Finite Group Databasebeta

← C2×#20254556 #27088308

Subgroups — C2×#20254556 #27088308

Order 12032.

1822 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689657C211yesmaximal, characteristic
2#20254556C2116yesmaximal, direct product factor
3C2×#15689522C212yesmaximal, complemented
4C2×#15689480C212yesmaximal, complemented
5C2×#15689458C211yesmaximal, characteristic, complemented
6C2×#15689441C211yesmaximal, characteristic
7C2×#15689453C211yesmaximal, characteristic, complemented
8C2×#15689436C211yesmaximal, characteristic
9C2×#15689427C212yesmaximal, complemented
10C2×#15689420C212yesmaximal, complemented
11C2×#15689176C211yesmaximal, characteristic, complemented
12C22×#125×C47C211yesmaximal, characteristic, complemented
13#15689657—24no—
14#15689657C2218yes—
15C2×#13094018—21no—
16C2×#13094016C2211yescharacteristic
17C2×#13094012C2211yescharacteristic
18C2×#13094007—21no—
19C2×#13094003C2211yescharacteristic
20C2×#13093998C2211yescharacteristic
21C2×C4×Dic94C2211yescharacteristic
22C2×C4×D188C2211yescharacteristic
23C2×C4×C22⋊D47—21no—
24C2×#13093981—21no—
25#15689522C22116yescomplemented
26#15689480C22116yescomplemented
27#15689458C2218yescomplemented
28#15689453C2218yescomplemented
29#15689441C2218yes—
30#15689436C2218yes—
31#15689427C22116yescomplemented
32#15689420C22116yescomplemented
33#15689176C2218yescomplemented
34C2×#13093957C2212yes—
35C2×#13093957C2212yes—
36C2×#13093955C2212yes—
37C2×#13093955C2212yes—
38C2×C4⋊(C47⋊C8)C2212yes—
39C2×#13093947C2212yes—
40C2×#13093947C2212yes—
41C2×#13093944C2212yes—
42C2×#13093944C2212yes—
43C2×C22⋊(C47⋊C8)C2211yescharacteristic
44C2×C22⋊(C47⋊C8)C2211yescharacteristic
45C2×C4×C47⋊C8C2212yes—
46C2×#125×C47—21no—
47C2×#125×C47—21no—
48C2×#125×C47—24no—
49C2×#125×C47C2212yes—
50C2×#125×C47C2212yes—
51C2×#125×C47C2218yescomplemented
52C22×C4⋊C4×C47—21no—
53C22×C4⋊C4×C47—21no—
54C22×C4⋊C4×C47C2211yescharacteristic
55C22×C4⋊C4×C47C2211yescharacteristic
56C22×C42×C47C2211yescharacteristic
57C22×Dic94C2311yescharacteristic
58C22×D188C2311yescharacteristic
59C22×C22⋊D47—21no—
60C22×C4×D47—21no—
61C22×C47⋊C8C2312yes—
62C22×C47⋊C8C2312yes—
63#13094018—28no—
64#13094016—22no—
65#13094016C2314yes—
66#13094012—22no—
67#13094012C2314yes—
68#13094007—28no—
69#13094003—22no—
70#13094003C2314yes—
71#13093998—22no—
72#13093998C2314yes—
73C4×Dic94—22no—
74C4×Dic94C2314yes—
75C4×D188—22no—
76C4×D188C2314yes—
77C4×C22⋊D47—28no—
78#13093981—28no—
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×C22⋊Dic47—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4—21no—
88C2×(C2×D47)⋊C4—21no—
89C2×(C2×D47)⋊C4—21no—
90C2×C4×Dic47—21no—
91#13093957C2318yes—
92#13093957C2318yes—
93#13093955C2318yes—
94#13093955C2318yes—
95C4⋊(C47⋊C8)C2318yes—
96#13093947C2318yes—
97#13093947C2318yes—
98#13093944C2318yes—
99#13093944C2318yes—
100C22⋊(C47⋊C8)C2314yes—
101C22⋊(C47⋊C8)C2314yes—
102C4×C47⋊C8C2318yes—
103C23×C4×C47—21no—
104C23×C4×C47—21no—
105C23×C4×C47D411yescharacteristic
106C23×C4×C47D411yescharacteristic
107C23×C4×C47C2311yescharacteristic
108#125×C47—24no—
109#125×C47—24no—
110#125×C47—28no—
111#125×C47—28no—
112#125×C47C2318yes—
113#125×C47C2318yes—
114C2×C4⋊C4×C47—22no—
115C2×C4⋊C4×C47—22no—
116C2×C4⋊C4×C47—22no—
117C2×C4⋊C4×C47—22no—
118C2×C4⋊C4×C47—28no—
119C2×C4⋊C4×C47—28no—
120C2×C4⋊C4×C47—21no—
121C2×C4⋊C4×C47—21no—
122C2×C4⋊C4×C47—22no—
123C2×C4⋊C4×C47—22no—
124C2×C4⋊C4×C47C2312yes—
125C2×C4⋊C4×C47C2312yes—
126C2×C4⋊C4×C47C2314yes—
127C2×C4⋊C4×C47C2314yes—
128C2×C42×C47D411yescharacteristic
129C2×C42×C47D411yescharacteristic
130C2×C42×C47D414yes—
131C2×C42×C47C2311yescharacteristic
132C2×C42×C47C2311yescharacteristic
133C2×C42×C47C2314yes—
134C24×C47C2×D411yescharacteristic
135C23×D47—21no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—24no—
139C22×C4×C47—24no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—42no—
143C22×C4×C47—44no—
144C22×C4×C47—44no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D414yes—
152C22×C4×C47C2×D414yes—
153C22×C4×C47C2412yes—
154C22×C4×C47C2411yescharacteristic
155C2×C22⋊D47—24no—
156C2×C22⋊D47—44no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C2×D188—24no—
161C2×D188—21no—
162C2×D188C2412yes—
163C2×C4×D47—24no—
164C2×C4×D47—44no—
165C2×Dic94—24no—
166C2×Dic94—21no—
167C2×Dic94C2412yes—
168C4⋊C4×C47—28no—
169C4⋊C4×C47—28no—
170C4⋊C4×C47—28no—
171C4⋊C4×C47—28no—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47C2414yes—
177C4⋊C4×C47C2414yes—
178C42×C47C2×D414yes—
179C42×C47C2×D412yes—
180C42×C47C2×D412yes—
181C42×C47C2×D414yes—
182C42×C47C2412yes—
183C42×C47C2412yes—
184C22⋊Dic47—24no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—24no—
187(C2×D47)⋊C4—24no—
188(C2×D47)⋊C4—24no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47C2412yes—
192C4⋊Dic47C2412yes—
193Dic47⋊C4—24no—
194Dic47⋊C4—24no—
195Dic47⋊C4—24no—
196Dic47⋊C4—24no—
197C4×Dic47—24no—
198C2×C47⋊C8—24no—
199C2×C47⋊C8—24no—
200C2×C47⋊C8C2414yes—
201C2×C47⋊C8C2414yes—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47#13514yes—
205C23×C47C22×D412yes—
206C23×C47C22×D411yescharacteristic
207C22×D47—22no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—42no—
211C2×C4×C47—22no—
212C2×C4×C47—28no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—44no—
222C2×C4×C47—48no—
223C2×C4×C47—48no—
224C2×C4×C47—44no—
225C2×C4×C47—44no—
226C2×C4×C47C22×D412yes—
227C2×C4×C47C22×D412yes—
228C2×C4×C47C22×D412yes—
229C2×C4×C47C22×D412yes—
230C2×C4×C47C22×D412yes—
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C472−1+414yes—
233C2×C4×C47C2511yescharacteristic
234C22⋊D47—48no—
235C2×Dic47—22no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241D188—24no—
242D188—42no—
243C4×D47—48no—
244Dic94—24no—
245Dic94—42no—
246C47⋊C8—24no—
247C47⋊C8—24no—
248C2×#3232—471nomaximal, Sylow
249C22×C47—22no—
250C22×C47—28no—
251C22×C47—22no—
252C22×C47#53612yes—
253C22×C47C2×#13514yes—
254C22×C47C2×#13512yes—
255C22×C47#57512yes—
256C22×C47C23×D411yescharacteristic
257C2×D47—42no—
258C2×D47—42no—
259C2×D47—42no—
260C2×D47—84no—
261C4×C47—22no—
262C4×C47—22no—
263C4×C47—22no—
264C4×C47—22no—
265C4×C47—22no—
266C4×C47—44no—
267C4×C47—44no—
268C4×C47#53612yes—
269C4×C47C2×2−1+412yes—
270Dic47—42no—
271Dic47—42no—
272Dic47—42no—
273C2×#529—471no—
274C22×#125D4711yescharacteristic, complemented
275#3232—4716no—
276C2×#499—472no—
277C2×#488—472no—
278C2×#484—471no—
279C2×#481—471no—
280C2×#483—471no—
281C2×#480—471no—
282C2×#479—472no—
283C2×#478—472no—
284C2×#431—471no—
285C2×C47—24no—
286C2×C47#323214yescomplemented
287C2×C47C2×#53611yescharacteristic
288C2×C47C22×#13511yescharacteristic
289C2×C47C2×#57511yescharacteristic
290D47—82no—
291#529—478no—
292#529—944no—
293C2×#126—941no—
294C2×#125—21no—
295C2×#125—21no—
296C2×#125—24no—
297C2×#125—471no—
298C2×#125C2×D4712yes—
299C2×#125C2×D4712yes—
300C2×#125C2×D4718yescomplemented
301C2×#124—471no—
302C2×#123—941no—
303C2×C22⋊Q8—471no—
304C2×#121—471no—
305C2×C4×Q8—471no—
306C2×C4×D4—471no—
307C2×C4×D4—941no—
308C2×#117—941no—
309C22×C4⋊C4—21no—
310C22×C4⋊C4—21no—
311C22×C4⋊C4C2×D4711yescharacteristic
312C22×C4⋊C4C2×D4711yescharacteristic
313C22×C42C2×D4711yescharacteristic
314#499—4716no—
315#488—4716no—
316#484—478no—
317#483—478no—
318#481—478no—
319#480—478no—
320#479—4716no—
321#478—4716no—
322#431—478no—
323C2×#107—472no—
324C2×#107—472no—
325C2×#106—472no—
326C2×#106—472no—
327C2×C4⋊C8—472no—
328C2×#103—472no—
329C2×#103—472no—
330C2×#102—472no—
331C2×#102—472no—
332C2×C22⋊C8—471no—
333C2×C22⋊C8—471no—
334C2×C4×C8—472no—
335C47C2×#323211yescharacteristic, complemented, Sylow
336C22×Q8—471no—
337C22×D4—471no—
338C22×D4—941no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—941no—
342C23×C4C22⋊D4711yescharacteristic
343C23×C4C22⋊D4711yescharacteristic
344C23×C4C22×D4711yescharacteristic
345C22×C8—472no—
346C22×C8—472no—
347#126—948no—
348#125—24no—
349#125—24no—
350#125—28no—
351#125—28no—
352#125—474no—
353#125—942no—
354#125C22×D4718yes—
355#125C22×D4718yes—
356#124—474no—
357#124—942no—
358#123—948no—
359C22⋊Q8—474no—
360C22⋊Q8—942no—
361#121—474no—
362#121—942no—
363C4×Q8—474no—
364C4×Q8—942no—
365C4×D4—474no—
366C4×D4—948no—
367C4×D4—942no—
368#117—948no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4C22×D4712yes—
386C2×C4⋊C4C22×D4712yes—
387C2×C4⋊C4C22×D4714yes—
388C2×C4⋊C4C22×D4714yes—
389C2×C22⋊C4—941no—
390C2×C22⋊C4—941no—
391C2×C22⋊C4—941no—
392C2×C22⋊C4—941no—
393C2×C22⋊C4—941no—
394C2×C42—941no—
395C2×C42C22⋊D4711yescharacteristic
396C2×C42C22⋊D4711yescharacteristic
397C2×C42C22⋊D4714yes—
398C2×C42C22×D4711yescharacteristic
399C2×C42C22×D4711yescharacteristic
400C2×C42C22×D4714yes—
401#107—478no—
402#107—478no—
403#106—478no—
404#106—478no—
405C4⋊C8—478no—
406#103—478no—
407#103—478no—
408#102—478no—
409#102—478no—
410C22⋊C8—474no—
411C22⋊C8—474no—
412C4×C8—478no—
413C24—941no—
414C24C2×C22⋊D4711yescharacteristic
415C2×Q8—472no—
416C2×Q8—941no—
417C2×Q8—944no—
418C2×D4—472no—
419C2×D4—944no—
420C2×D4—941no—
421C2×D4—944no—
422C2×D4—1884no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—941no—
435C22×C4—944no—
436C22×C4—1884no—
437C22×C4C2×C22⋊D4712yes—
438C22×C4C2×C22⋊D4714yes—
439C22×C4C2×C22⋊D4711yescharacteristic
440C22×C4C2×C22⋊D4711yescharacteristic
441C22×C4C2×C22⋊D4711yescharacteristic
442C22×C4C2×C22⋊D4711yescharacteristic
443C22×C4C2×C22⋊D4711yescharacteristic
444C22×C4C2×C22⋊D4714yes—
445C22×C4C23×D4712yes—
446C22×C4C23×D4711yescharacteristic
447C2×C8—474no—
448C2×C8—474no—
449C2×C8—944no—
450C2×C8—944no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—472no—
460C4⋊C4—472no—
461C4⋊C4—944no—
462C4⋊C4—944no—
463C4⋊C4—944no—
464C4⋊C4—944no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4C23×D4714yes—
468C4⋊C4C23×D4714yes—
469C22⋊C4—944no—
470C22⋊C4—944no—
471C22⋊C4—944no—
472C22⋊C4—944no—
473C22⋊C4—944no—
474C42—944no—
475C42C2×C22⋊D4714yes—
476C42C2×C22⋊D4712yes—
477C42C2×C22⋊D4712yes—
478C42C2×C22⋊D4714yes—
479C42C23×D4712yes—
480C42C23×D4712yes—
481C23—22no—
482C23—22no—
483C23—942no—
484C23—1882no—
485C23—1882no—
486C23—1882no—
487C23#1309404414yes—
488C23C22×C22⋊D4712yes—
489C23C22×C22⋊D4711yescharacteristic
490Q8—944no—
491Q8—1882no—
492D4—944no—
493D4—1888no—
494D4—1882no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—942no—
511C2×C4—942no—
512C2×C4—942no—
513C2×C4—1888no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4C22×C22⋊D4712yes—
518C2×C4C22×C22⋊D4712yes—
519C2×C4C22×C22⋊D4712yes—
520C2×C4C22×C22⋊D4712yes—
521C2×C4C22×C22⋊D4712yes—
522C2×C4C22×C22⋊D4711yescharacteristic
523C2×C4#1309407214yes—
524C2×C4C24×D4711yescharacteristic
525C8—944no—
526C8—944no—
527C22—22no—
528C22—28no—
529C22—22no—
530C22—3764no—
531C22—1882no—
532C22—1882no—
533C22—1882no—
534C22#1568969712yes—
535C22C2×#1309404414yes—
536C22C2×#1309404412yes—
537C22#1568992112yes—
538C22C23×C22⋊D4711yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—44no—
545C4—44no—
546C4—1882no—
547C4—1882no—
548C4—1882no—
549C4#1568969712yes—
550C4C2×#1309407212yes—
551C2—24no—
552C2—3762no—
553C2#2025455614yesdirect product factor
554C2C2×#1568969711yescharacteristic
555C2C22×#1309404411yescharacteristic
556C2C2×#1568992111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC22×#125×C47C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenterC22×#125D47
hypocenterC47C2×#3232
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #20254556C2direct (N × H)
3. C2×#15689522C2semidirect
4. C2×#15689480C2semidirect
5. C2×#15689458C2semidirect
7. C2×#15689453C2semidirect
9. C2×#15689427C2semidirect
10. C2×#15689420C2semidirect
11. C2×#15689176C2semidirect
12. C22×#125×C47C2semidirect
25. #15689522C22semidirect
26. #15689480C22semidirect
27. #15689458C22semidirect
28. #15689453C22semidirect
31. #15689427C22semidirect
32. #15689420C22semidirect
33. #15689176C22semidirect
51. C2×#125×C47C22semidirect
274. C22×#125D47semidirect
286. C2×C47#3232semidirect
300. C2×#125C2×D47semidirect
335. C47C2×#3232semidirect
553. C2#20254556direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896572
#202545562
C2×#156895222
C2×#156894802
C2×#156894582
C2×#156894412
C2×#156894532
C2×#156894362
C2×#156894272
C2×#156894202
C2×#156891762
C22×#125×C472
C2×#323247