Finite Group Databasebeta

← C2×#20254573 #27088378

Subgroups — C2×#20254573 #27088378

Order 12032.

2050 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689671C211yesmaximal, characteristic
2C2×#125×D47C211yesmaximal, characteristic, complemented
3#20254573C2116yesmaximal, direct product factor
4C2×#15689533C211yesmaximal, characteristic, complemented
5C2×#15689532C211yesmaximal, characteristic, complemented
6C2×#15689487C211yesmaximal, characteristic, complemented
7C2×#15689440C212yesmaximal, complemented
8C2×#15689435C212yesmaximal, complemented
9C2×#15689425C212yesmaximal, complemented
10C2×#15689421C212yesmaximal, complemented
11C2×#15689178C211yesmaximal, characteristic, complemented
12C2×C47×#490C211yesmaximal, characteristic, complemented
13C22×#10625352—21no—
14#15689671—24no—
15#15689671C2218yes—
16C2×#13094024C2211yescharacteristic
17C2×#13094016C2211yescharacteristic
18C2×#13094015C2211yescharacteristic
19C2×#13094015C2211yescharacteristic
20C2×#13094014—21no—
21C2×#13094014—21no—
22#125×D47—24no—
23#125×D47C2218yescomplemented
24C2×#13094011—21no—
25C2×#13093995—21no—
26C2×#13093995C2212yes—
27C2×D4×Dic47—21no—
28C2×D4×Dic47C2212yes—
29C2×C4⋊C4×D47—21no—
30C2×C4⋊C4×D47—21no—
31C2×C4⋊C4×D47C2211yescharacteristic
32C2×C4⋊C4×D47C2211yescharacteristic
33C2×C42×D47C2211yescharacteristic
34#15689533C2218yescomplemented
35#15689532C2218yescomplemented
36#15689487C2218yescomplemented
37#15689440C22116yescomplemented
38#15689435C22116yescomplemented
39#15689425C22116yescomplemented
40#15689421C22116yescomplemented
41#15689178C2218yescomplemented
42C2×#13093958C2212yes—
43C2×#13093957C2212yes—
44C2×#13093956C2212yes—
45C2×#13093955C2212yes—
46C2×Dic47⋊C8C2212yes—
47C2×#13093945C2212yes—
48C2×#13093945C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×#13093935C2211yescharacteristic
51C2×#127×C47C2211yescharacteristic
52C2×#125×C47C2211yescharacteristic
53C47×#490C2218yescomplemented
54C2×#102×C47C2212yes—
55C2×#102×C47C2212yes—
56C2×#97×C47C2211yescharacteristic
57C2×#10625352—28no—
58C2×#10625352—48no—
59C22×Dic94—21no—
60C22×C22⋊D47—22no—
61C23×Dic47—22no—
62C22×C4×D47—21no—
63C22×C4×D47—21no—
64C22×C4×D47D411yescharacteristic
65C22×C4×D47D411yescharacteristic
66C22×C4×D47C2311yescharacteristic
67C22×C47⋊C8C2312yes—
68#13094024—22no—
69#13094024C2314yes—
70#13094016—22no—
71#13094016C2314yes—
72#13094015—22no—
73#13094015—22no—
74#13094015C2314yes—
75#13094015C2314yes—
76#13094014—28no—
77#13094014—28no—
78#13094011—28no—
79#13093995—28no—
80#13093995—24no—
81#13093995C2318yes—
82D4×Dic47—28no—
83D4×Dic47—24no—
84D4×Dic47C2318yes—
85C2×C4⋊Dic47—21no—
86C2×C4⋊Dic47—21no—
87C2×C4⋊Dic47—21no—
88C2×C4⋊Dic47C2311yescharacteristic
89C2×C4⋊Dic47C2311yescharacteristic
90C2×C4⋊Dic47C2312yes—
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C4⋊C4×D47—28no—
98C4⋊C4×D47—28no—
99C4⋊C4×D47—22no—
100C4⋊C4×D47—22no—
101C4⋊C4×D47C2314yes—
102C4⋊C4×D47C2314yes—
103C2×C22⋊Dic47—22no—
104C2×C22⋊Dic47—22no—
105C2×C4×Dic47D411yescharacteristic
106C2×C4×Dic47D411yescharacteristic
107C2×C4×Dic47C2311yescharacteristic
108C42×D47D414yes—
109C42×D47C2314yes—
110#13093958C2318yes—
111#13093957C2318yes—
112#13093956C2318yes—
113#13093955C2318yes—
114Dic47⋊C8C2318yes—
115#13093945C2318yes—
116#13093945C2318yes—
117(C2×D47)⋊C8C2318yes—
118#13093935C2314yes—
119C22×D4×C47—21no—
120C22×D4×C47—21no—
121C22×D4×C47C2312yes—
122C22×C8×C47C2312yes—
123#127×C47—22no—
124#127×C47C2314yes—
125#125×C47—22no—
126#125×C47C2314yes—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47C2311yescharacteristic
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C42×C47C2311yescharacteristic
132#102×C47C2318yes—
133#102×C47C2318yes—
134#97×C47C2314yes—
135C24×C47—22no—
136C23×D47C2×D411yescharacteristic
137C2×D4×C47—22no—
138C2×D4×C47—24no—
139C2×D4×C47—24no—
140C2×D4×C47—28no—
141C2×D4×C47—44no—
142C2×D4×C47—44no—
143C2×D4×C47C2414yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C2×C22⋊D47—28no—
150C2×C22⋊D47—48no—
151C22×Dic47—22no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—28no—
155C22×Dic47—48no—
156C22×Dic47C2×D411yescharacteristic
157C22×Dic47C2×D411yescharacteristic
158C22×Dic47C2×D411yescharacteristic
159#10625352—432no—
160C2×C4×D47—24no—
161C2×C4×D47—24no—
162C2×C4×D47—24no—
163C2×C4×D47—24no—
164C2×C4×D47—42no—
165C2×C4×D47—44no—
166C2×C4×D47—44no—
167C2×C4×D47C2×D412yes—
168C2×C4×D47C2×D414yes—
169C2×C4×D47C2×D414yes—
170C2×C4×D47C2412yes—
171C2×Dic94—24no—
172C2×Dic94—44no—
173C2×C8×C47—24no—
174C2×C8×C47C2414yes—
175C4⋊C4×C47—24no—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47C2412yes—
180C4⋊C4×C47C2412yes—
181C42×C47C2×D412yes—
182C42×C47C2412yes—
183C22⋊Dic47—28no—
184C22⋊Dic47—28no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—24no—
188C4⋊Dic47—22no—
189C4⋊Dic47—24no—
190C4⋊Dic47—24no—
191C4⋊Dic47C2412yes—
192C4⋊Dic47C2412yes—
193C4⋊Dic47C2414yes—
194Dic47⋊C4—24no—
195Dic47⋊C4—24no—
196Dic47⋊C4—24no—
197Dic47⋊C4—24no—
198Dic47⋊C4—24no—
199Dic47⋊C4—24no—
200C4×Dic47C2×D412yes—
201C4×Dic47C2×D414yes—
202C4×Dic47C2×D414yes—
203C4×Dic47C2412yes—
204C2×C47⋊C8—24no—
205C2×C47⋊C8C2414yes—
206C23×C47—24no—
207C23×C47—44no—
208C23×C47—44no—
209C23×C47—44no—
210C23×C47C22×D411yescharacteristic
211C22×D47—22no—
212C22×D47—22no—
213C22×D47AGL(1,ℤ/8)14yes—
214C22×D47C22×D412yes—
215D4×C47—28no—
216D4×C47—48no—
217D4×C47—48no—
218D4×C47—44no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47C22×D412yes—
226C2×C4×C47C22×D412yes—
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C472−1+414yes—
229C2×C4×C47C2511yescharacteristic
230C22⋊D47—416no—
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—416no—
238C2×Dic47—44no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47C22×D412yes—
242C2×Dic47C22×D412yes—
243C2×Dic47C22×D412yes—
244C4×D47—28no—
245C4×D47—24no—
246C4×D47—24no—
247C4×D47—44no—
248C4×D47—48no—
249C4×D47—48no—
250Dic94—48no—
251C8×C47—24no—
252C47⋊C8—24no—
253C2×#3234—471nomaximal, Sylow
254C22×C47—44no—
255C22×C47—44no—
256C22×C47—44no—
257C22×C47—88no—
258C22×C47#53612yes—
259C22×C47C2×AGL(1,ℤ/8)12yes—
260C22×C47#57512yes—
261C22×C47C23×D411yescharacteristic
262C2×D47—22no—
263C2×D47—28no—
264C2×D47—22no—
265C2×D47C2×AGL(1,ℤ/8)14yes—
266C4×C47—22no—
267C4×C47—22no—
268C4×C47—42no—
269C4×C47—42no—
270C4×C47#53612yes—
271C4×C47C2×2−1+412yes—
272Dic47—22no—
273Dic47—22no—
274Dic47—22no—
275Dic47—44no—
276Dic47—42no—
277Dic47—42no—
278C2×#532—471no—
279C22×#125—471no—
280#3234—4716no—
281C2×#501—471no—
282C2×#501—471no—
283C2×#490—471no—
284C2×#490D4711yescharacteristic, complemented
285C2×#481—472no—
286C2×#480—472no—
287C2×#478—472no—
288C2×#478—472no—
289C2×#432—471no—
290C2×C47—84no—
291C2×C47#323414yescomplemented
292C2×C47C2×#53611yescharacteristic
293C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
294C2×C47C2×#57511yescharacteristic
295D47—24no—
296C22×#41—941no—
297#532—478no—
298#532—944no—
299C2×C4⋊Q8—471no—
300C2×#127C2×D4711yescharacteristic
301C2×#125—478no—
302C2×#125—471no—
303C2×#125—471no—
304C2×#125—471no—
305C2×#125—941no—
306C2×#125—941no—
307C2×#125—944no—
308C2×#125C2×D4711yescharacteristic
309C2×#124—941no—
310C2×#121—472no—
311C2×#121—941no—
312C2×C4×D4—472no—
313C2×C4×D4—941no—
314C22×C4⋊C4—471no—
315C22×C4⋊C4—471no—
316C22×C4⋊C4—941no—
317C22×C4⋊C4—941no—
318C22×C42—471no—
319#501—478no—
320#501—478no—
321#490—478no—
322#490C2×D4718yescomplemented
323#481—4716no—
324#480—4716no—
325#478—4716no—
326#478—4716no—
327#432—478no—
328C2×#107—472no—
329C2×#107—472no—
330C2×#106—472no—
331C2×#106—472no—
332C2×C4⋊C8—472no—
333C2×#102—472no—
334C2×#102—472no—
335C2×#102C2×D4712yes—
336C2×#102C2×D4712yes—
337C2×C22⋊C8—472no—
338C2×#97—471no—
339C2×#97C2×D4711yescharacteristic
340C47C2×#323411yescharacteristic, complemented, Sylow
341C2×#41—948no—
342C2×#41—1888no—
343C22×Q8—941no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—942no—
347C22×D4C22×D4712yes—
348C23×C4—471no—
349C23×C4—471no—
350C23×C4—471no—
351C23×C4—941no—
352C23×C4—941no—
353C23×C4—942no—
354C22×C8—472no—
355C22×C8C22×D4712yes—
356C4⋊Q8—474no—
357C4⋊Q8—942no—
358#127—22no—
359#127C22×D4714yes—
360#125—22no—
361#125—474no—
362#125—474no—
363#125—474no—
364#125—948no—
365#125—948no—
366#125—942no—
367#125—942no—
368#125—942no—
369#125C22×D4714yes—
370#124—948no—
371#121—478no—
372#121—944no—
373#121—948no—
374C4×D4—478no—
375C4×D4—944no—
376C4×D4—948no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—472no—
382C2×C4⋊C4—474no—
383C2×C4⋊C4—474no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—948no—
387C2×C4⋊C4—948no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C22⋊C4—942no—
400C2×C22⋊C4—942no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—474no—
405C2×C42—474no—
406C2×C42C22×D4711yescharacteristic
407#107—478no—
408#107—478no—
409#106—478no—
410#106—478no—
411C4⋊C8—478no—
412#102—478no—
413#102—478no—
414#102C22×D4718yes—
415#102C22×D4718yes—
416C22⋊C8—478no—
417#97—474no—
418#97C22×D4714yes—
419C24—22no—
420C24—471no—
421#41—18832no—
422C2×Q8—944no—
423C2×Q8—1884no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—28no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—948no—
431C2×D4—1888no—
432C2×D4C23×D4714yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—471no—
436C22×C4—471no—
437C22×C4—471no—
438C22×C4—472no—
439C22×C4—474no—
440C22×C4—472no—
441C22×C4—474no—
442C22×C4—944no—
443C22×C4—944no—
444C22×C4—944no—
445C22×C4—948no—
446C22×C4—944no—
447C22×C4—942no—
448C22×C4—941no—
449C22×C4—941no—
450C22×C4—1882no—
451C22×C4—1884no—
452C22×C4—1884no—
453C22×C4—1888no—
454C22×C4D4×D4711yescharacteristic
455C22×C4D4×D4711yescharacteristic
456C22×C4C23×D4711yescharacteristic
457C2×C8—24no—
458C2×C8—474no—
459C2×C8—944no—
460C2×C8C23×D4714yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—472no—
466C4⋊C4—472no—
467C4⋊C4—474no—
468C4⋊C4—944no—
469C4⋊C4—944no—
470C4⋊C4—944no—
471C4⋊C4—944no—
472C4⋊C4—944no—
473C4⋊C4—944no—
474C4⋊C4—944no—
475C4⋊C4—944no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—942no—
479C4⋊C4—944no—
480C4⋊C4C23×D4712yes—
481C4⋊C4C23×D4712yes—
482C22⋊C4—948no—
483C22⋊C4—948no—
484C42—474no—
485C42—472no—
486C42—474no—
487C42—472no—
488C42D4×D4712yes—
489C42C23×D4712yes—
490C23—24no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—474no—
495C23—472no—
496C23—942no—
497C23—942no—
498C23C2×D4×D4711yescharacteristic
499Q8—1888no—
500D4—28no—
501D4—48no—
502D4—48no—
503D4—44no—
504D4—18816no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—472no—
512C2×C4—472no—
513C2×C4—472no—
514C2×C4—948no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—944no—
519C2×C4—944no—
520C2×C4—944no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—1884no—
524C2×C4—1888no—
525C2×C4—1888no—
526C2×C4—18816no—
527C2×C4—1882no—
528C2×C4—1884no—
529C2×C4—1882no—
530C2×C4C2×D4×D4712yes—
531C2×C4C2×D4×D4712yes—
532C2×C4C2×D4×D4711yescharacteristic
533C2×C4#1309407114yes—
534C2×C4C24×D4711yescharacteristic
535C8—24no—
536C8—944no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—88no—
541C22—474no—
542C22—942no—
543C22—948no—
544C22—942no—
545C22#1568969312yes—
546C22AGL(1,ℤ/8)×D4712yes—
547C22#1568991912yes—
548C22C22×D4×D4711yescharacteristic
549C4—22no—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—942no—
554C4—942no—
555C4—942no—
556C4—1884no—
557C4—1882no—
558C4—1882no—
559C4#1568969312yes—
560C4C2×#1309407112yes—
561C2—84no—
562C2—944no—
563C2#2025457314yesdirect product factor
564C2C2×#1568969311yescharacteristic
565C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
566C2C2×#1568991911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#490C2
socleC23×C47C22×D4
prime radicalC2×#15689671C2
hypercenterC2×#490D47
hypocenterC47C2×#3234
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
2. C2×#125×D47C2semidirect
3. #20254573C2direct (N × H)
4. C2×#15689533C2semidirect
5. C2×#15689532C2semidirect
6. C2×#15689487C2semidirect
7. C2×#15689440C2semidirect
8. C2×#15689435C2semidirect
9. C2×#15689425C2semidirect
10. C2×#15689421C2semidirect
11. C2×#15689178C2semidirect
12. C2×C47×#490C2semidirect
23. #125×D47C22semidirect
34. #15689533C22semidirect
35. #15689532C22semidirect
36. #15689487C22semidirect
37. #15689440C22semidirect
38. #15689435C22semidirect
39. #15689425C22semidirect
40. #15689421C22semidirect
41. #15689178C22semidirect
53. C47×#490C22semidirect
284. C2×#490D47semidirect
291. C2×C47#3234semidirect
322. #490C2×D47semidirect
340. C47C2×#3234semidirect
563. C2#20254573direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896712
C2×#125×D472
#202545732
C2×#156895332
C2×#156895322
C2×#156894872
C2×#156894402
C2×#156894352
C2×#156894252
C2×#156894212
C2×#156891782
C2×C47×#4902
C2×#323447