yes—
57C22×Dic94—21no—
58C22×Dic94C2311yescharacteristic
59C22×C22⋊D47—21no—
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2312yes—
64#13094024—22no—
65#13094024C2314yes—
66#13094017—28no—
67#13094015—22no—
68#13094015—24no—
69#13094015C2318yes—
70#13094015C2314yes—
71#13094011—22no—
72#13094011C2314yes—
73#13094006—28no—
74#13094004—28no—
75#13094004—22no—
76#13094004C2314yes—
77#13094000—28no—
78#13093995—22no—
79#13093995C2314yes—
80Q8×Dic47—28no—
81C4×Dic94—22no—
82C4×Dic94C2314yes—
83#13093989—28no—
84C4×C22⋊D47—28no—
85#13093983—28no—
86#13093982—24no—
87#13093982C2318yes—
88C2×C4⋊Dic47—21no—
89C2×C4⋊Dic47—21no—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47—22no—
92C2×C4⋊Dic47C2312yes—
93C2×C4⋊Dic47C2311yescharacteristic
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—22no—
97C2×C22⋊Dic47—21no—
98C2×C22⋊Dic47—21no—
99C2×C22⋊Dic47—21no—
100C2×(C2×D47)⋊C4—22no—
101C2×(C2×D47)⋊C4—21no—
102C2×C4×Dic47—21no—
103C2×C4×Dic47—22no—
104C2×C4×Dic47D411yescharacteristic
105C2×C4×Dic47D411yescharacteristic
106C2×C4×Dic47C2311yescharacteristic
107C42×D47D414yes—
108C42×D47C2314yes—
109#13093958C2318yes—
110#13093955C2318yes—
111#13093955C2318yes—
112C4⋊(C47⋊C8)C2314yes—
113#13093949C2318yes—
114#13093947C2318yes—
115#13093945C2318yes—
116(C2×D47)⋊C8C2318yes—
117C8×Dic47C2318yes—
118C22×Q8×C47—21no—
119C22×D4×C47C2311yescharacteristic
120C23×C4×C47—21no—
121C22×C8×C47C2312yes—
122C4⋊Q8×C47—22no—
123C4⋊Q8×C47C2314yes—
124C4×D4×C47—22no—
125C4×D4×C47C2314yes—
126C2×C4⋊C4×C47—21no—
127C2×C4⋊C4×C47C2311yescharacteristic
128C2×C4⋊C4×C47C2312yes—
129C2×C22⋊C4×C47—21no—
130C2×C42×C47C2311yescharacteristic
131#107×C47C2318yes—
132C4⋊C8×C47C2314yes—
133#102×C47C2318yes—
134C24×C47—21no—
135C23×D47C2×D411yescharacteristic
136C2×Q8×C47—24no—
137C2×Q8×C47—44no—
138C2×D4×C47—21no—
139C2×D4×C47—22no—
140C2×D4×C47—22no—
141C2×D4×C47C2412yes—
142C22×C4×C47—21no—
143C22×C4×C47—22no—
144C22×C4×C47—24no—
145C22×C4×C47—44no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C2×C22⋊D47—24no—
150C2×C22⋊D47—44no—
151C22×Dic47—21no—
152C22×Dic47—22no—
153C22×Dic47—21no—
154C22×Dic47C2×D411yescharacteristic
155C22×Dic47C2×D411yescharacteristic
156C22×Dic47C2×D411yescharacteristic
157C2×C4×D47—24no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—42no—
161C2×C4×D47C2×D412yes—
162C2×C4×D47C2×D414yes—
163C2×C4×D47C2×D414yes—
164C2×C4×D47C2412yes—
165C2×Dic94—22no—
166C2×Dic94—22no—
167C2×Dic94—24no—
168C2×Dic94—21no—
169C2×Dic94—44no—
170C2×Dic94C2412yes—
171C2×C8×C47—24no—
172C2×C8×C47C2414yes—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47C2412yes—
177C4⋊C4×C47C2414yes—
178C22⋊C4×C47—24no—
179C42×C47C2×D412yes—
180C42×C47C2412yes—
181C22⋊Dic47—24no—
182C22⋊Dic47—24no—
183C22⋊Dic47—24no—
184(C2×D47)⋊C4—28no—
185(C2×D47)⋊C4—24no—
186C4⋊Dic47—24no—
187C4⋊Dic47—22no—
188C4⋊Dic47—21no—
189C4⋊Dic47—24no—
190C4⋊Dic47—24no—
191C4⋊Dic47—28no—
192C4⋊Dic47C2414yes—
193C4⋊Dic47C2412yes—
194Dic47⋊C4—24no—
195Dic47⋊C4—24no—
196Dic47⋊C4—28no—
197C4×Dic47—24no—
198C4×Dic47—28no—
199C4×Dic47C2×D412yes—
200C4×Dic47C2×D414yes—
201C4×Dic47C2×D414yes—
202C4×Dic47C2412yes—
203C2×C47⋊C8—24no—
204C2×C47⋊C8C2414yes—
205C23×C47—22no—
206C23×C47—42no—
207C23×C47—42no—
208C23×C47—42no—
209C23×C47C22×D411yescharacteristic
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47C22×D412yes—
214Q8×C47—48no—
215D4×C47—22no—
216D4×C47—22no—
217D4×C47—42no—
218C2×C4×C47—22no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—42no—
222C2×C4×C47—44no—
223C2×C4×C47—48no—
224C2×C4×C47#13514yes—
225C2×C4×C47C22×D412yes—
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×D412yes—
228C2×C4×C472−1+412yes—
229C2×C4×C472−1+412yes—
230C2×C4×C47C2511yescharacteristic
231C22⋊D47—48no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—22no—
238C2×Dic47—42no—
239C2×Dic47—44no—
240C2×Dic47—42no—
241C2×Dic47C22×D412yes—
242C2×Dic47C22×D412yes—
243C2×Dic47C22×D412yes—
244C4×D47—24no—
245C4×D47—28no—
246C4×D47—22no—
247C4×D47—22no—
248C4×D47—44no—
249Dic94—22no—
250Dic94—22no—
251Dic94—48no—
252Dic94—42no—
253C8×C47—24no—
254C47⋊C8—24no—
255C2×#3233—471nomaximal, Sylow
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—84no—
260C22×C47#53612yes—
261C22×C47C2×#13512yes—
262C22×C47#57512yes—
263C22×C47C23×D411yescharacteristic
264C2×D47—22no—
265C2×D47—22no—
266C2×D47—22no—
267C2×D47—44no—
268C4×C47—22no—
269C4×C47—42no—
270C4×C47—44no—
271C4×C47#53612yes—
272C4×C47C2×#13514yes—
273C4×C47C2×2−1+411yescharacteristic
274C4×C47C2×2−1+411yescharacteristic
275Dic47—22no—
276Dic47—22no—
277Dic47—22no—
278Dic47—42no—
279Dic47—44no—
280Dic47—42no—
281C2×#533—471no—
282C2×#529—471no—
283#3233—4716no—
284C2×#499—472no—
285C2×#488—472no—
286C2×#485—472no—
287C2×#482—472no—
288C2×#477—471no—
289C2×#475—471no—
290C2×#476—471no—
291C2×#474D4711yescharacteristic, complemented
292C2×#431—471no—
293C2×C47—82no—
294C2×C47#323314yescomplemented
295C2×C47C2×#53611yescharacteristic
296C2×C47C22×#13511yescharacteristic
297C2×C47C2×#57511yescharacteristic
298D47—42no—
299#533—478no—
300#533—944no—
301C2×C4⋊Q8—471no—
302C2×C4⋊Q8C2×D4711yescharacteristic
303#529—478no—
304#529—944no—
305C2×#126—941no—
306C2×#125—472no—
307C2×#125—471no—
308C2×#124—471no—
309C2×#123—941no—
310C2×C22⋊Q8—471no—
311C2×C22⋊Q8—941no—
312C2×C22⋊Q8—941no—
313C2×#121—471no—
314C2×C4×Q8—471no—
315C2×C4×Q8—941no—
316C2×C4×Q8—941no—
317C2×C4×D4—941no—
318C2×C4×D4C2×D4711yescharacteristic
319C2×#117—472no—
320C2×#117—941no—
321C22×C42—471no—
322#499—4716no—
323#488—4716no—
324#485—4716no—
325#482—4716no—
326#477—478no—
327#476—478no—
328#475—478no—
329#474C2×D4718yescomplemented
330#431—478no—
331C2×#107—472no—
332C2×#107C2×D4712yes—
333C2×#106—472no—
334C2×#106—472no—
335C2×C4⋊C8—471no—
336C2×C4⋊C8C2×D4711yescharacteristic
337C2×#103—472no—
338C2×#103—472no—
339C2×#102—472no—
340C2×#102C2×D4712yes—
341C2×C22⋊C8—472no—
342C2×C4×C8—472no—
343C47C2×#323311yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C22×Q8—471no—
346C22×Q8—941no—
347C22×D4—941no—
348C22×D4C22×D4711yescharacteristic
349C23×C4—21no—
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—471no—
353C22×C8—472no—
354C22×C8C22×D4712yes—
355C4⋊Q8—22no—
356C4⋊Q8—474no—
357C4⋊Q8—942no—
358C4⋊Q8C22×D4714yes—
359#126—948no—
360#125—474no—
361#125—478no—
362#125—942no—
363#125—944no—
364#124—474no—
365#124—942no—
366#123—948no—
367C22⋊Q8—474no—
368C22⋊Q8—942no—
369C22⋊Q8—948no—
370C22⋊Q8—948no—
371#121—474no—
372#121—942no—
373C4×Q8—474no—
374C4×Q8—948no—
375C4×Q8—948no—
376C4×Q8—942no—
377C4×D4—22no—
378C4×D4—948no—
379C4×D4C22×D4714yes—
380#117—478no—
381#117—948no—
382#117—944no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—472no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4C22×D4712yes—
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—942no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—474no—
405C2×C42—474no—
406C2×C42—942no—
407C2×C42—941no—
408C2×C42C22×D4711yescharacteristic
409#107—478no—
410#107C22×D4718yes—
411#106—478no—
412#106—478no—
413C4⋊C8—474no—
414C4⋊C8C22×D4714yes—
415#103—478no—
416#103—478no—
417#102—478no—
418#102C22×D4718yes—
419C22⋊C8—478no—
420C4×C8—478no—
421C24—21no—
422C24—471no—
423C2×Q8—24no—
424C2×Q8—44no—
425C2×Q8—472no—
426C2×Q8—941no—
427C2×Q8—944no—
428C2×Q8—942no—
429C2×Q8—942no—
430C2×Q8—1884no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—944no—
435C2×D4—1884no—
436C2×D4C23×D4712yes—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—471no—
442C22×C4—471no—
443C22×C4—471no—
444C22×C4—472no—
445C22×C4—474no—
446C22×C4—472no—
447C22×C4—474no—
448C22×C4—944no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—942no—
454C22×C4—1882no—
455C22×C4D4×D4711yescharacteristic
456C22×C4D4×D4711yescharacteristic
457C22×C4C23×D4711yescharacteristic
458C2×C8—24no—
459C2×C8—474no—
460C2×C8—944no—
461C2×C8C23×D4714yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—474no—
466C4⋊C4—472no—
467C4⋊C4—944no—
468C4⋊C4—948no—
469C4⋊C4—944no—
470C4⋊C4—944no—
471C4⋊C4—948no—
472C4⋊C4—944no—
473C4⋊C4—942no—
474C4⋊C4—941no—
475C4⋊C4—944no—
476C4⋊C4C23×D4714yes—
477C4⋊C4C23×D4712yes—
478C22⋊C4—24no—
479C22⋊C4—948no—
480C22⋊C4—944no—
481C22⋊C4—944no—
482C22⋊C4—944no—
483C22⋊C4—944no—
484C42—474no—
485C42—474no—
486C42—472no—
487C42—472no—
488C42—948no—
489C42—944no—
490C42D4×D4712yes—
491C42C23×D4712yes—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—472no—
497C23—942no—
498C23—942no—
499C23—942no—
500C23C2×D4×D4711yescharacteristic
501Q8—48no—
502Q8—942no—
503Q8—942no—
504Q8—1888no—
505Q8—1882no—
506D4—22no—
507D4—22no—
508D4—42no—
509D4—1888no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—472no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—948no—
523C2×C4—944no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—944no—
529C2×C4—1884no—
530C2×C4—1882no—
531C2×C4—1884no—
532C2×C4—1882no—
533C2×C4#1309404314yes—
534C2×C4C2×D4×D4712yes—
535C2×C4C2×D4×D4712yes—
536C2×C4C2×D4×D4711yescharacteristic
537C2×C4#1309407112yes—
538C2×C4#1309407212yes—
539C2×C4C24×D4711yescharacteristic
540C8—24no—
541C8—944no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—84no—
546C22—942no—
547C22—942no—
548C22—942no—
549C22—1884no—
550C22#1568969612yes—
551C22C2×#1309404312yes—
552C22#1568992212yes—
553C22C22×D4×D4711yescharacteristic
554C4—22no—
555C4—42no—
556C4—44no—
557C4—942no—
558C4—942no—
559C4—942no—
560C4—1882no—
561C4—1884no—
562C4—1882no—
563C4#1568969612yes—
564C4C2×#1309404314yes—
565C4C2×#1309407111yescharacteristic
566C4C2×#1309407211yescharacteristic
567C2—82no—
568C2—1882no—
569C2#2025456514yesdirect product factor
570C2C2×#1568969611yescharacteristic
571C2C22×#1309404311yescharacteristic
572C2C2×#1568992211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#474C2
socleC23×C47C22×D4
prime radicalC2×#13093995C22
hypercenterC2×#474D47
hypocenterC47C2×#3233
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#15689678C2semidirect
3. #20254565C2direct (N × H)
4. C2×#15689524C2semidirect
5. C2×#15689476C2semidirect
6. C2×#15689466C2semidirect
8. C2×#15689419C2semidirect
9. C2×#15689405C2semidirect
10. C2×#15689413C2semidirect
11. C2×#15689175C2semidirect
12. C2×C47×#474C2semidirect
14. #15689678C22semidirect
34. #15689524C22semidirect
35. #15689476C22semidirect
36. #15689466C22semidirect
38. #15689419C22semidirect
39. #15689413C22semidirect
40. #15689405C22semidirect
41. #15689175C22semidirect
53. C47×#474C22semidirect
291. C2×#474D47semidirect
294. C2×C47#3233semidirect
329. #474C2×D47semidirect
343. C47C2×#3233semidirect
569. C2#20254565direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896782
C2×#156896552
#202545652
C2×#156895242
C2×#156894762
C2×#156894662
C2×#156894452
C2×#156894192
C2×#156894052
C2×#156894132
C2×#156891752
C2×C47×#4742
C2×#323347