Finite Group Databasebeta

← C2×#20246896 #26384149

Subgroups — C2×#20246896 #26384149

Order 12032.

3830 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094007C212yesmaximal, complemented
2C22×#13093995C211yesmaximal, characteristic
3#20246896C2116yesmaximal, direct product factor
4C2×#15689051C212yesmaximal, complemented
5C2×#15689038C212yesmaximal, complemented
6C2×#15689026C211yesmaximal, characteristic, complemented
7C2×#15689021C211yesmaximal, characteristic, complemented
8C2×#15688996C212yesmaximal, complemented
9C2×#15688992C211yesmaximal, characteristic, complemented
10C2×#15688980C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic, complemented
12C22×C4×D4×C47C211yesmaximal, characteristic, complemented
13C23×C22⋊D47C2212yes—
14C23×C4×D47C2211yescharacteristic
15C2×#13094007—216no—
16C2×#13094007C22116yescomplemented
17C2×#13093995—28no—
18C2×#13093995C2218yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C4⋊Dic47C2211yescharacteristic
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2212yes—
23C22×Dic47⋊C4C2212yes—
24C22×C22⋊Dic47C2212yes—
25C22×C22⋊Dic47C2212yes—
26C22×C22⋊Dic47C2212yes—
27C22×(C2×D47)⋊C4C2211yescharacteristic
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2211yescharacteristic
30C22×C4×Dic47C2211yescharacteristic
31#15689051C22116yescomplemented
32#15689038C22116yescomplemented
33#15689026C2218yescomplemented
34#15689021C2218yescomplemented
35#15688996C22116yescomplemented
36#15688992C2218yescomplemented
37#15688980C2218yescomplemented
38C4×(C2×D47)⋊C4C2218yescomplemented
39C2×C47⋊#95C2211yescharacteristic
40C2×C47⋊#95C2212yes—
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2211yescharacteristic
43C23×D4×C47C2211yescharacteristic
44C24×C4×C47C2212yes—
45C2×C4×D4×C47—28no—
46C2×C4×D4×C47C2218yescomplemented
47C22×C4⋊C4×C47C2211yescharacteristic
48C22×C22⋊C4×C47C2212yes—
49C22×C42×C47C2211yescharacteristic
50C24×D47C2311yescharacteristic
51C22×C22⋊D47—28no—
52C22×C22⋊D47—28no—
53C22×C22⋊D47—44no—
54C22×C22⋊D47C2318yes—
55C23×Dic47C2311yescharacteristic
56C23×Dic47C2312yes—
57C23×Dic47C2312yes—
58C23×Dic47C2311yescharacteristic
59C23×Dic47C2311yescharacteristic
60C22×C4×D47—24no—
61C22×C4×D47—24no—
62C22×C4×D47—24no—
63C22×C4×D47C2314yes—
64#13094007—264no—
65#13093995—232no—
66C2×C4⋊Dic47—24no—
67C2×C4⋊Dic47—24no—
68C2×C4⋊Dic47C2314yes—
69C2×C4⋊Dic47C2314yes—
70C2×Dic47⋊C4—28no—
71C2×Dic47⋊C4—28no—
72C2×Dic47⋊C4—44no—
73C2×Dic47⋊C4C2318yes—
74C2×Dic47⋊C4C2318yes—
75C2×Dic47⋊C4C2318yes—
76C2×C22⋊Dic47—28no—
77C2×C22⋊Dic47—28no—
78C2×C22⋊Dic47—28no—
79C2×C22⋊Dic47C2318yes—
80C2×C22⋊Dic47C2318yes—
81C2×C22⋊Dic47C2318yes—
82C2×(C2×D47)⋊C4—28no—
83C2×(C2×D47)⋊C4—24no—
84C2×(C2×D47)⋊C4D418yescomplemented
85C2×(C2×D47)⋊C4C2314yes—
86C2×(C2×D47)⋊C4C2318yes—
87C2×(C2×D47)⋊C4C2314yes—
88C2×C4×Dic47—24no—
89C2×C4×Dic47C2314yes—
90C47⋊#95C2314yes—
91C47⋊#95C2314yes—
92C47⋊#95C2318yes—
93C47⋊#95C2318yes—
94C25×C47C2312yes—
95C22×D4×C47—24no—
96C22×D4×C47—24no—
97C22×D4×C47D418yes—
98C22×D4×C47C2314yes—
99C23×C4×C47—28no—
100C23×C4×C47—28no—
101C23×C4×C47—44no—
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2312yes—
104C23×C4×C47C2318yes—
105C23×C4×C47C2311yescharacteristic
106C23×C4×C47C2311yescharacteristic
107C4×D4×C47—232no—
108C2×C4⋊C4×C47—24no—
109C2×C4⋊C4×C47C2314yes—
110C2×C22⋊C4×C47—28no—
111C2×C22⋊C4×C47C2318yes—
112C2×C42×C47—24no—
113C2×C42×C47C2314yes—
114C24×C47—24no—
115C24×C47—24no—
116C24×C47—24no—
117C24×C47—24no—
118C24×C47—24no—
119C24×C47—24no—
120C24×C47C2×D418yes—
121C24×C47C2414yes—
122C24×C47C2411yescharacteristic
123C23×D47—22no—
124C23×D47—22no—
125C23×D47—22no—
126C23×D47—22no—
127C23×D47—22no—
128C23×D47—22no—
129C23×D47C2×D414yes—
130C23×D47C2412yes—
131C2×D4×C47—232no—
132C2×D4×C47—28no—
133C2×D4×C47—28no—
134C2×D4×C47C2×D4116yes—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—24no—
139C22×C4×C47—216no—
140C22×C4×C47—24no—
141C22×C4×C47—22no—
142C22×C4×C47—24no—
143C22×C4×C47—22no—
144C22×C4×C47—216no—
145C22×C4×C47—48no—
146C22×C4×C47—432no—
147C22×C4×C47C2×D414yes—
148C22×C4×C47C2×D414yes—
149C22×C4×C47#4114yes—
150C22×C4×C47#4114yes—
151C22×C4×C47C2412yes—
152C22×C4×C47C2414yes—
153C22×C4×C47C2412yes—
154C22×C4×C47C2412yes—
155C2×C22⋊D47—216no—
156C2×C22⋊D47—216no—
157C2×C22⋊D47—48no—
158C2×C22⋊D47—432no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—24no—
164C22×Dic47—22no—
165C22×Dic47—24no—
166C22×Dic47—24no—
167C22×Dic47—24no—
168C22×Dic47—24no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47—24no—
172C22×Dic47C2×D414yes—
173C22×Dic47#4114yes—
174C22×Dic47C2412yes—
175C22×Dic47C2414yes—
176C22×Dic47C2414yes—
177C22×Dic47C2412yes—
178C22×Dic47C2412yes—
179C2×C4×D47—28no—
180C2×C4×D47—28no—
181C2×C4×D47—28no—
182C2×C4×D47—416no—
183C4⋊C4×C47—28no—
184C22⋊C4×C47—216no—
185C42×C47—28no—
186C22⋊Dic47—216no—
187C22⋊Dic47—216no—
188C22⋊Dic47—216no—
189(C2×D47)⋊C4—216no—
190(C2×D47)⋊C4—28no—
191(C2×D47)⋊C4C2×D4116yescomplemented
192C4⋊Dic47—28no—
193C4⋊Dic47—28no—
194Dic47⋊C4—216no—
195Dic47⋊C4—216no—
196Dic47⋊C4—48no—
197C4×Dic47—28no—
198C23×C47—216no—
199C23×C47—216no—
200C23×C47—216no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—48no—
208C23×C47—48no—
209C23×C47—48no—
210C23×C47—48no—
211C23×C47C22×D412yes—
212C23×C47C22×D418yes—
213C23×C47C22×D412yes—
214C23×C47C2×#4112yes—
215C23×C47C2×#4112yes—
216C23×C47C2×#4112yes—
217C23×C47H(5,𝔽2)12yes—
218C23×C472−1+412yes—
219C23×C47C2511yescharacteristic
220C22×D47—28no—
221C22×D47—28no—
222C22×D47—28no—
223C22×D47—22no—
224C22×D47—22no—
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—44no—
230C22×D47—44no—
231C22×D47—44no—
232C22×D47—44no—
233C22×D47C22×D414yes—
234D4×C47—232no—
235C2×C4×C47—28no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—22no—
240C2×C4×C47—24no—
241C2×C4×C47—22no—
242C2×C4×C47—24no—
243C2×C4×C47—22no—
244C2×C4×C47—432no—
245C2×C4×C47—48no—
246C2×C4×C47—44no—
247C2×C4×C47#123116yescomplemented
248C2×C4×C47C22×D414yes—
249C2×C4×C47C22×D414yes—
250C2×C4×C47C2×#4114yes—
251C2×C4×C47C2×#4114yes—
252C22⋊D47—432no—
253C2×Dic47—28no—
254C2×Dic47—28no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—22no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—24no—
268C2×Dic47—44no—
269C2×Dic47—48no—
270C2×Dic47—48no—
271C2×Dic47C22×D414yes—
272C2×Dic47C2×#4114yes—
273C4×D47—416no—
274C2×#2422—471nomaximal, Sylow
275C22×C47—28no—
276C22×C47—28no—
277C22×C47—28no—
278C22×C47—44no—
279C22×C47—432no—
280C22×C47—44no—
281C22×C47—44no—
282C22×C47—44no—
283C22×C47C2×#12314yes—
284C22×C47#53214yes—
285C22×C47#54214yes—
286C22×C47D4214yes—
287C22×C47#54714yes—
288C22×C47#54714yes—
289C22×C47#55314yes—
290C22×C47C23×D411yescharacteristic
291C22×C47C23×D411yescharacteristic
292C22×C47C22×#4111yescharacteristic
293C22×C47C22×#4111yescharacteristic
294C22×C47C22×#4111yescharacteristic
295C22×C47C2×H(5,𝔽2)11yescharacteristic
296C22×C47C2×2−1+411yescharacteristic
297C2×D47—24no—
298C2×D47—24no—
299C2×D47—24no—
300C2×D47—42no—
301C2×D47—416no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—42no—
305C4×C47—24no—
306C4×C47—44no—
307C4×C47—42no—
308C4×C47C2×#12318yescomplemented
309Dic47—24no—
310Dic47—24no—
311Dic47—42no—
312Dic47—44no—
313Dic47—44no—
314C22×#123—472no—
315C22×#121—471no—
316C22×C4×D4D4711yescharacteristic, complemented
317#2422—4716no—
318C2×#397—472no—
319C2×#394—472no—
320C2×#390—471no—
321C2×#389—471no—
322C2×#385—472no—
323C2×#384—471no—
324C2×#380—471no—
325C2×C4×C22⋊C4—471no—
326C2×C47—48no—
327C2×C47#242218yescomplemented
328C2×C47C22×#12311yescharacteristic
329C2×C47C2×#53211yescharacteristic
330C2×C47C2×#54211yescharacteristic
331C2×C47C2×D4211yescharacteristic
332C2×C47C2×#54711yescharacteristic
333C2×C47C2×#54711yescharacteristic
334C2×C47C2×#55311yescharacteristic
335D47—44no—
336C23×D4—472no—
337C23×D4C2×D4711yescharacteristic
338C24×C4—471no—
339C24×C4C2×D4712yes—
340C2×#123—4716no—
341C2×#123—9416no—
342C2×#121—478no—
343C2×#121—948no—
344C2×C4×D4—28no—
345C2×C4×D4C2×D4718yescomplemented
346C22×C4⋊C4—472no—
347C22×C4⋊C4—472no—
348C22×C4⋊C4—471no—
349C22×C4⋊C4—471no—
350C22×C4⋊C4—472no—
351C22×C4⋊C4C2×D4711yescharacteristic
352C22×C22⋊C4—471no—
353C22×C22⋊C4—472no—
354C22×C22⋊C4—472no—
355C22×C22⋊C4—472no—
356C22×C22⋊C4—471no—
357C22×C22⋊C4—472no—
358C22×C22⋊C4C2×D4712yes—
359C22×C42—471no—
360C22×C42C2×D4711yescharacteristic
361#397—4716no—
362#394—4716no—
363#390—478no—
364#389—478no—
365#385—4716no—
366#384—478no—
367#380—478no—
368C4×C22⋊C4—478no—
369C2×#95—472no—
370C2×#95—471no—
371C2×#95—472no—
372C2×#95—471no—
373C47C2×#242211yescharacteristic, complemented, Sylow
374C25—471no—
375C25C22×D4712yes—
376C22×D4—24no—
377C22×D4—24no—
378C22×D4—478no—
379C22×D4—948no—
380C22×D4—948no—
381C22×D4—1884no—
382C22×D4C22⋊D4718yes—
383C22×D4C22×D4714yes—
384C23×C4—28no—
385C23×C4—28no—
386C23×C4—44no—
387C23×C4—471no—
388C23×C4—471no—
389C23×C4—472no—
390C23×C4—471no—
391C23×C4—472no—
392C23×C4—474no—
393C23×C4—944no—
394C23×C4—944no—
395C23×C4—944no—
396C23×C4C22×D4711yescharacteristic
397C23×C4C22×D4712yes—
398C23×C4C22×D4718yes—
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401#123—9464no—
402#121—9432no—
403C4×D4—232no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—478no—
407C2×C4⋊C4—478no—
408C2×C4⋊C4—474no—
409C2×C4⋊C4—478no—
410C2×C4⋊C4—948no—
411C2×C4⋊C4—944no—
412C2×C4⋊C4—948no—
413C2×C4⋊C4—944no—
414C2×C4⋊C4—1884no—
415C2×C4⋊C4C22×D4714yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—478no—
418C2×C22⋊C4—474no—
419C2×C22⋊C4—478no—
420C2×C22⋊C4—478no—
421C2×C22⋊C4—478no—
422C2×C22⋊C4—478no—
423C2×C22⋊C4—474no—
424C2×C22⋊C4—944no—
425C2×C22⋊C4—948no—
426C2×C22⋊C4—948no—
427C2×C22⋊C4—948no—
428C2×C22⋊C4—948no—
429C2×C22⋊C4C22×D4718yes—
430C2×C42—24no—
431C2×C42—474no—
432C2×C42—944no—
433C2×C42C22×D4714yes—
434#95—474no—
435#95—474no—
436#95—478no—
437#95—478no—
438C24—24no—
439C24—24no—
440C24—24no—
441C24—24no—
442C24—24no—
443C24—24no—
444C24—474no—
445C24—472no—
446C24—942no—
447C24—942no—
448C24—942no—
449C24—942no—
450C24—942no—
451C24—942no—
452C24C2×C22⋊D4718yes—
453C24C23×D4714yes—
454C24C23×D4711yescharacteristic
455C2×D4—232no—
456C2×D4—28no—
457C2×D4—28no—
458C2×D4—9416no—
459C2×D4—9416no—
460C2×D4—1888no—
461C2×D4—18832no—
462C2×D4C2×C22⋊D47116yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—216no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—216no—
473C22×C4—48no—
474C22×C4—432no—
475C22×C4—474no—
476C22×C4—474no—
477C22×C4—472no—
478C22×C4—472no—
479C22×C4—472no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—944no—
487C22×C4—944no—
488C22×C4—942no—
489C22×C4—944no—
490C22×C4—944no—
491C22×C4—944no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—944no—
495C22×C4—948no—
496C22×C4—948no—
497C22×C4—948no—
498C22×C4—18816no—
499C22×C4#1062535014yes—
500C22×C4#1062535014yes—
501C22×C4D4×D4714yes—
502C22×C4C2×C22⋊D4714yes—
503C22×C4C23×D4712yes—
504C22×C4C23×D4714yes—
505C22×C4C23×D4712yes—
506C22×C4C23×D4712yes—
507C4⋊C4—28no—
508C4⋊C4—9416no—
509C4⋊C4—9416no—
510C4⋊C4—948no—
511C4⋊C4—948no—
512C4⋊C4—1888no—
513C22⋊C4—216no—
514C22⋊C4—4716no—
515C22⋊C4—948no—
516C22⋊C4—9416no—
517C22⋊C4—9416no—
518C22⋊C4—9416no—
519C22⋊C4—9416no—
520C42—28no—
521C42—948no—
522C23—216no—
523C23—216no—
524C23—216no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—474no—
536C23—948no—
537C23—948no—
538C23—948no—
539C23—942no—
540C23—942no—
541C23—942no—
542C23—942no—
543C23—942no—
544C23—942no—
545C23—1884no—
546C23—1884no—
547C23—1884no—
548C23—1884no—
549C23C2×D4×D4712yes—
550C23C22×C22⋊D4718yes—
551C23C22×C22⋊D4712yes—
552C23#41×D4712yes—
553C23C2×#1062535012yes—
554C23C2×#1062535012yes—
555C23#1309406912yes—
556C23#1309407112yes—
557C23C24×D4711yescharacteristic
558D4—232no—
559D4—18832no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—432no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—474no—
573C2×C4—474no—
574C2×C4—948no—
575C2×C4—948no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—944no—
581C2×C4—944no—
582C2×C4—942no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—944no—
586C2×C4—942no—
587C2×C4—942no—
588C2×C4—944no—
589C2×C4—1888no—
590C2×C4—1884no—
591C2×C4—1888no—
592C2×C4—18816no—
593C2×C4#13094007116yescomplemented
594C2×C4C2×D4×D4714yes—
595C2×C4C22×C22⋊D4714yes—
596C2×C4C2×#1062535014yes—
597C2×C4C2×#1062535014yes—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—44no—
602C22—432no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—944no—
607C22—944no—
608C22—944no—
609C22—1882no—
610C22—1882no—
611C22—18816no—
612C22—1882no—
613C22—1882no—
614C22C2×#1309400714yes—
615C22#1568967014yes—
616C22#1568973814yes—
617C22D4×C22⋊D4714yes—
618C22#1568977214yes—
619C22#1568977814yes—
620C22#1568982314yes—
621C22C22×D4×D4711yescharacteristic
622C22C23×C22⋊D4711yescharacteristic
623C22C2×#41×D4711yescharacteristic
624C22C22×#1062535011yescharacteristic
625C22C22×#1062535011yescharacteristic
626C22C2×#1309406911yescharacteristic
627C22C2×#1309407111yescharacteristic
628C4—24no—
629C4—44no—
630C4—42no—
631C4—944no—
632C4—944no—
633C4—1884no—
634C4—1882no—
635C4—1884no—
636C4C2×#1309400718yescomplemented
637C2—48no—
638C2—1884no—
639C2#2024689618yesdirect product factor
640C2C22×#1309400711yescharacteristic
641C2C2×#1568967011yescharacteristic
642C2C2×#1568973811yescharacteristic
643C2C2×D4×C22⋊D4711yescharacteristic
644C2C2×#1568977211yescharacteristic
645C2C2×#1568977811yescharacteristic
646C2C2×#1568982311yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×C4×D4×C47C2
socleC24×C47C24
prime radicalC22×#13093995C2
hypercenterC22×C4×D4D47
hypocenterC47C2×#2422
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#13094007C2semidirect
3. #20246896C2direct (N × H)
4. C2×#15689051C2semidirect
5. C2×#15689038C2semidirect
6. C2×#15689026C2semidirect
7. C2×#15689021C2semidirect
8. C2×#15688996C2semidirect
9. C2×#15688992C2semidirect
10. C2×#15688980C2semidirect
11. C2×C4×(C2×D47)⋊C4C2semidirect
12. C22×C4×D4×C47C2semidirect
16. C2×#13094007C22semidirect
31. #15689051C22semidirect
32. #15689038C22semidirect
33. #15689026C22semidirect
34. #15689021C22semidirect
35. #15688996C22semidirect
36. #15688992C22semidirect
37. #15688980C22semidirect
38. C4×(C2×D47)⋊C4C22semidirect
46. C2×C4×D4×C47C22semidirect
84. C2×(C2×D47)⋊C4D4semidirect
191. (C2×D47)⋊C4C2×D4semidirect
247. C2×C4×C47#123semidirect
308. C4×C47C2×#123semidirect
316. C22×C4×D4D47semidirect
327. C2×C47#2422semidirect
345. C2×C4×D4C2×D47semidirect
373. C47C2×#2422semidirect
593. C2×C4#13094007semidirect
636. C4C2×#13094007semidirect
639. C2#20246896direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940072
C22×#130939952
#202468962
C2×#156890512
C2×#156890382
C2×#156890262
C2×#156890212
C2×#156889962
C2×#156889922
C2×#156889802
C2×C4×(C2×D47)⋊C42
C22×C4×D4×C472
C2×#242247