Finite Group Databasebeta

← C2×#20253315 #27084060

Subgroups — C2×#20253315 #27084060

Order 12032.

2650 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689636C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
3#20253315C2116yesmaximal, direct product factor
4C2×#15689456C211yesmaximal, characteristic, complemented
5C2×#15689442C211yesmaximal, characteristic, complemented
6C2×#15689441C211yesmaximal, characteristic, complemented
7C2×#15689452C212yesmaximal, complemented
8C2×#15689438C212yesmaximal, complemented
9C2×#15689164C212yesmaximal, complemented
10C2×#15689123C212yesmaximal, complemented
11C2×#15689085C211yesmaximal, characteristic, complemented
12C2×C47×#484C211yesmaximal, characteristic, complemented
13C22×#10625354C2211yescharacteristic
14C23×C4×D47—21no—
15C23×C4×D47C2211yescharacteristic
16C22×#10625324C2212yes—
17#15689636—24no—
18#15689636C2218yes—
19C2×#13094009—21no—
20C2×#13094000C2211yescharacteristic
21C2×#13093999C2211yescharacteristic
22C2×#13093998C2211yescharacteristic
23C2×#13093994—21no—
24C2×#13093986—21no—
25C2×(C22⋊D47)⋊C4—21no—
26C2×#13093982C2211yescharacteristic
27C22×C4⋊Dic47C2211yescharacteristic
28C22×Dic47⋊C4—21no—
29C2×C4⋊C4×D47—24no—
30C2×C4⋊C4×D47—21no—
31C2×C4⋊C4×D47—21no—
32C2×C4⋊C4×D47C2212yes—
33C2×C4⋊C4×D47C2212yes—
34C2×C4⋊C4×D47C2218yescomplemented
35#15689456C2218yescomplemented
36#15689452C22116yescomplemented
37#15689442C2218yescomplemented
38#15689441C2218yescomplemented
39#15689438C22116yescomplemented
40#15689164C22116yescomplemented
41C2×#13093957C2212yescomplemented
42C2×#13093956C2212yescomplemented
43C2×#13093955C2212yescomplemented
44#15689123C22116yescomplemented
45C2×#13093948C2212yescomplemented
46C2×#13093946C2212yescomplemented
47C2×#13093944C2212yescomplemented
48#15689085C2218yescomplemented
49C2×C22⋊(C47⋊C8)C2211yescharacteristic
50C2×(C2×D47)⋊C8C2212yes—
51C2×C22⋊Q8×C47C2211yescharacteristic
52C22×C4⋊C4×C47C2211yescharacteristic
53C47×#484C2218yescomplemented
54C2×#107×C47C2212yescomplemented
55C2×#103×C47C2212yescomplemented
56C2×C22⋊C8×C47C2211yescharacteristic
57C24×D47D411yescharacteristic
58C2×#10625354—22no—
59C2×#10625354—28no—
60C2×#10625354C2314yes—
61C22×D188—21no—
62C22×D188—21no—
63C22×D188C2311yescharacteristic
64C22×C22⋊D47—21no—
65C23×Dic47—21no—
66C23×Dic47D411yescharacteristic
67C22×C4×D47—28no—
68C22×C4×D47—22no—
69C22×C4×D47—22no—
70C22×C4×D47—28no—
71C22×C4×D47—21no—
72C22×C4×D47—48no—
73C22×C4×D47D411yescharacteristic
74C22×C4×D47D411yescharacteristic
75C22×C4×D47D414yes—
76C22×C4×D47C2311yescharacteristic
77C22×C4×D47C2314yes—
78C22×C4×D47C2311yescharacteristic
79C2×#10625324—28no—
80C2×#10625324C2318yes—
81C22×C47⋊C8C2312yes—
82#13094009—28no—
83#13094000—22no—
84#13094000C2314yes—
85#13093999—22no—
86#13093999C2314yes—
87#13093998—22no—
88#13093998C2314yes—
89#13093994—28no—
90#13093986—28no—
91(C22⋊D47)⋊C4—28no—
92#13093982—22no—
93#13093982C2314yes—
94C2×C4⋊Dic47—21no—
95C2×C4⋊Dic47—22no—
96C2×C4⋊Dic47C2312yes—
97C2×C4⋊Dic47C2314yes—
98C2×C4⋊Dic47C2311yescharacteristic
99C2×Dic47⋊C4—22no—
100C2×Dic47⋊C4—22no—
101C2×Dic47⋊C4—28no—
102C2×Dic47⋊C4—21no—
103C4⋊C4×D47—24no—
104C4⋊C4×D47—24no—
105C4⋊C4×D47—28no—
106C4⋊C4×D47—28no—
107C4⋊C4×D47C2318yes—
108C4⋊C4×D47C2318yes—
109C2×(C2×D47)⋊C4—21no—
110C2×(C2×D47)⋊C4—21no—
111C2×(C2×D47)⋊C4—21no—
112C2×(C2×D47)⋊C4—21no—
113C2×C4×Dic47—21no—
114#13093957C2318yescomplemented
115#13093956C2318yescomplemented
116#13093955C2318yescomplemented
117#13093948C2318yescomplemented
118#13093946C2318yescomplemented
119#13093944C2318yescomplemented
120C22⋊(C47⋊C8)C2314yes—
121(C2×D47)⋊C8C2318yes—
122C22×Q8×C47C2311yescharacteristic
123C23×C4×C47—21no—
124C23×C4×C47C2311yescharacteristic
125C22×C8×C47C2312yes—
126C22⋊Q8×C47—22no—
127C22⋊Q8×C47C2314yes—
128C2×C4⋊C4×C47—21no—
129C2×C4⋊C4×C47—22no—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47C2314yes—
132C2×C4⋊C4×C47C2312yes—
133C2×C4⋊C4×C47C2311yescharacteristic
134C2×C22⋊C4×C47—21no—
135#107×C47C2318yescomplemented
136#103×C47C2318yescomplemented
137C22⋊C8×C47C2314yes—
138C24×C47C2×D411yescharacteristic
139C23×D47—24no—
140C23×D47—24no—
141C23×D47—24no—
142C23×D47—21no—
143C23×D47C2×D414yes—
144C23×D47C2×D411yescharacteristic
145C23×D47C2×D411yescharacteristic
146C2×Q8×C47—21no—
147C2×Q8×C47—24no—
148C2×Q8×C47C2412yes—
149C22×C4×C47—24no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—24no—
153C22×C4×C47—22no—
154C22×C4×C47—44no—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D412yes—
157C22×C4×C47C2412yes—
158C22×C4×C47C2411yescharacteristic
159C2×C22⋊D47—24no—
160C2×C22⋊D47—44no—
161C22×Dic47—24no—
162C22×Dic47—21no—
163C22×Dic47—24no—
164C22×Dic47—22no—
165C22×Dic47—44no—
166C22×Dic47C2×D411yescharacteristic
167C22×Dic47C2×D414yes—
168C22×Dic47C2×D411yescharacteristic
169#10625354—216no—
170#10625354—48no—
171C2×D188—24no—
172C2×D188—24no—
173C2×D188—24no—
174C2×D188—21no—
175C2×D188—44no—
176C2×D188—44no—
177C2×D188C2412yes—
178C2×C4×D47—24no—
179C2×C4×D47—24no—
180C2×C4×D47—24no—
181C2×C4×D47—216no—
182C2×C4×D47—216no—
183C2×C4×D47—28no—
184C2×C4×D47—28no—
185C2×C4×D47—24no—
186C2×C4×D47—48no—
187C2×C4×D47—432no—
188C2×C4×D47—48no—
189C2×C4×D47—44no—
190C2×C4×D47C2×D414yes—
191C2×C4×D47C2×D412yes—
192C2×C4×D47C2×D414yes—
193C2×C4×D47C2×D412yes—
194C2×C4×D47#4118yes—
195C2×C4×D47C2412yes—
196C2×C4×D47C2412yes—
197C2×C8×C47—24no—
198C2×C8×C47C2414yes—
199C4⋊C4×C47—24no—
200C4⋊C4×C47—22no—
201C4⋊C4×C47—24no—
202C4⋊C4×C47—21no—
203C4⋊C4×C47C2414yes—
204C4⋊C4×C47C2412yes—
205C22⋊C4×C47—24no—
206(C2×D47)⋊C4—24no—
207(C2×D47)⋊C4—24no—
208(C2×D47)⋊C4—24no—
209(C2×D47)⋊C4—24no—
210C4⋊Dic47—24no—
211C4⋊Dic47—22no—
212C4⋊Dic47—21no—
213C4⋊Dic47C2414yes—
214C4⋊Dic47C2412yes—
215Dic47⋊C4—28no—
216Dic47⋊C4—28no—
217Dic47⋊C4—24no—
218C4×Dic47—24no—
219C2×C47⋊C8—24no—
220C2×C47⋊C8C2414yes—
221#10625324—216no—
222C23×C47—22no—
223C23×C47—22no—
224C23×C47#13714yes—
225C23×C47C22×D412yes—
226C23×C47C22×D411yescharacteristic
227C22×D47—232no—
228C22×D47—22no—
229C22×D47—22no—
230C22×D47—28no—
231C22×D47—22no—
232C22×D47—22no—
233C22×D47—28no—
234C22×D47—28no—
235C22×D47—22no—
236C22×D47—42no—
237C22×D47—42no—
238C22×D47—42no—
239C22×D47#12314yes—
240C22×D47AGL(1,ℤ/8)14yes—
241C22×D47C22×D412yes—
242C22×D47C22×D412yes—
243Q8×C47—24no—
244Q8×C47—42no—
245C2×C4×C47—22no—
246C2×C4×C47—24no—
247C2×C4×C47—24no—
248C2×C4×C47—22no—
249C2×C4×C47—24no—
250C2×C4×C47—22no—
251C2×C4×C47—48no—
252C2×C4×C47—42no—
253C2×C4×C47—44no—
254C2×C4×C47—42no—
255C2×C4×C47C22×D412yes—
256C2×C4×C47C22×D411yescharacteristic
257C2×C4×C47C2×#4114yes—
258C2×C4×C47C2511yescharacteristic
259C22⋊D47—48no—
260C2×Dic47—22no—
261C2×Dic47—28no—
262C2×Dic47—22no—
263C2×Dic47—24no—
264C2×Dic47—48no—
265C2×Dic47—44no—
266C2×Dic47—42no—
267C2×Dic47#12314yes—
268C2×Dic47C22×D412yes—
269C2×Dic47C22×D412yes—
270D188—24no—
271D188—48no—
272D188—48no—
273D188—42no—
274C4×D47—28no—
275C4×D47—24no—
276C4×D47—24no—
277C4×D47—28no—
278C4×D47—416no—
279C4×D47—416no—
280C4×D47—48no—
281C4×D47#12318yes—
282C4×D47C2×#4118yes—
283C8×C47—24no—
284C47⋊C8—24no—
285C2×#3092—471nomaximal, Sylow
286C22×C47—28no—
287C22×C47—22no—
288C22×C47—22no—
289C22×C47C2×#12312yes—
290C22×C47C2×AGL(1,ℤ/8)12yes—
291C22×C47C2×#13714yes—
292C22×C47C2×#13712yes—
293C22×C47C23×D411yescharacteristic
294C2×D47—232no—
295C2×D47—28no—
296C2×D47—28no—
297C2×D47—22no—
298C2×D47—22no—
299C2×D47—22no—
300C2×D47—22no—
301C2×D47—42no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—84no—
305C2×D47C2×#12314yes—
306C2×D47C2×AGL(1,ℤ/8)14yes—
307C4×C47—22no—
308C4×C47—42no—
309C4×C47—44no—
310C4×C47—42no—
311C4×C47C2×#12312yes—
312C4×C47C22×#4112yes—
313Dic47—22no—
314Dic47—44no—
315Dic47—42no—
316Dic47C2×#12314yes—
317C2×#525—471no—
318C23×C4⋊C4—471no—
319#3092—4716no—
320C2×#484—471no—
321C2×#484D4711yescharacteristic, complemented
322C2×#481—471no—
323C2×#481—471no—
324C2×#483—472no—
325C2×#480—472no—
326C2×#428—472no—
327C2×#417—472no—
328C2×#407—471no—
329C2×C47—24no—
330C2×C47#309214yescomplemented
331C2×C47C22×#12311yescharacteristic
332C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
333C2×C47C22×#13711yescharacteristic
334D47—24no—
335D47—24no—
336D47—82no—
337C22×#41—471no—
338C24×C4—471no—
339C24×C4—941no—
340C22×M16—472no—
341#525—478no—
342#525—944no—
343C2×#123—941no—
344C2×C22⋊Q8—471no—
345C2×C22⋊Q8C2×D4711yescharacteristic
346C2×#121—471no—
347C2×#121—471no—
348C2×#120—941no—
349C2×C4×D4—941no—
350C2×C4×D4—941no—
351C2×#117—471no—
352C22×C4⋊C4—471no—
353C22×C4⋊C4—472no—
354C22×C4⋊C4—478no—
355C22×C4⋊C4—472no—
356C22×C4⋊C4—941no—
357C22×C4⋊C4—941no—
358C22×C4⋊C4—944no—
359C22×C4⋊C4—941no—
360C22×C4⋊C4C2×D4711yescharacteristic
361#484—478no—
362#484C2×D4718yescomplemented
363#483—4716no—
364#481—478no—
365#481—478no—
366#480—4716no—
367#428—4716no—
368C2×#107—472no—
369C2×#107C2×D4712yescomplemented
370C2×#106—472no—
371C2×#106—472no—
372#417—4716no—
373C2×#103—472no—
374C2×#103C2×D4712yescomplemented
375C2×#102—472no—
376C2×#102—472no—
377#407—478no—
378C2×C22⋊C8—471no—
379C2×C22⋊C8—472no—
380C2×C22⋊C8C2×D4711yescharacteristic
381C47C2×#309211yescharacteristic, complemented, Sylow
382C25—471no—
383C2×#41—474no—
384C2×#41—942no—
385C2×#41—948no—
386C22×Q8C22×D4711yescharacteristic
387C22×D4—471no—
388C22×D4—941no—
389C22×D4—941no—
390C22×D4—941no—
391C23×C4—21no—
392C23×C4—471no—
393C23×C4—471no—
394C23×C4—471no—
395C23×C4—471no—
396C23×C4—474no—
397C23×C4—471no—
398C23×C4—474no—
399C23×C4—948no—
400C23×C4—941no—
401C23×C4—948no—
402C23×C4—942no—
403C23×C4—942no—
404C23×C4—941no—
405C23×C4—1888no—
406C23×C4C22×D4711yescharacteristic
407C2×M16—478no—
408C2×M16—948no—
409C22×C8—472no—
410C22×C8C22×D4712yes—
411#123—948no—
412C22⋊Q8—22no—
413C22⋊Q8—474no—
414C22⋊Q8—942no—
415C22⋊Q8C22×D4714yes—
416#121—474no—
417#121—474no—
418#121—942no—
419#121—942no—
420#120—948no—
421C4×D4—948no—
422C4×D4—948no—
423#117—474no—
424#117—942no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—472no—
429C2×C4⋊C4—474no—
430C2×C4⋊C4—471no—
431C2×C4⋊C4—478no—
432C2×C4⋊C4—478no—
433C2×C4⋊C4—944no—
434C2×C4⋊C4—948no—
435C2×C4⋊C4—944no—
436C2×C4⋊C4—948no—
437C2×C4⋊C4—948no—
438C2×C4⋊C4—942no—
439C2×C4⋊C4—942no—
440C2×C4⋊C4—941no—
441C2×C4⋊C4—942no—
442C2×C4⋊C4—941no—
443C2×C4⋊C4C22×D4712yes—
444C2×C4⋊C4C22×D4714yes—
445C2×C4⋊C4C22×D4711yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—941no—
448C2×C22⋊C4—941no—
449C2×C22⋊C4—941no—
450C2×C22⋊C4—941no—
451C2×C42—941no—
452#107—478no—
453#107C22×D4718yescomplemented
454#106—478no—
455#106—478no—
456#103—478no—
457#103C22×D4718yescomplemented
458#102—478no—
459#102—478no—
460C22⋊C8—474no—
461C22⋊C8—478no—
462C22⋊C8C22×D4714yes—
463C24—471no—
464C24—471no—
465C24—474no—
466C24—944no—
467C24—944no—
468C24—944no—
469C24—941no—
470C24D4×D4711yescharacteristic
471#41—9416no—
472#41—1888no—
473C2×Q8—21no—
474C2×Q8—24no—
475C2×Q8C23×D4712yes—
476C2×D4—472no—
477C2×D4—944no—
478C2×D4—944no—
479C2×D4—944no—
480C2×D4—941no—
481C2×D4—944no—
482C2×D4—1884no—
483C2×D4—1884no—
484C2×D4—1884no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—471no—
492C22×C4—472no—
493C22×C4—478no—
494C22×C4—472no—
495C22×C4—474no—
496C22×C4—471no—
497C22×C4—474no—
498C22×C4—472no—
499C22×C4—474no—
500C22×C4—472no—
501C22×C4—944no—
502C22×C4—9416no—
503C22×C4—9416no—
504C22×C4—944no—
505C22×C4—944no—
506C22×C4—948no—
507C22×C4—948no—
508C22×C4—944no—
509C22×C4—944no—
510C22×C4—944no—
511C22×C4—941no—
512C22×C4—942no—
513C22×C4—1884no—
514C22×C4—1888no—
515C22×C4—18832no—
516C22×C4—1888no—
517C22×C4—1884no—
518C22×C4D4×D4712yes—
519C22×C4D4×D4711yescharacteristic
520C22×C4C23×D4712yes—
521C22×C4C23×D4711yescharacteristic
522M16—9416no—
523C2×C8—24no—
524C2×C8—474no—
525C2×C8—944no—
526C2×C8C23×D4714yes—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—21no—
531C4⋊C4—474no—
532C4⋊C4—472no—
533C4⋊C4—948no—
534C4⋊C4—948no—
535C4⋊C4—944no—
536C4⋊C4—944no—
537C4⋊C4—942no—
538C4⋊C4—941no—
539C4⋊C4C23×D4714yes—
540C4⋊C4C23×D4712yes—
541C22⋊C4—24no—
542C22⋊C4—944no—
543C22⋊C4—944no—
544C22⋊C4—944no—
545C22⋊C4—944no—
546C42—944no—
547C23—22no—
548C23—22no—
549C23—474no—
550C23—474no—
551C23—472no—
552C23—472no—
553C23—948no—
554C23—948no—
555C23—9432no—
556C23—942no—
557C23—942no—
558C23—948no—
559C23—942no—
560C23—942no—
561C23—942no—
562C23—1882no—
563C23—1882no—
564C23—1882no—
565C23#1309405614yes—
566C23C2×D4×D4712yes—
567C23C2×D4×D4711yescharacteristic
568Q8—24no—
569Q8—42no—
570D4—944no—
571D4—1888no—
572D4—1888no—
573D4—1888no—
574D4—1882no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—474no—
586C2×C4—472no—
587C2×C4—478no—
588C2×C4—478no—
589C2×C4—472no—
590C2×C4—948no—
591C2×C4—948no—
592C2×C4—948no—
593C2×C4—942no—
594C2×C4—944no—
595C2×C4—944no—
596C2×C4—942no—
597C2×C4—944no—
598C2×C4—1888no—
599C2×C4—18816no—
600C2×C4—18816no—
601C2×C4—1888no—
602C2×C4—1884no—
603C2×C4—1882no—
604C2×C4C2×D4×D4712yes—
605C2×C4C2×D4×D4711yescharacteristic
606C2×C4#41×D4714yes—
607C2×C4C24×D4711yescharacteristic
608C8—24no—
609C8—944no—
610C22—22no—
611C22—22no—
612C22—28no—
613C22—474no—
614C22—474no—
615C22—9432no—
616C22—942no—
617C22—942no—
618C22—942no—
619C22—942no—
620C22—948no—
621C22—948no—
622C22—3764no—
623C22—1882no—
624C22—1882no—
625C22—1882no—
626C22#123×D4712yes—
627C22AGL(1,ℤ/8)×D4712yes—
628C22C2×#1309405614yes—
629C22C2×#1309405612yes—
630C22C22×D4×D4711yescharacteristic
631C4—22no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—474no—
636C4—942no—
637C4—1884no—
638C4—1882no—
639C4#123×D4712yes—
640C4C2×#41×D4712yes—
641C2—24no—
642C2—944no—
643C2—944no—
644C2—3762no—
645C2#2025331514yesdirect product factor
646C2C2×#123×D4711yescharacteristic
647C2C2×AGL(1,ℤ/8)×D4711yescharacteristic
648C2C22×#1309405611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#484C2
socleC23×C47C22×D4
prime radicalC2×#15689636C2
hypercenterC2×#484D47
hypocenterC47C2×#3092
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
2. C22×C4⋊C4×D47C2semidirect
3. #20253315C2direct (N × H)
4. C2×#15689456C2semidirect
5. C2×#15689442C2semidirect
6. C2×#15689441C2semidirect
7. C2×#15689452C2semidirect
8. C2×#15689438C2semidirect
9. C2×#15689164C2semidirect
10. C2×#15689123C2semidirect
11. C2×#15689085C2semidirect
12. C2×C47×#484C2semidirect
34. C2×C4⋊C4×D47C22semidirect
35. #15689456C22semidirect
36. #15689452C22semidirect
37. #15689442C22semidirect
38. #15689441C22semidirect
39. #15689438C22semidirect
40. #15689164C22semidirect
41. C2×#13093957C22semidirect
42. C2×#13093956C22semidirect
43. C2×#13093955C22semidirect
44. #15689123C22semidirect
45. C2×#13093948C22semidirect
46. C2×#13093946C22semidirect
47. C2×#13093944C22semidirect
48. #15689085C22semidirect
53. C47×#484C22semidirect
54. C2×#107×C47C22semidirect
55. C2×#103×C47C22semidirect
114. #13093957C23semidirect
115. #13093956C23semidirect
116. #13093955C23semidirect
117. #13093948C23semidirect
118. #13093946C23semidirect
119. #13093944C23semidirect
135. #107×C47C23semidirect
136. #103×C47C23semidirect
321. C2×#484D47semidirect
330. C2×C47#3092semidirect
362. #484C2×D47semidirect
369. C2×#107C2×D47semidirect
374. C2×#103C2×D47semidirect
381. C47C2×#3092semidirect
453. #107C22×D47semidirect
457. #103C22×D47semidirect
645. C2#20253315direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896362
C22×C4⋊C4×D472
#202533152
C2×#156894562
C2×#156894422
C2×#156894412
C2×#156894522
C2×#156894382
C2×#156891642
C2×#156891232
C2×#156890852
C2×C47×#4842
C2×#309247