Finite Group Databasebeta

← C2×#20253465 #27084358

Subgroups — C2×#20253465 #27084358

Order 12032.

2170 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689654C211yesmaximal, characteristic
2C22×#13093989C211yesmaximal, characteristic, complemented
3#20253465C2116yesmaximal, direct product factor
4C2×#15689453C211yesmaximal, characteristic, complemented
5C2×#15689429C212yesmaximal, complemented
6C2×#15689344C212yesmaximal, complemented
7C2×#15689281C211yesmaximal, characteristic, complemented
8C2×#15689261C211yesmaximal, characteristic
9C2×#15689217C212yesmaximal, complemented
10C2×#15689214C212yesmaximal, complemented
11C2×#15689173C211yesmaximal, characteristic, complemented
12C2×C47×#480C211yesmaximal, characteristic
13C23×Dic94C2211yescharacteristic
14C22×Dic188C2212yes—
15C22×#10625335C2212yes—
16#15689654—24no—
17#15689654C2218yes—
18C2×#13094017—21no—
19C2×#13094015C2211yescharacteristic
20C2×#13094011C2211yescharacteristic
21C2×#13094005—21no—
22C2×#13094003C2211yescharacteristic
23C2×#13093989—21no—
24C2×#13093989—21no—
25C2×#13093989—24no—
26C2×#13093989C2212yes—
27C2×#13093989C2212yes—
28C2×#13093989C2211yescharacteristic
29C2×#13093989C2218yescomplemented
30C2×D4×Dic47—21no—
31C2×D4×Dic47C2211yescharacteristic
32C2×#13093980—21no—
33C22×Dic47⋊C4—21no—
34C22×C4×Dic47—21no—
35C22×C4×Dic47C2211yescharacteristic
36#15689453C2218yescomplemented
37#15689429C22116yescomplemented
38#15689344C22116yescomplemented
39#15689281C2218yescomplemented
40#15689261C2218yes—
41#15689217C22116yescomplemented
42#15689214C22116yescomplemented
43#15689173C2218yescomplemented
44C2×#13093955C2212yes—
45C2×Dic47⋊C8C2212yes—
46C2×#13093949C2212yes—
47C2×#13093947C2212yes—
48C2×#13093947C2212yes—
49C2×C22⋊(C47⋊C8)C2211yescharacteristic
50C2×C8×Dic47C2212yes—
51C2×#121×C47C2211yescharacteristic
52C22×C4⋊C4×C47C2211yescharacteristic
53C47×#480C2218yes—
54C2×#107×C47C2212yes—
55C2×#102×C47C2212yes—
56C2×C22⋊C8×C47C2211yescharacteristic
57C22×Dic94—21no—
58C22×Dic94—22no—
59C22×Dic94—28no—
60C22×Dic94C2314yes—
61C22×Dic94C2311yescharacteristic
62C22×Dic94C2312yes—
63C23×Dic47—21no—
64C23×Dic47—21no—
65C23×Dic47D411yescharacteristic
66C23×Dic47D411yescharacteristic
67C2×Dic188—28no—
68C2×Dic188C2318yes—
69C2×#10625335—28no—
70C2×#10625335C2318yes—
71C22×C47⋊C8C2312yes—
72#13094017—28no—
73#13094015—22no—
74#13094015C2314yes—
75#13094011—22no—
76#13094011C2314yes—
77#13094005—28no—
78#13094003—22no—
79#13094003C2314yes—
80#13093989—24no—
81#13093989—24no—
82#13093989—28no—
83#13093989—28no—
84#13093989—22no—
85#13093989C2314yes—
86#13093989C2318yes—
87#13093989C2318yes—
88D4×Dic47—28no—
89D4×Dic47—22no—
90D4×Dic47C2314yes—
91#13093980—28no—
92C2×C4⋊Dic47—21no—
93C2×C4⋊Dic47C2311yescharacteristic
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—22no—
96C2×Dic47⋊C4—22no—
97C2×Dic47⋊C4—28no—
98C2×Dic47⋊C4—21no—
99C2×Dic47⋊C4—21no—
100C2×C22⋊Dic47—21no—
101C2×C22⋊Dic47—21no—
102C2×C22⋊Dic47—21no—
103C2×C22⋊Dic47—21no—
104C2×C4×Dic47—22no—
105C2×C4×Dic47—22no—
106C2×C4×Dic47—28no—
107C2×C4×Dic47—21no—
108C2×C4×Dic47D411yescharacteristic
109C2×C4×Dic47D411yescharacteristic
110C2×C4×Dic47D414yes—
111C2×C4×Dic47C2311yescharacteristic
112C2×C4×Dic47C2311yescharacteristic
113C2×C4×Dic47C2314yes—
114#13093955C2318yes—
115Dic47⋊C8C2318yes—
116#13093949C2318yes—
117#13093947C2318yes—
118#13093947C2318yes—
119C22⋊(C47⋊C8)C2314yes—
120C8×Dic47C2318yes—
121C22×D4×C47—21no—
122C22×D4×C47C2311yescharacteristic
123C23×C4×C47—21no—
124C23×C4×C47C2311yescharacteristic
125C22×C8×C47C2312yes—
126#121×C47—22no—
127#121×C47C2314yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47—22no—
130C2×C4⋊C4×C47C2314yes—
131C2×C4⋊C4×C47C2312yes—
132C2×C4⋊C4×C47C2311yescharacteristic
133C2×C22⋊C4×C47—21no—
134#107×C47C2318yes—
135#102×C47C2318yes—
136C22⋊C8×C47C2314yes—
137C24×C47—21no—
138C24×C47C2×D411yescharacteristic
139C2×D4×C47—24no—
140C2×D4×C47—21no—
141C2×D4×C47—24no—
142C2×D4×C47—44no—
143C2×D4×C47C2412yes—
144C22×C4×C47—24no—
145C22×C4×C47—21no—
146C22×C4×C47—24no—
147C22×C4×C47—22no—
148C22×C4×C47—44no—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2412yes—
152C22×C4×C47C2411yescharacteristic
153C22×Dic47—28no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—24no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—24no—
161C22×Dic47—44no—
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D414yes—
164C22×Dic47C2×D411yescharacteristic
165C22×Dic47C2×D411yescharacteristic
166C22×Dic47C2×D414yes—
167C22×Dic47C2×D411yescharacteristic
168C2×Dic94—28no—
169C2×Dic94—216no—
170C2×Dic94—24no—
171C2×Dic94—22no—
172C2×Dic94—24no—
173C2×Dic94—216no—
174C2×Dic94—24no—
175C2×Dic94—21no—
176C2×Dic94#4118yes—
177C2×Dic94C2412yes—
178C2×Dic94C2414yes—
179C2×C8×C47—24no—
180C2×C8×C47C2414yes—
181C4⋊C4×C47—24no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47C2412yes—
185C4⋊C4×C47C2414yes—
186C22⋊C4×C47—24no—
187C22⋊Dic47—24no—
188C22⋊Dic47—24no—
189C22⋊Dic47—24no—
190C22⋊Dic47—24no—
191#10625335—216no—
192C4⋊Dic47—21no—
193C4⋊Dic47—24no—
194C4⋊Dic47C2412yes—
195Dic47⋊C4—24no—
196Dic47⋊C4—28no—
197Dic47⋊C4—28no—
198Dic47⋊C4—24no—
199Dic47⋊C4—24no—
200C4×Dic47—28no—
201C4×Dic47—28no—
202C4×Dic47—24no—
203C4×Dic47C2×D414yes—
204C4×Dic47C2×D412yes—
205C4×Dic47C2×D412yes—
206C4×Dic47C2×D414yes—
207C4×Dic47C2412yes—
208C4×Dic47C2412yes—
209C2×C47⋊C8—24no—
210C2×C47⋊C8C2414yes—
211Dic188—216no—
212C23×C47—22no—
213C23×C47—22no—
214C23×C47—22no—
215C23×C47—42no—
216C23×C47—42no—
217C23×C47—42no—
218C23×C47#13514yes—
219C23×C47C22×D412yes—
220C23×C47C22×D411yescharacteristic
221D4×C47—24no—
222D4×C47—48no—
223D4×C47—42no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—48no—
230C2×C4×C47—44no—
231C2×C4×C47—42no—
232C2×C4×C47C22×D412yes—
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C2×#4114yes—
235C2×C4×C47C2511yescharacteristic
236C2×Dic47—216no—
237C2×Dic47—28no—
238C2×Dic47—28no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—24no—
243C2×Dic47—28no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—48no—
249C2×Dic47#13714yes—
250C2×Dic47C22×D412yes—
251C2×Dic47C22×D412yes—
252C2×Dic47C22×D412yes—
253C2×Dic47C22×D412yes—
254Dic94—216no—
255Dic94—28no—
256Dic94—24no—
257Dic94—24no—
258Dic94—42no—
259Dic94C2×#4118yes—
260C8×C47—24no—
261C47⋊C8—24no—
262C2×#3108—471nomaximal, Sylow
263C22×C47—28no—
264C22×C47—22no—
265C22×C47—22no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—84no—
270C22×C47#52512yes—
271C22×C47C2×#13514yes—
272C22×C47C2×#13512yes—
273C22×C47C2×#13712yes—
274C22×C47C23×D411yescharacteristic
275C4×C47—22no—
276C4×C47—44no—
277C4×C47—42no—
278C4×C47#52512yes—
279C4×C47C22×#4112yes—
280Dic47—28no—
281Dic47—22no—
282Dic47—22no—
283Dic47—22no—
284Dic47—42no—
285Dic47—42no—
286Dic47C2×#13714yes—
287C2×#529—471no—
288C22×C4×Q8—471no—
289#3108—4716no—
290C2×#483—471no—
291C2×#480D4711yescharacteristic
292C2×#479—472no—
293C2×#464—472no—
294C2×#451—471no—
295C2×#448—471no—
296C2×C4×Q16—472no—
297C2×C4×SD16—472no—
298C2×#431—471no—
299C2×C47—24no—
300C2×C47—82no—
301C2×C47#310814yescomplemented
302C2×C47C2×#52511yescharacteristic
303C2×C47C22×#13511yescharacteristic
304C2×C47C22×#13711yescharacteristic
305C23×Q8—471no—
306C22×Q16—472no—
307C22×SD16—472no—
308#529—478no—
309#529—944no—
310C2×#126—941no—
311C2×#125—471no—
312C2×#124—471no—
313C2×#123—941no—
314C2×C22⋊Q8—471no—
315C2×#121C2×D4711yescharacteristic
316C2×C4×Q8—472no—
317C2×C4×Q8—478no—
318C2×C4×Q8—472no—
319C2×C4×Q8—471no—
320C2×C4×Q8—941no—
321C2×C4×Q8—941no—
322C2×C4×Q8—944no—
323C2×C4×D4—471no—
324C2×C4×D4—941no—
325C2×#117—941no—
326C22×C4⋊C4—941no—
327C22×C4⋊C4C2×D4711yescharacteristic
328C22×C42—471no—
329C22×C42—941no—
330#483—478no—
331#480C2×D4718yes—
332#479—4716no—
333#464—4716no—
334#451—478no—
335#448—478no—
336C4×Q16—4716no—
337C4×SD16—4716no—
338#431—478no—
339C2×#107C2×D4712yes—
340C2×#106—472no—
341C2×C4⋊C8—472no—
342C2×#103—472no—
343C2×#103—472no—
344C2×#103—472no—
345C2×#102C2×D4712yes—
346C2×C22⋊C8—471no—
347C2×C22⋊C8C2×D4711yescharacteristic
348C2×C4×C8—472no—
349C47C2×#310811yescharacteristic, complemented, Sylow
350C22×Q8—472no—
351C22×Q8—474no—
352C22×Q8—471no—
353C22×Q8—948no—
354C22×Q8—942no—
355C22×Q8—941no—
356C22×D4—21no—
357C22×D4C22×D4711yescharacteristic
358C23×C4—21no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—941no—
362C23×C4—941no—
363C23×C4C22×D4711yescharacteristic
364C2×Q16—478no—
365C2×Q16—948no—
366C2×SD16—478no—
367C2×SD16—948no—
368C22×C8—472no—
369C22×C8C22×D4712yes—
370#126—948no—
371#125—474no—
372#125—942no—
373#124—474no—
374#124—942no—
375#123—948no—
376C22⋊Q8—474no—
377C22⋊Q8—942no—
378#121—22no—
379#121C22×D4714yes—
380C4×Q8—478no—
381C4×Q8—478no—
382C4×Q8—474no—
383C4×Q8—948no—
384C4×Q8—944no—
385C4×Q8—944no—
386C4×Q8—948no—
387C4×Q8—942no—
388C4×D4—474no—
389C4×D4—942no—
390C4×D4—948no—
391#117—948no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—948no—
396C2×C4⋊C4—942no—
397C2×C4⋊C4—942no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4C22×D4712yes—
403C2×C4⋊C4C22×D4714yes—
404C2×C4⋊C4C22×D4711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—941no—
408C2×C22⋊C4—941no—
409C2×C22⋊C4—941no—
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—474no—
413C2×C42—471no—
414C2×C42—471no—
415C2×C42—474no—
416C2×C42—941no—
417C2×C42—948no—
418C2×C42—942no—
419C2×C42—942no—
420#107C22×D4718yes—
421#106—478no—
422C4⋊C8—478no—
423#103—478no—
424#103—478no—
425#103—478no—
426#102C22×D4718yes—
427C22⋊C8—474no—
428C22⋊C8C22×D4714yes—
429C4×C8—478no—
430C24—21no—
431C24D4×D4711yescharacteristic
432C2×Q8—478no—
433C2×Q8—474no—
434C2×Q8—472no—
435C2×Q8—9416no—
436C2×Q8—948no—
437C2×Q8—9416no—
438C2×Q8—944no—
439C2×Q8—942no—
440C2×Q8—941no—
441C2×Q8—944no—
442C2×Q8—944no—
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—24no—
446C2×D4—44no—
447C2×D4C23×D4712yes—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—44no—
453C22×C4—471no—
454C22×C4—474no—
455C22×C4—474no—
456C22×C4—471no—
457C22×C4—471no—
458C22×C4—471no—
459C22×C4—944no—
460C22×C4—944no—
461C22×C4—948no—
462C22×C4—944no—
463C22×C4—944no—
464C22×C4—942no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—1884no—
468C22×C4D4×D4712yes—
469C22×C4D4×D4711yescharacteristic
470C22×C4C23×D4712yes—
471C22×C4C23×D4711yescharacteristic
472Q16—9416no—
473SD16—9416no—
474C2×C8—24no—
475C2×C8—474no—
476C2×C8—944no—
477C2×C8C23×D4714yes—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—472no—
482C4⋊C4—948no—
483C4⋊C4—948no—
484C4⋊C4—944no—
485C4⋊C4—944no—
486C4⋊C4—944no—
487C4⋊C4—944no—
488C4⋊C4—941no—
489C4⋊C4C23×D4714yes—
490C4⋊C4C23×D4712yes—
491C22⋊C4—24no—
492C22⋊C4—944no—
493C22⋊C4—944no—
494C22⋊C4—944no—
495C22⋊C4—944no—
496C42—474no—
497C42—472no—
498C42—472no—
499C42—474no—
500C42—472no—
501C42—472no—
502C42—948no—
503C42—948no—
504C42—944no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23#1309404314yes—
512C23C2×D4×D4712yes—
513C23C2×D4×D4711yescharacteristic
514Q8—478no—
515Q8—9416no—
516Q8—948no—
517Q8—944no—
518Q8—944no—
519Q8—1882no—
520D4—24no—
521D4—48no—
522D4—42no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—474no—
532C2×C4—472no—
533C2×C4—472no—
534C2×C4—472no—
535C2×C4—472no—
536C2×C4—948no—
537C2×C4—9416no—
538C2×C4—948no—
539C2×C4—948no—
540C2×C4—942no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—944no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—1888no—
547C2×C4—1882no—
548C2×C4—1882no—
549C2×C4C2×D4×D4712yes—
550C2×C4C2×D4×D4711yescharacteristic
551C2×C4#41×D4714yes—
552C2×C4C24×D4711yescharacteristic
553C8—24no—
554C8—944no—
555C22—28no—
556C22—22no—
557C22—22no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—84no—
562C22#1568963112yes—
563C22C2×#1309404314yes—
564C22C2×#1309404312yes—
565C22#137×D4712yes—
566C22C22×D4×D4711yescharacteristic
567C4—22no—
568C4—44no—
569C4—42no—
570C4—474no—
571C4—942no—
572C4—942no—
573C4—942no—
574C4—948no—
575C4—1882no—
576C4—1882no—
577C4#1568963112yes—
578C4C2×#41×D4712yes—
579C2—24no—
580C2—82no—
581C2#2025346514yesdirect product factor
582C2C2×#1568963111yescharacteristic
583C2C22×#1309404311yescharacteristic
584C2C2×#137×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#480C2
socleC23×C47C22×D4
prime radicalC2×#121×C47C22
hypercenterC2×#480D47
hypocenterC47C2×#3108
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. C22×#13093989C2semidirect
3. #20253465C2direct (N × H)
4. C2×#15689453C2semidirect
5. C2×#15689429C2semidirect
6. C2×#15689344C2semidirect
7. C2×#15689281C2semidirect
9. C2×#15689217C2semidirect
10. C2×#15689214C2semidirect
11. C2×#15689173C2semidirect
29. C2×#13093989C22semidirect
36. #15689453C22semidirect
37. #15689429C22semidirect
38. #15689344C22semidirect
39. #15689281C22semidirect
41. #15689217C22semidirect
42. #15689214C22semidirect
43. #15689173C22semidirect
301. C2×C47#3108semidirect
349. C47C2×#3108semidirect
581. C2#20253465direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896542
C22×#130939892
#202534652
C2×#156894532
C2×#156894292
C2×#156893442
C2×#156892812
C2×#156892612
C2×#156892172
C2×#156892142
C2×#156891732
C2×C47×#4802
C2×#310847