Finite Group Databasebeta

← C2×#20246905 #26384156

Subgroups — C2×#20246905 #26384156

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×#13094009C212yesmaximal, complemented
2C22×#13093999C211yesmaximal, characteristic
3C22×#13093986C211yesmaximal, characteristic, complemented
4#20246905C2116yesmaximal, direct product factor
5C2×#15689049C212yesmaximal, complemented
6C2×#15689040C212yesmaximal, complemented
7C2×#15689025C211yesmaximal, characteristic, complemented
8C2×#15689019C211yesmaximal, characteristic, complemented
9C2×#15688994C212yesmaximal, complemented
10C2×#15688980C211yesmaximal, characteristic, complemented
11C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic, complemented
12C2×C47×#384C211yesmaximal, characteristic, complemented
13C23×D188C2212yes—
14C23×D188C2211yescharacteristic
15C23×C4×D47C2211yescharacteristic
16C23×C4×D47C2212yes—
17C2×#13094009—216no—
18C2×#13094009C22116yescomplemented
19C2×#13093999—28no—
20C2×#13093999C2218yes—
21C2×#13093986—28no—
22C2×#13093986C2218yescomplemented
23C22×C4⋊Dic47C2211yescharacteristic
24C22×Dic47⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2212yes—
26C22×(C2×D47)⋊C4C2212yes—
27C22×(C2×D47)⋊C4C2212yes—
28C22×(C2×D47)⋊C4C2211yescharacteristic
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2211yescharacteristic
32C22×C4×Dic47C2211yescharacteristic
33#15689049C22116yescomplemented
34#15689040C22116yescomplemented
35#15689025C2218yescomplemented
36#15689019C2218yescomplemented
37#15688994C22116yescomplemented
38#15688980C2218yescomplemented
39C4×(C2×D47)⋊C4C2218yescomplemented
40C2×C47⋊#95C2211yescharacteristic
41C2×C47⋊#95C2212yes—
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2212yes—
47C22×C4⋊C4×C47C2212yes—
48C22×C42×C47C2211yescharacteristic
49C47×#384C2218yescomplemented
50C24×D47C2312yes—
51C24×D47C2311yescharacteristic
52C22×D188—24no—
53C22×D188—24no—
54C22×D188—28no—
55C22×D188—28no—
56C22×D188—44no—
57C22×D188D418yes—
58C22×D188C2318yes—
59C22×D188C2314yes—
60C23×Dic47C2311yescharacteristic
61C23×Dic47C2311yescharacteristic
62C23×Dic47C2312yes—
63C23×Dic47C2311yescharacteristic
64C22×C4×D47—28no—
65C22×C4×D47—24no—
66C22×C4×D47—24no—
67C22×C4×D47—24no—
68C22×C4×D47—28no—
69C22×C4×D47—44no—
70C22×C4×D47C2314yes—
71C22×C4×D47C2318yes—
72#13094009—264no—
73#13093999—232no—
74#13093986—232no—
75C2×C4⋊Dic47—24no—
76C2×C4⋊Dic47C2314yes—
77C2×Dic47⋊C4—28no—
78C2×Dic47⋊C4C2318yes—
79C2×(C2×D47)⋊C4—28no—
80C2×(C2×D47)⋊C4—28no—
81C2×(C2×D47)⋊C4—28no—
82C2×(C2×D47)⋊C4—28no—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—24no—
85C2×(C2×D47)⋊C4D418yescomplemented
86C2×(C2×D47)⋊C4C2314yes—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×(C2×D47)⋊C4C2318yes—
92C2×(C2×D47)⋊C4C2314yes—
93C2×C4×Dic47—24no—
94C2×C4×Dic47C2314yes—
95C47⋊#95C2314yes—
96C47⋊#95C2314yes—
97C47⋊#95C2318yes—
98C47⋊#95C2318yes—
99C23×C4×C47C2312yes—
100C23×C4×C47C2312yes—
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47—24no—
105C2×C4⋊C4×C47—24no—
106C2×C4⋊C4×C47—28no—
107C2×C4⋊C4×C47—44no—
108C2×C4⋊C4×C47C2314yes—
109C2×C4⋊C4×C47C2314yes—
110C2×C4⋊C4×C47C2318yes—
111C2×C4⋊C4×C47C2318yes—
112C2×C42×C47—24no—
113C2×C42×C47C2314yes—
114C24×C47C2411yescharacteristic
115C23×D47—24no—
116C23×D47—22no—
117C23×D47—22no—
118C23×D47—24no—
119C23×D47—22no—
120C23×D47—22no—
121C23×D47—24no—
122C23×D47—24no—
123C23×D47—24no—
124C23×D47—22no—
125C23×D47—22no—
126C23×D47—24no—
127C23×D47C2×D414yes—
128C23×D47C2×D418yes—
129C23×D47C2414yes—
130C23×D47C2412yes—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—24no—
139C22×C4×C47—24no—
140C22×C4×C47—24no—
141C22×C4×C47C2×D414yes—
142C22×C4×C47C2×D414yes—
143C22×C4×C47#4114yes—
144C22×C4×C47#4114yes—
145C22×C4×C47#4114yes—
146C22×C4×C47C2414yes—
147C22×C4×C47C2414yes—
148C22×C4×C47C2412yes—
149C22×C4×C47C2412yes—
150C22×C4×C47C2412yes—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—24no—
155C22×Dic47—22no—
156C22×Dic47—24no—
157C22×Dic47—22no—
158C22×Dic47—24no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47C2×D414yes—
163C22×Dic47C2412yes—
164C22×Dic47C2412yes—
165C22×Dic47C2414yes—
166C22×Dic47C2412yes—
167C2×D188—232no—
168C2×D188—28no—
169C2×D188—28no—
170C2×D188—216no—
171C2×D188—216no—
172C2×D188—48no—
173C2×D188—432no—
174C2×D188C2×D4116yes—
175C2×C4×D47—216no—
176C2×C4×D47—28no—
177C2×C4×D47—28no—
178C2×C4×D47—28no—
179C2×C4×D47—216no—
180C2×C4×D47—48no—
181C2×C4×D47—432no—
182C2×C4×D47—416no—
183C4⋊C4×C47—28no—
184C4⋊C4×C47—28no—
185C4⋊C4×C47—216no—
186C4⋊C4×C47—48no—
187C42×C47—28no—
188(C2×D47)⋊C4—216no—
189(C2×D47)⋊C4—216no—
190(C2×D47)⋊C4—216no—
191(C2×D47)⋊C4—216no—
192(C2×D47)⋊C4—28no—
193(C2×D47)⋊C4—216no—
194(C2×D47)⋊C4C2×D4116yescomplemented
195C4⋊Dic47—28no—
196Dic47⋊C4—216no—
197C4×Dic47—28no—
198C23×C47C22×D412yes—
199C23×C47C22×D412yes—
200C23×C47C2×#4112yes—
201C23×C47C2×#4112yes—
202C23×C47C2×#4112yes—
203C23×C47H(5,𝔽2)12yes—
204C23×C472−1+412yes—
205C23×C47C2511yescharacteristic
206C22×D47—216no—
207C22×D47—28no—
208C22×D47—216no—
209C22×D47—28no—
210C22×D47—28no—
211C22×D47—216no—
212C22×D47—24no—
213C22×D47—22no—
214C22×D47—22no—
215C22×D47—24no—
216C22×D47—22no—
217C22×D47—22no—
218C22×D47—24no—
219C22×D47—24no—
220C22×D47—24no—
221C22×D47—22no—
222C22×D47—22no—
223C22×D47—24no—
224C22×D47—44no—
225C22×D47—48no—
226C22×D47—44no—
227C22×D47—48no—
228C22×D47—48no—
229C22×D47—44no—
230C22×D47—44no—
231C22×D47—48no—
232C22×D47C22×D414yes—
233C22×D47C22×D418yes—
234C2×C4×C47—28no—
235C2×C4×C47—28no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—24no—
241C2×C4×C47—24no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—24no—
246C2×C4×C47—48no—
247C2×C4×C47—48no—
248C2×C4×C47#123116yescomplemented
249C2×C4×C47C22×D414yes—
250C2×C4×C47C22×D414yes—
251C2×C4×C47C2×#4114yes—
252C2×C4×C47C2×#4114yes—
253C2×C4×C47C2×#4114yes—
254C2×Dic47—28no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—24no—
259C2×Dic47—22no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—22no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—44no—
267C2×Dic47—48no—
268C2×Dic47—44no—
269C2×Dic47C22×D414yes—
270D188—232no—
271D188—432no—
272C4×D47—432no—
273C4×D47—416no—
274C2×#2422—471nomaximal, Sylow
275C22×C47C2×#12314yes—
276C22×C47#53214yes—
277C22×C47#54214yes—
278C22×C47D4214yes—
279C22×C47#54714yes—
280C22×C47#54714yes—
281C22×C47#55314yes—
282C22×C47C23×D411yescharacteristic
283C22×C47C23×D411yescharacteristic
284C22×C47C22×#4111yescharacteristic
285C22×C47C22×#4111yescharacteristic
286C22×C47C22×#4111yescharacteristic
287C22×C47C2×H(5,𝔽2)11yescharacteristic
288C22×C47C2×2−1+411yescharacteristic
289C2×D47—28no—
290C2×D47—24no—
291C2×D47—28no—
292C2×D47—24no—
293C2×D47—24no—
294C2×D47—28no—
295C2×D47—42no—
296C2×D47—44no—
297C2×D47—432no—
298C2×D47—416no—
299C2×D47—42no—
300C2×D47—44no—
301C2×D47—44no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—44no—
305C4×C47—24no—
306C4×C47—24no—
307C4×C47—44no—
308C4×C47—44no—
309C4×C47C2×#12318yescomplemented
310Dic47—24no—
311Dic47—42no—
312Dic47—44no—
313Dic47—42no—
314C22×#123—472no—
315C22×#121—471no—
316C22×C4×D4—471no—
317#2422—4716no—
318C2×#397—472no—
319C2×#394—472no—
320C2×#390—471no—
321C2×#389—471no—
322C2×#385—472no—
323C2×#384D4711yescharacteristic, complemented
324C2×#380—471no—
325C2×C4×C22⋊C4—471no—
326C2×C47#242218yescomplemented
327C2×C47C22×#12311yescharacteristic
328C2×C47C2×#53211yescharacteristic
329C2×C47C2×#54211yescharacteristic
330C2×C47C2×D4211yescharacteristic
331C2×C47C2×#54711yescharacteristic
332C2×C47C2×#54711yescharacteristic
333C2×C47C2×#55311yescharacteristic
334D47—48no—
335D47—44no—
336C23×D4—471no—
337C23×D4—472no—
338C24×C4—471no—
339C24×C4—472no—
340C2×#123—4716no—
341C2×#123—9416no—
342C2×#121—478no—
343C2×#121—948no—
344C2×C4×D4—478no—
345C2×C4×D4—948no—
346C22×C4⋊C4—472no—
347C22×C4⋊C4—471no—
348C22×C4⋊C4C2×D4712yes—
349C22×C4⋊C4C2×D4711yescharacteristic
350C22×C4⋊C4C2×D4711yescharacteristic
351C22×C4⋊C4C2×D4712yes—
352C22×C22⋊C4—471no—
353C22×C22⋊C4—471no—
354C22×C22⋊C4—472no—
355C22×C22⋊C4—472no—
356C22×C22⋊C4—472no—
357C22×C22⋊C4—472no—
358C22×C22⋊C4—472no—
359C22×C42—471no—
360C22×C42C2×D4711yescharacteristic
361#397—4716no—
362#394—4716no—
363#390—478no—
364#389—478no—
365#385—4716no—
366#384C2×D4718yescomplemented
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—
375C25—472no—
376C22×D4—478no—
377C22×D4—474no—
378C22×D4—478no—
379C22×D4—948no—
380C22×D4—948no—
381C22×D4—944no—
382C22×D4—944no—
383C22×D4—1884no—
384C23×C4—471no—
385C23×C4—472no—
386C23×C4—471no—
387C23×C4—471no—
388C23×C4—474no—
389C23×C4—478no—
390C23×C4—948no—
391C23×C4—948no—
392C23×C4—944no—
393C23×C4—944no—
394C23×C4—944no—
395C23×C4—1884no—
396C23×C4C22×D4712yes—
397C23×C4C22×D4712yes—
398C23×C4C22×D4711yescharacteristic
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401#123—9464no—
402#121—9432no—
403C4×D4—9432no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—28no—
407C2×C4⋊C4—44no—
408C2×C4⋊C4—478no—
409C2×C4⋊C4—474no—
410C2×C4⋊C4—948no—
411C2×C4⋊C4—944no—
412C2×C4⋊C4C22×D4714yes—
413C2×C4⋊C4C22×D4714yes—
414C2×C4⋊C4C22×D4718yes—
415C2×C4⋊C4C22×D4718yes—
416C2×C22⋊C4—478no—
417C2×C22⋊C4—474no—
418C2×C22⋊C4—478no—
419C2×C22⋊C4—478no—
420C2×C22⋊C4—478no—
421C2×C22⋊C4—478no—
422C2×C22⋊C4—474no—
423C2×C22⋊C4—478no—
424C2×C22⋊C4—948no—
425C2×C22⋊C4—948no—
426C2×C22⋊C4—948no—
427C2×C22⋊C4—948no—
428C2×C22⋊C4—944no—
429C2×C22⋊C4—948no—
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—474no—
439C24—478no—
440C24—472no—
441C24—474no—
442C24—944no—
443C24—942no—
444C24—942no—
445C24—944no—
446C24—944no—
447C24—942no—
448C24—942no—
449C24—944no—
450C24—944no—
451C24—942no—
452C24—942no—
453C24—944no—
454C24C23×D4711yescharacteristic
455C2×D4—4716no—
456C2×D4—9416no—
457C2×D4—9416no—
458C2×D4—9432no—
459C2×D4—948no—
460C2×D4—948no—
461C2×D4—1888no—
462C2×D4—18832no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—474no—
474C22×C4—472no—
475C22×C4—472no—
476C22×C4—472no—
477C22×C4—474no—
478C22×C4—9416no—
479C22×C4—9416no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—944no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—944no—
488C22×C4—944no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—948no—
492C22×C4—948no—
493C22×C4—948no—
494C22×C4—1888no—
495C22×C4—18832no—
496C22×C4—18816no—
497C22×C4#1062535014yes—
498C22×C4D4×D4714yes—
499C22×C4D4×D4714yes—
500C22×C4#1062535414yes—
501C22×C4#1062535414yes—
502C22×C4C23×D4714yes—
503C22×C4C23×D4714yes—
504C22×C4C23×D4712yes—
505C22×C4C23×D4712yes—
506C22×C4C23×D4712yes—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—216no—
510C4⋊C4—48no—
511C4⋊C4—9416no—
512C4⋊C4—948no—
513C22⋊C4—4716no—
514C22⋊C4—9416no—
515C22⋊C4—9416no—
516C22⋊C4—9416no—
517C22⋊C4—9416no—
518C22⋊C4—948no—
519C22⋊C4—9416no—
520C42—28no—
521C42—948no—
522C23—474no—
523C23—478no—
524C23—9416no—
525C23—948no—
526C23—948no—
527C23—9416no—
528C23—9416no—
529C23—948no—
530C23—944no—
531C23—942no—
532C23—942no—
533C23—944no—
534C23—944no—
535C23—942no—
536C23—942no—
537C23—944no—
538C23—944no—
539C23—942no—
540C23—942no—
541C23—944no—
542C23—1884no—
543C23—1888no—
544C23—1888no—
545C23—1884no—
546C23—1884no—
547C23—1888no—
548C23—1884no—
549C23—1888no—
550C23C2×D4×D4712yes—
551C23C2×D4×D4712yes—
552C23C2×#1062535012yes—
553C23C2×#1062535412yes—
554C23C2×#1062535412yes—
555C23#1309406912yes—
556C23#1309407212yes—
557C23C24×D4711yescharacteristic
558D4—9432no—
559D4—18832no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—474no—
575C2×C4—948no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—942no—
579C2×C4—944no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—942no—
586C2×C4—942no—
587C2×C4—18832no—
588C2×C4—1884no—
589C2×C4—1884no—
590C2×C4—1888no—
591C2×C4—18816no—
592C2×C4#13094009116yescomplemented
593C2×C4C2×D4×D4714yes—
594C2×C4C2×D4×D4714yes—
595C2×C4C2×#1062535014yes—
596C2×C4C2×#1062535414yes—
597C2×C4C2×#1062535414yes—
598C22—948no—
599C22—944no—
600C22—944no—
601C22—948no—
602C22—948no—
603C22—944no—
604C22—1882no—
605C22—1884no—
606C22—1884no—
607C22—1882no—
608C22—1882no—
609C22—18832no—
610C22—18816no—
611C22—1882no—
612C22—1884no—
613C22—1884no—
614C22C2×#1309400914yes—
615C22#1568967414yes—
616C22#1568974714yes—
617C22#1568975814yes—
618C22#1568977414yes—
619C22#1568977614yes—
620C22#1568982114yes—
621C22C22×D4×D4711yescharacteristic
622C22C22×D4×D4711yescharacteristic
623C22C22×#1062535011yescharacteristic
624C22C22×#1062535411yescharacteristic
625C22C22×#1062535411yescharacteristic
626C22C2×#1309406911yescharacteristic
627C22C2×#1309407211yescharacteristic
628C4—24no—
629C4—24no—
630C4—44no—
631C4—44no—
632C4—944no—
633C4—1882no—
634C4—1882no—
635C4—1884no—
636C4C2×#1309400918yescomplemented
637C2—1888no—
638C2—1884no—
639C2#2024690518yesdirect product factor
640C2C22×#1309400911yescharacteristic
641C2C2×#1568967411yescharacteristic
642C2C2×#1568974711yescharacteristic
643C2C2×#1568975811yescharacteristic
644C2C2×#1568977411yescharacteristic
645C2C2×#1568977611yescharacteristic
646C2C2×#1568982111yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#384C2
socleC24×C47C24
prime radicalC22×#13093999C2
hypercenterC2×#384D47
hypocenterC47C2×#2422
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
1. C22×#13094009C2semidirect
3. C22×#13093986C2semidirect
4. #20246905C2direct (N × H)
5. C2×#15689049C2semidirect
6. C2×#15689040C2semidirect
7. C2×#15689025C2semidirect
8. C2×#15689019C2semidirect
9. C2×#15688994C2semidirect
10. C2×#15688980C2semidirect
11. C2×C4×(C2×D47)⋊C4C2semidirect
12. C2×C47×#384C2semidirect
18. C2×#13094009C22semidirect
22. C2×#13093986C22semidirect
33. #15689049C22semidirect
34. #15689040C22semidirect
35. #15689025C22semidirect
36. #15689019C22semidirect
37. #15688994C22semidirect
38. #15688980C22semidirect
39. C4×(C2×D47)⋊C4C22semidirect
49. C47×#384C22semidirect
85. C2×(C2×D47)⋊C4D4semidirect
194. (C2×D47)⋊C4C2×D4semidirect
248. C2×C4×C47#123semidirect
309. C4×C47C2×#123semidirect
323. C2×#384D47semidirect
326. C2×C47#2422semidirect
366. #384C2×D47semidirect
373. C47C2×#2422semidirect
592. C2×C4#13094009semidirect
636. C4C2×#13094009semidirect
639. C2#20246905direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940092
C22×#130939992
C22×#130939862
#202469052
C2×#156890492
C2×#156890402
C2×#156890252
C2×#156890192
C2×#156889942
C2×#156889802
C2×C4×(C2×D47)⋊C42
C2×C47×#3842
C2×#242247