Finite Group Databasebeta

← C2×#20254621 #27088413

Subgroups — C2×#20254621 #27088413

Order 12032.

2018 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094023C211yesmaximal, characteristic, complemented
2C2×#15689659C211yesmaximal, characteristic
3#20254621C2116yesmaximal, direct product factor
4C2×#15689531C212yesmaximal, complemented
5C2×#15689484C212yesmaximal, complemented
6C2×#15689456C211yesmaximal, characteristic, complemented
7C2×#15689442C211yesmaximal, characteristic
8C2×#15689441C211yesmaximal, characteristic
9C2×#15689406C212yesmaximal, complemented
10C2×#15689402C212yesmaximal, complemented
11C2×#15689177C211yesmaximal, characteristic, complemented
12C2×C47×#484C211yesmaximal, characteristic, complemented
13C23×Dic94—21no—
14C2×#13094023—21no—
15C2×#13094023—21no—
16C2×#13094023—24no—
17C2×#13094023C2212yes—
18C2×#13094023C2212yes—
19C2×#13094023C2218yescomplemented
20#15689659—24no—
21#15689659C2218yes—
22C2×#13094019—21no—
23C2×#13094015C2211yescharacteristic
24C2×#13094013C2211yescharacteristic
25C2×#13094008—21no—
26C2×#13093998C2211yescharacteristic
27C2×Q8×Dic47C2211yescharacteristic
28C2×#13093986C2211yescharacteristic
29C2×(C22⋊D47)⋊C4—21no—
30C2×#13093982—21no—
31C22×C4⋊Dic47C2211yescharacteristic
32C22×Dic47⋊C4—21no—
33C22×C4×Dic47C2211yescharacteristic
34#15689531C22116yescomplemented
35#15689484C22116yescomplemented
36#15689456C2218yescomplemented
37#15689442C2218yes—
38#15689441C2218yes—
39#15689406C22116yescomplemented
40#15689402C22116yescomplemented
41#15689177C2218yescomplemented
42C2×#13093957C2212yes—
43C2×#13093956C2212yes—
44C2×#13093955C2212yes—
45C2×Dic47⋊C8C2212yes—
46C2×#13093948C2212yes—
47C2×#13093946C2212yes—
48C2×#13093944C2212yes—
49C2×C22⋊(C47⋊C8)C2211yescharacteristic
50C2×#13093934C2212yes—
51C2×C22⋊Q8×C47C2211yescharacteristic
52C22×C4⋊C4×C47C2211yescharacteristic
53C47×#484C2218yescomplemented
54C2×#107×C47C2212yes—
55C2×#103×C47C2212yes—
56C2×C22⋊C8×C47C2211yescharacteristic
57C22×Dic94—22no—
58C22×Dic94—22no—
59C22×Dic94—28no—
60C22×Dic94—48no—
61C22×D188C2311yescharacteristic
62C22×C22⋊D47—21no—
63C23×Dic47—21no—
64C23×Dic47D411yescharacteristic
65C23×Dic47D411yescharacteristic
66C22×C4×D47—21no—
67C22×C47⋊C8C2312yes—
68#13094023—24no—
69#13094023—24no—
70#13094023—28no—
71#13094023—28no—
72#13094023C2318yes—
73#13094023C2318yes—
74#13094019—28no—
75#13094015—22no—
76#13094015C2314yes—
77#13094013—22no—
78#13094013C2314yes—
79#13094008—28no—
80#13093998—22no—
81#13093998C2314yes—
82Q8×Dic47—22no—
83Q8×Dic47C2314yes—
84#13093986—22no—
85#13093986C2314yes—
86(C22⋊D47)⋊C4—28no—
87#13093982—28no—
88C2×C4⋊Dic47—21no—
89C2×C4⋊Dic47—22no—
90C2×C4⋊Dic47—21no—
91C2×C4⋊Dic47—21no—
92C2×C4⋊Dic47C2312yes—
93C2×C4⋊Dic47C2314yes—
94C2×C4⋊Dic47C2311yescharacteristic
95C2×Dic47⋊C4—22no—
96C2×Dic47⋊C4—22no—
97C2×Dic47⋊C4—28no—
98C2×Dic47⋊C4—21no—
99C2×(C2×D47)⋊C4—21no—
100C2×(C2×D47)⋊C4—21no—
101C2×(C2×D47)⋊C4—21no—
102C2×(C2×D47)⋊C4—21no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47D411yescharacteristic
105C2×C4×Dic47D411yescharacteristic
106C2×C4×Dic47D414yes—
107C2×C4×Dic47C2311yescharacteristic
108C2×C4×Dic47C2311yescharacteristic
109C2×C4×Dic47C2314yes—
110#13093957C2318yes—
111#13093956C2318yes—
112#13093955C2318yes—
113Dic47⋊C8C2318yes—
114#13093948C2318yes—
115#13093946C2318yes—
116#13093944C2318yes—
117C22⋊(C47⋊C8)C2314yes—
118#13093934C2318yes—
119C22×Q8×C47C2311yescharacteristic
120C23×C4×C47—21no—
121C23×C4×C47C2311yescharacteristic
122C22×C8×C47C2312yes—
123C22⋊Q8×C47—22no—
124C22⋊Q8×C47C2314yes—
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47—22no—
127C2×C4⋊C4×C47—21no—
128C2×C4⋊C4×C47C2314yes—
129C2×C4⋊C4×C47C2312yes—
130C2×C4⋊C4×C47C2311yescharacteristic
131C2×C22⋊C4×C47—21no—
132#107×C47C2318yes—
133#103×C47C2318yes—
134C22⋊C8×C47C2314yes—
135C24×C47C2×D411yescharacteristic
136C23×D47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47—24no—
139C2×Q8×C47C2412yes—
140C22×C4×C47—24no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—24no—
144C22×C4×C47—22no—
145C22×C4×C47—44no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2411yescharacteristic
150C2×C22⋊D47—24no—
151C2×C22⋊D47—44no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—21no—
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—44no—
158C22×Dic47C2×D411yescharacteristic
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D414yes—
162C22×Dic47C2×D414yes—
163C22×Dic47C2×D411yescharacteristic
164C2×D188—24no—
165C2×D188—21no—
166C2×D188C2412yes—
167C2×C4×D47—24no—
168C2×C4×D47—44no—
169C2×Dic94—28no—
170C2×Dic94—28no—
171C2×Dic94—48no—
172C2×Dic94—432no—
173C2×Dic94—48no—
174C2×C8×C47—24no—
175C2×C8×C47C2414yes—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—24no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47C2414yes—
181C4⋊C4×C47C2412yes—
182C22⋊C4×C47—24no—
183(C2×D47)⋊C4—24no—
184(C2×D47)⋊C4—24no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—24no—
187C4⋊Dic47—24no—
188C4⋊Dic47—22no—
189C4⋊Dic47—21no—
190C4⋊Dic47—24no—
191C4⋊Dic47—24no—
192C4⋊Dic47C2414yes—
193C4⋊Dic47C2412yes—
194Dic47⋊C4—28no—
195Dic47⋊C4—28no—
196Dic47⋊C4—24no—
197C4×Dic47—24no—
198C4×Dic47C2×D414yes—
199C4×Dic47C2×D412yes—
200C4×Dic47C2×D412yes—
201C4×Dic47C2×D414yes—
202C4×Dic47C2412yes—
203C4×Dic47C2412yes—
204C2×C47⋊C8—24no—
205C2×C47⋊C8C2414yes—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47#13714yes—
209C23×C47C22×D412yes—
210C23×C47C22×D411yescharacteristic
211C22×D47—22no—
212C22×D47—42no—
213C22×D47—42no—
214C22×D47—42no—
215Q8×C47—24no—
216Q8×C47—42no—
217C2×C4×C47—22no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—22no—
221C2×C4×C47—24no—
222C2×C4×C47—22no—
223C2×C4×C47—48no—
224C2×C4×C47—42no—
225C2×C4×C47—44no—
226C2×C4×C47—42no—
227C2×C4×C47C22×D412yes—
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C472−1+414yes—
230C2×C4×C47C2511yescharacteristic
231C22⋊D47—48no—
232C2×Dic47—22no—
233C2×Dic47—28no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—28no—
237C2×Dic47—22no—
238C2×Dic47—24no—
239C2×Dic47—48no—
240C2×Dic47—44no—
241C2×Dic47—42no—
242C2×Dic47AGL(1,ℤ/8)14yes—
243C2×Dic47C22×D412yes—
244C2×Dic47C22×D412yes—
245C2×Dic47C22×D412yes—
246C2×Dic47C22×D412yes—
247D188—24no—
248D188—42no—
249C4×D47—48no—
250Dic94—416no—
251Dic94—416no—
252C8×C47—24no—
253C47⋊C8—24no—
254C2×#3239—471nomaximal, Sylow
255C22×C47—22no—
256C22×C47—28no—
257C22×C47—22no—
258C22×C47#53612yes—
259C22×C47C2×AGL(1,ℤ/8)12yes—
260C22×C47C2×#13714yes—
261C22×C47C2×#13712yes—
262C22×C47C23×D411yescharacteristic
263C2×D47—42no—
264C2×D47—42no—
265C2×D47—42no—
266C2×D47—84no—
267C4×C47—22no—
268C4×C47—42no—
269C4×C47—44no—
270C4×C47—42no—
271C4×C47#53612yes—
272C4×C47C2×2−1+412yes—
273Dic47—22no—
274Dic47—22no—
275Dic47—22no—
276Dic47—44no—
277Dic47—42no—
278Dic47C2×AGL(1,ℤ/8)14yes—
279C22×C4⋊Q8—471no—
280C2×#529—471no—
281#3239—4716no—
282C2×#501—472no—
283C2×#489—472no—
284C2×#484—471no—
285C2×#484D4711yescharacteristic, complemented
286C2×#481—471no—
287C2×#481—471no—
288C2×#475—472no—
289C2×#474—472no—
290C2×#432—471no—
291C2×C47—24no—
292C2×C47#323914yescomplemented
293C2×C47C2×#53611yescharacteristic
294C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
295C2×C47C22×#13711yescharacteristic
296D47—82no—
297C23×Q8—941no—
298C2×C4⋊Q8—472no—
299C2×C4⋊Q8—472no—
300C2×C4⋊Q8—478no—
301C2×C4⋊Q8—941no—
302C2×C4⋊Q8—941no—
303C2×C4⋊Q8—944no—
304#529—478no—
305#529—944no—
306C2×#126—941no—
307C2×#125—471no—
308C2×#124—471no—
309C2×#123—941no—
310C2×C22⋊Q8C2×D4711yescharacteristic
311C2×#121—471no—
312C2×C4×Q8—471no—
313C2×C4×D4—471no—
314C2×C4×D4—941no—
315C2×#117—941no—
316C22×C4⋊C4—471no—
317C22×C4⋊C4—941no—
318C22×C4⋊C4C2×D4711yescharacteristic
319C22×C42—471no—
320#501—4716no—
321#489—4716no—
322#484—478no—
323#484C2×D4718yescomplemented
324#481—478no—
325#481—478no—
326#475—4716no—
327#474—4716no—
328#432—478no—
329C2×#107—472no—
330C2×#107C2×D4712yes—
331C2×#106—472no—
332C2×#106—472no—
333C2×C4⋊C8—472no—
334C2×#103—472no—
335C2×#103C2×D4712yes—
336C2×#102—472no—
337C2×#102—472no—
338C2×C22⋊C8—471no—
339C2×C22⋊C8C2×D4711yescharacteristic
340C2×#97—472no—
341C47C2×#323911yescharacteristic, complemented, Sylow
342C22×Q8—948no—
343C22×Q8—942no—
344C22×Q8—942no—
345C22×Q8—1888no—
346C22×Q8C22×D4711yescharacteristic
347C22×D4—471no—
348C22×D4—941no—
349C23×C4—21no—
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—941no—
353C23×C4—941no—
354C23×C4C22×D4711yescharacteristic
355C22×C8—472no—
356C22×C8C22×D4712yes—
357C4⋊Q8—478no—
358C4⋊Q8—478no—
359C4⋊Q8—948no—
360C4⋊Q8—944no—
361C4⋊Q8—944no—
362C4⋊Q8—948no—
363#126—948no—
364#125—474no—
365#125—942no—
366#124—474no—
367#124—942no—
368#123—948no—
369C22⋊Q8—22no—
370C22⋊Q8C22×D4714yes—
371#121—474no—
372#121—942no—
373C4×Q8—474no—
374C4×Q8—942no—
375C4×D4—474no—
376C4×D4—942no—
377C4×D4—948no—
378#117—948no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—472no—
383C2×C4⋊C4—474no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—948no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4C22×D4712yes—
394C2×C4⋊C4C22×D4714yes—
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—474no—
404C2×C42—471no—
405C2×C42—471no—
406C2×C42—474no—
407C2×C42—941no—
408#107—478no—
409#107C22×D4718yes—
410#106—478no—
411#106—478no—
412C4⋊C8—478no—
413#103—478no—
414#103C22×D4718yes—
415#102—478no—
416#102—478no—
417C22⋊C8—474no—
418C22⋊C8C22×D4714yes—
419#97—478no—
420C24—941no—
421C24D4×D4711yescharacteristic
422C2×Q8—21no—
423C2×Q8—24no—
424C2×Q8—948no—
425C2×Q8—948no—
426C2×Q8—1888no—
427C2×Q8—18832no—
428C2×Q8—1888no—
429C2×Q8C23×D4712yes—
430C2×D4—472no—
431C2×D4—944no—
432C2×D4—941no—
433C2×D4—944no—
434C2×D4—1884no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—44no—
441C22×C4—471no—
442C22×C4—474no—
443C22×C4—474no—
444C22×C4—471no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—941no—
452C22×C4—942no—
453C22×C4—1884no—
454C22×C4—1884no—
455C22×C4D4×D4712yes—
456C22×C4D4×D4711yescharacteristic
457C22×C4C23×D4712yes—
458C22×C4C23×D4711yescharacteristic
459C2×C8—24no—
460C2×C8—474no—
461C2×C8—944no—
462C2×C8C23×D4714yes—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—474no—
468C4⋊C4—472no—
469C4⋊C4—948no—
470C4⋊C4—948no—
471C4⋊C4—944no—
472C4⋊C4—944no—
473C4⋊C4—944no—
474C4⋊C4—944no—
475C4⋊C4—942no—
476C4⋊C4—941no—
477C4⋊C4C23×D4714yes—
478C4⋊C4C23×D4712yes—
479C22⋊C4—24no—
480C22⋊C4—944no—
481C22⋊C4—944no—
482C22⋊C4—944no—
483C22⋊C4—944no—
484C42—474no—
485C42—472no—
486C42—472no—
487C42—474no—
488C42—472no—
489C42—472no—
490C42—944no—
491C23—22no—
492C23—22no—
493C23—942no—
494C23—1882no—
495C23—1882no—
496C23—1882no—
497C23#1309405614yes—
498C23C2×D4×D4712yes—
499C23C2×D4×D4711yescharacteristic
500Q8—24no—
501Q8—42no—
502Q8—18816no—
503Q8—18816no—
504D4—944no—
505D4—1888no—
506D4—1882no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—474no—
518C2×C4—472no—
519C2×C4—472no—
520C2×C4—472no—
521C2×C4—472no—
522C2×C4—948no—
523C2×C4—942no—
524C2×C4—948no—
525C2×C4—942no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—944no—
529C2×C4—1888no—
530C2×C4—1888no—
531C2×C4—1884no—
532C2×C4—1882no—
533C2×C4C2×D4×D4712yes—
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4#1309407114yes—
536C2×C4C24×D4711yescharacteristic
537C8—24no—
538C8—944no—
539C22—22no—
540C22—28no—
541C22—22no—
542C22—3764no—
543C22—1882no—
544C22—1882no—
545C22—1882no—
546C22#1568969512yes—
547C22AGL(1,ℤ/8)×D4712yes—
548C22C2×#1309405614yes—
549C22C2×#1309405612yes—
550C22C22×D4×D4711yescharacteristic
551C4—22no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—474no—
556C4—942no—
557C4—942no—
558C4—942no—
559C4—1884no—
560C4—1882no—
561C4#1568969512yes—
562C4C2×#1309407112yes—
563C2—24no—
564C2—3762no—
565C2#2025462114yesdirect product factor
566C2C2×#1568969511yescharacteristic
567C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
568C2C22×#1309405611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#484C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
hypercenterC2×#484D47
hypocenterC47C2×#3239
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
3. #20254621C2direct (N × H)
4. C2×#15689531C2semidirect
5. C2×#15689484C2semidirect
6. C2×#15689456C2semidirect
9. C2×#15689406C2semidirect
10. C2×#15689402C2semidirect
11. C2×#15689177C2semidirect
12. C2×C47×#484C2semidirect
19. C2×#13094023C22semidirect
34. #15689531C22semidirect
35. #15689484C22semidirect
36. #15689456C22semidirect
39. #15689406C22semidirect
40. #15689402C22semidirect
41. #15689177C22semidirect
53. C47×#484C22semidirect
285. C2×#484D47semidirect
292. C2×C47#3239semidirect
323. #484C2×D47semidirect
341. C47C2×#3239semidirect
565. C2#20254621direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940232
C2×#156896592
#202546212
C2×#156895312
C2×#156894842
C2×#156894562
C2×#156894422
C2×#156894412
C2×#156894062
C2×#156894022
C2×#156891772
C2×C47×#4842
C2×#323947