Finite Group Databasebeta

← C2×#20255692 #27093186

Subgroups — C2×#20255692 #27093186

Order 12032.

2374 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689776C211yesmaximal, characteristic, complemented
2C2×#15689776C211yesmaximal, characteristic, complemented
3#20255692C2116yesmaximal, direct product factor
4C2×#15689457C212yesmaximal, complemented
5C2×#15689400C211yesmaximal, characteristic, complemented
6C2×#15689372C212yesmaximal, complemented
7C2×#15689325C211yesmaximal, characteristic, complemented
8C2×#15689271C212yesmaximal, complemented
9C2×C4×D376C211yesmaximal, characteristic, complemented
10C2×#15689202C211yesmaximal, characteristic, complemented
11C2×#15689123C212yesmaximal, complemented
12C2×C47×#487C211yesmaximal, characteristic, complemented
13C22×#10625354—21no—
14C22×#10625354C2212yes—
15C22×#10625336C2212yes—
16C22×D376C2211yescharacteristic
17C22×#10625324C2212yes—
18#15689776—24no—
19#15689776—24no—
20#15689776C2218yescomplemented
21#15689776C2218yescomplemented
22C2×#13094009—22no—
23C2×#13094009—22no—
24C2×#13094000—21no—
25C2×#13094000C2212yes—
26C2×#13093999—21no—
27C2×#13093999C2212yes—
28C2×C4×D188C2211yescharacteristic
29C2×C4×D188C2211yescharacteristic
30C2×C4×D188C2211yescharacteristic
31C2×C4⋊C4×D47—21no—
32C2×C4⋊C4×D47C2212yes—
33#15689457C22116yescomplemented
34#15689400C2218yescomplemented
35#15689372C22116yescomplemented
36#15689325C2218yescomplemented
37#15689271C22116yescomplemented
38C4×D376C2218yescomplemented
39#15689202C2218yescomplemented
40C2×#13093958C2211yescharacteristic
41C2×#13093957C2212yes—
42C2×C4⋊(C47⋊C8)C2211yescharacteristic
43#15689123C22116yescomplemented
44C2×#13093948C2212yes—
45C2×#13093946C2211yescharacteristic
46C2×#13093946C2211yescharacteristic
47C2×#13093944C2212yes—
48C2×(C2×D47)⋊C8C2212yes—
49C2×C4⋊Q8×C47C2211yescharacteristic
50C2×C4⋊Q8×C47C2211yescharacteristic
51C47×#487C2218yescomplemented
52C2×#103×C47C2212yes—
53C2×#103×C47C2212yes—
54C2×C4×C8×C47C2211yescharacteristic
55C2×#10625354—28no—
56C2×#10625354—24no—
57C2×#10625354—28no—
58C2×#10625354—44no—
59C2×#10625354—48no—
60C2×#10625354C2318yes—
61C22×D188—22no—
62C22×D188—22no—
63C22×D188D411yescharacteristic
64C22×D188C2311yescharacteristic
65C22×D188C2311yescharacteristic
66C22×C4×D47—22no—
67C22×C4×D47—21no—
68C22×C4×D47—21no—
69C22×C4×D47—22no—
70C22×C4×D47C2312yes—
71C2×#10625336—44no—
72C2×#10625336C2318yes—
73C2×D376—24no—
74C2×D376C2314yes—
75C2×#10625324—44no—
76C2×#10625324C2318yes—
77C22×C47⋊C8C2312yes—
78#13094009—216no—
79#13094009—216no—
80#13094000—24no—
81#13094000—28no—
82#13094000C2318yes—
83#13093999—28no—
84#13093999—24no—
85#13093999C2318yes—
86C4×D188—22no—
87C4×D188—22no—
88C4×D188D414yes—
89C4×D188C2314yes—
90C4×D188C2314yes—
91C4×D188C2314yes—
92C2×C4⋊Dic47D411yescharacteristic
93C2×C4⋊Dic47C2311yescharacteristic
94C2×C4⋊Dic47C2311yescharacteristic
95C2×Dic47⋊C4—22no—
96C2×Dic47⋊C4—22no—
97C4⋊C4×D47—24no—
98C4⋊C4×D47—28no—
99C4⋊C4×D47C2318yes—
100C2×(C2×D47)⋊C4—22no—
101C2×(C2×D47)⋊C4—21no—
102C2×(C2×D47)⋊C4—21no—
103C2×(C2×D47)⋊C4—22no—
104C2×(C2×D47)⋊C4D412yes—
105#13093958C2314yes—
106#13093957C2318yes—
107C4⋊(C47⋊C8)C2314yes—
108#13093948C2318yes—
109#13093946C2314yes—
110#13093946C2314yes—
111#13093944C2318yes—
112(C2×D47)⋊C8C2318yes—
113C22×Q8×C47—21no—
114C22×Q8×C47C2312yes—
115C22×C8×C47C2311yescharacteristic
116C22×C8×C47C2311yescharacteristic
117C4⋊Q8×C47—22no—
118C4⋊Q8×C47—22no—
119C4⋊Q8×C47C2314yes—
120C4⋊Q8×C47C2314yes—
121C2×C4⋊C4×C47—21no—
122C2×C4⋊C4×C47—21no—
123C2×C4⋊C4×C47—21no—
124C2×C4⋊C4×C47C2312yes—
125C2×C42×C47C2311yescharacteristic
126#103×C47C2318yes—
127#103×C47C2318yes—
128C4×C8×C47C2314yes—
129C23×D47—21no—
130C23×D47—21no—
131C23×D47C2×D412yes—
132C2×Q8×C47—22no—
133C2×Q8×C47—24no—
134C2×Q8×C47—24no—
135C2×Q8×C47—24no—
136C2×Q8×C47—44no—
137C2×Q8×C47C2414yes—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47C2×D412yes—
146#10625354—216no—
147#10625354—48no—
148#10625354—416no—
149#10625354—432no—
150C2×D188—24no—
151C2×D188—21no—
152C2×D188—22no—
153C2×D188—22no—
154C2×D188—22no—
155C2×D188—28no—
156C2×D188—28no—
157C2×D188—21no—
158C2×D188—48no—
159C2×D188—48no—
160C2×D188C2×D414yes—
161C2×D188#4114yescomplemented
162C2×D188C2412yes—
163C2×D188C2412yes—
164C2×C4×D47—24no—
165C2×C4×D47—24no—
166C2×C4×D47—28no—
167C2×C4×D47—24no—
168C2×C4×D47—24no—
169C2×C4×D47—28no—
170C2×C4×D47—48no—
171C2×C4×D47—48no—
172C2×C4×D47—44no—
173C2×C4×D47—44no—
174C2×C4×D47C2×D414yes—
175C2×C4×D47C2414yes—
176C2×C8×C47—22no—
177C2×C8×C47—22no—
178C2×C8×C47C2412yes—
179C2×C8×C47C2412yes—
180C4⋊C4×C47—24no—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—24no—
183C4⋊C4×C47—22no—
184C4⋊C4×C47C2414yes—
185C42×C47C2×D412yes—
186C42×C47C2412yes—
187#10625336—48no—
188(C2×D47)⋊C4—28no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191(C2×D47)⋊C4—28no—
192(C2×D47)⋊C4C2×D418yes—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47C2×D414yes—
196C4⋊Dic47C2412yes—
197C4⋊Dic47C2412yes—
198Dic47⋊C4—28no—
199Dic47⋊C4—28no—
200C2×C47⋊C8—24no—
201C2×C47⋊C8C2414yes—
202D376—28no—
203#10625324—48no—
204C23×C47C22×D411yescharacteristic
205C22×D47—24no—
206C22×D47—24no—
207C22×D47—24no—
208C22×D47—24no—
209C22×D47—22no—
210C22×D47—22no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47—42no—
214C22×D47—42no—
215C22×D47—42no—
216C22×D47C22×D414yes—
217Q8×C47—24no—
218Q8×C47—24no—
219Q8×C47—44no—
220Q8×C47—48no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—44no—
225C2×C4×C47—44no—
226C2×C4×C47AGL(1,ℤ/8)14yes—
227C2×C4×C47C22×D412yes—
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D412yes—
230C2×C4×C47C2×#4112yes—
231C2×C4×C47H(5,𝔽2)12yes—
232C2×C4×C47C2511yescharacteristic
233C2×Dic47—24no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47C22×D414yes—
239D188—24no—
240D188—28no—
241D188—22no—
242D188—22no—
243D188—22no—
244D188—416no—
245D188—416no—
246D188—42no—
247D188C2×#4114yescomplemented
248C4×D47—28no—
249C4×D47—24no—
250C4×D47—24no—
251C4×D47—416no—
252C4×D47—416no—
253C4×D47—48no—
254C4×D47—48no—
255C8×C47—22no—
256C8×C47—22no—
257C47⋊C8—24no—
258C2×#3330—471nomaximal, Sylow
259C22×C47#54612yes—
260C22×C47C2×AGL(1,ℤ/8)12yes—
261C22×C47#57812yes—
262C22×C47C23×D411yescharacteristic
263C2×D47—24no—
264C2×D47—24no—
265C2×D47—24no—
266C2×D47—24no—
267C2×D47—48no—
268C2×D47—48no—
269C2×D47—42no—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47—84no—
275C4×C47—22no—
276C4×C47—44no—
277C4×C47—44no—
278C4×C47#54612yes—
279C4×C47C2×AGL(1,ℤ/8)14yes—
280C4×C47C22×#4111yescharacteristic
281C4×C47C2×H(5,𝔽2)11yescharacteristic
282Dic47—24no—
283Dic47—42no—
284Dic47—42no—
285C2×#547—471no—
286C2×#547—471no—
287#3330—4716no—
288C2×#487D4711yescharacteristic, complemented
289C2×#484—472no—
290C2×#474—471no—
291C2×#468—472no—
292C2×#461—471no—
293C2×#449—472no—
294C2×C4×D8—471no—
295C2×#436—471no—
296C2×#417—472no—
297C2×C47#333014yescomplemented
298C2×C47C2×#54611yescharacteristic
299C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
300C2×C47C2×#57811yescharacteristic
301D47—44no—
302D47—44no—
303D47—82no—
304C22×#41—472no—
305C22×#41—941no—
306C22×SD16—472no—
307C22×D8—471no—
308C22×M16—472no—
309#547—478no—
310#547—478no—
311#547—944no—
312#547—944no—
313C2×C4⋊Q8C2×D4711yescharacteristic
314C2×C4⋊Q8C2×D4711yescharacteristic
315C2×#123—942no—
316C2×#123—942no—
317C2×C22⋊Q8—472no—
318C2×C22⋊Q8—941no—
319C2×#121—472no—
320C2×#121—941no—
321C2×C4×D4—471no—
322C2×C4×D4—471no—
323C2×C4×D4—471no—
324C22×C4⋊C4—472no—
325C22×C4⋊C4—941no—
326#487C2×D4718yescomplemented
327#484—4716no—
328#474—478no—
329#468—4716no—
330#461—478no—
331#449—4716no—
332C4×D8—478no—
333#436—478no—
334C2×#107—471no—
335C2×#107—472no—
336C2×C4⋊C8—471no—
337#417—4716no—
338C2×#103—472no—
339C2×#103C2×D4712yes—
340C2×#103C2×D4712yes—
341C2×#102—471no—
342C2×#102—471no—
343C2×#102—472no—
344C2×C22⋊C8—472no—
345C2×C4×C8C2×D4711yescharacteristic
346C47C2×#333011yescharacteristic, complemented, Sylow
347C2×#41—478no—
348C2×#41—944no—
349C2×#41—948no—
350C2×#41—948no—
351C2×#41—1884no—
352C2×#41—1888no—
353C22×Q8—21no—
354C22×Q8C22×D4712yes—
355C22×D4—471no—
356C22×D4—471no—
357C22×D4—471no—
358C22×D4—942no—
359C22×D4—942no—
360C23×C4—472no—
361C23×C4—942no—
362C23×C4—942no—
363C23×C4—941no—
364C23×C4—941no—
365C2×SD16—478no—
366C2×SD16—1884no—
367C2×D8—474no—
368C2×D8—944no—
369C2×M16—478no—
370C2×M16—1884no—
371C22×C8—472no—
372C22×C8C22×D4711yescharacteristic
373C22×C8C22×D4711yescharacteristic
374C4⋊Q8—22no—
375C4⋊Q8—22no—
376C4⋊Q8C22×D4714yes—
377C4⋊Q8C22×D4714yes—
378#123—9416no—
379#123—9416no—
380C22⋊Q8—478no—
381C22⋊Q8—948no—
382C22⋊Q8—944no—
383#121—478no—
384#121—944no—
385#121—948no—
386C4×D4—474no—
387C4×D4—474no—
388C4×D4—474no—
389C4×D4—474no—
390C4×D4—942no—
391C4×D4—942no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—471no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—478no—
399C2×C4⋊C4—948no—
400C2×C4⋊C4—944no—
401C2×C4⋊C4—942no—
402C2×C4⋊C4—942no—
403C2×C4⋊C4C22×D4712yes—
404C2×C22⋊C4—472no—
405C2×C22⋊C4—942no—
406C2×C22⋊C4—942no—
407C2×C22⋊C4—941no—
408C2×C22⋊C4—941no—
409C2×C42C22×D4711yescharacteristic
410#107—474no—
411#107—478no—
412C4⋊C8—474no—
413#103—478no—
414#103C22×D4718yes—
415#103C22×D4718yes—
416#102—474no—
417#102—474no—
418#102—478no—
419C22⋊C8—478no—
420C4×C8C22×D4714yes—
421C24—472no—
422C24—941no—
423C24—941no—
424#41—9416no—
425#41—1888no—
426#41—18816no—
427#41—18832no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—44no—
433C2×Q8C23×D4714yes—
434C2×D4—474no—
435C2×D4—474no—
436C2×D4—472no—
437C2×D4—472no—
438C2×D4—948no—
439C2×D4—948no—
440C2×D4—944no—
441C2×D4—941no—
442C2×D4—941no—
443C2×D4—942no—
444C2×D4—942no—
445C2×D4—942no—
446C2×D4—1888no—
447C2×D4—1888no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—472no—
451C22×C4—474no—
452C22×C4—474no—
453C22×C4—948no—
454C22×C4—948no—
455C22×C4—944no—
456C22×C4—944no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—944no—
460C22×C4—944no—
461C22×C4—1888no—
462C22×C4—1888no—
463C22×C4—1884no—
464C22×C4—1884no—
465C22×C4D4×D4711yescharacteristic
466C22×C4D4×D4711yescharacteristic
467C22×C4C23×D4711yescharacteristic
468SD16—1888no—
469D8—948no—
470M16—1888no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—474no—
474C2×C8—944no—
475C2×C8C23×D4712yes—
476C2×C8C23×D4712yes—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—474no—
482C4⋊C4—472no—
483C4⋊C4—472no—
484C4⋊C4—948no—
485C4⋊C4—948no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4C23×D4714yes—
489C22⋊C4—478no—
490C22⋊C4—948no—
491C22⋊C4—948no—
492C22⋊C4—944no—
493C22⋊C4—944no—
494C42D4×D4712yes—
495C42C23×D4712yes—
496C23—474no—
497C23—944no—
498C23—944no—
499C23—944no—
500C23—944no—
501C23—942no—
502C23—942no—
503C23—1882no—
504C23—1882no—
505C23—1882no—
506C23—1882no—
507C23—1882no—
508C23C2×D4×D4711yescharacteristic
509Q8—24no—
510Q8—24no—
511Q8—44no—
512Q8—48no—
513D4—474no—
514D4—944no—
515D4—948no—
516D4—942no—
517D4—942no—
518D4—942no—
519D4—18816no—
520D4—18816no—
521D4—1882no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—474no—
528C2×C4—944no—
529C2×C4—948no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—18816no—
535C2×C4—18816no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1888no—
539C2×C4—1888no—
540C2×C4#1309405114yes—
541C2×C4C2×D4×D4712yes—
542C2×C4C2×D4×D4711yescharacteristic
543C2×C4C2×D4×D4712yes—
544C2×C4#41×D4712yes—
545C2×C4#1309407012yes—
546C2×C4C24×D4711yescharacteristic
547C8—22no—
548C8—22no—
549C8—944no—
550C22—944no—
551C22—944no—
552C22—944no—
553C22—944no—
554C22—3764no—
555C22—1882no—
556C22—1882no—
557C22—1882no—
558C22—1882no—
559C22—1882no—
560C22—1888no—
561C22—1888no—
562C22#1568976612yes—
563C22C2×#1309405112yes—
564C22#1568993612yes—
565C22C22×D4×D4711yescharacteristic
566C4—22no—
567C4—44no—
568C4—44no—
569C4—944no—
570C4—1882no—
571C4—1882no—
572C4#1568976612yes—
573C4C2×#1309405114yes—
574C4C2×#41×D4711yescharacteristic
575C4C2×#1309407011yescharacteristic
576C2—3762no—
577C2—1884no—
578C2—1884no—
579C2#2025569214yesdirect product factor
580C2C2×#1568976611yescharacteristic
581C2C22×#1309405111yescharacteristic
582C2C2×#1568993611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#487C2
socleC23×C47C22×D4
hypercenterC2×#487D47
hypocenterC47C2×#3330
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
1. C2×#15689776C2semidirect
2. C2×#15689776C2semidirect
3. #20255692C2direct (N × H)
4. C2×#15689457C2semidirect
5. C2×#15689400C2semidirect
6. C2×#15689372C2semidirect
7. C2×#15689325C2semidirect
8. C2×#15689271C2semidirect
9. C2×C4×D376C2semidirect
10. C2×#15689202C2semidirect
11. C2×#15689123C2semidirect
12. C2×C47×#487C2semidirect
20. #15689776C22semidirect
21. #15689776C22semidirect
33. #15689457C22semidirect
34. #15689400C22semidirect
35. #15689372C22semidirect
36. #15689325C22semidirect
37. #15689271C22semidirect
38. C4×D376C22semidirect
39. #15689202C22semidirect
43. #15689123C22semidirect
51. C47×#487C22semidirect
161. C2×D188#41semidirect
247. D188C2×#41semidirect
288. C2×#487D47semidirect
297. C2×C47#3330semidirect
326. #487C2×D47semidirect
346. C47C2×#3330semidirect
579. C2#20255692direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897762
#202556922
C2×#156894572
C2×#156894002
C2×#156893722
C2×#156893252
C2×#156892712
C2×C4×D3762
C2×#156892022
C2×#156891232
C2×C47×#4872
C2×#333047