Finite Group Databasebeta

← C2×#20246904 #26384154

Subgroups — C2×#20246904 #26384154

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×#13094006C212yesmaximal, complemented
2C22×#13093998C211yesmaximal, characteristic
3C22×#13093986C211yesmaximal, characteristic, complemented
4#20246904C2116yesmaximal, direct product factor
5C2×#15689052C212yesmaximal, complemented
6C2×#15689038C212yesmaximal, complemented
7C2×#15689024C211yesmaximal, characteristic, complemented
8C2×#15689022C211yesmaximal, characteristic, complemented
9C2×#15688997C212yesmaximal, complemented
10C2×#15688992C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊Dic47C211yesmaximal, characteristic, complemented
12C2×C47×#380C211yesmaximal, characteristic, complemented
13C23×D188C2211yescharacteristic
14C23×C22⋊D47C2212yes—
15C23×C4×D47C2212yes—
16C2×#13094006—216no—
17C2×#13094006C22116yescomplemented
18C2×#13093998—28no—
19C2×#13093998C2218yes—
20C2×#13093986—28no—
21C2×#13093986C2218yescomplemented
22C22×C4⋊Dic47C2211yescharacteristic
23C22×C4⋊Dic47C2212yes—
24C22×Dic47⋊C4C2212yes—
25C22×Dic47⋊C4C2212yes—
26C22×C22⋊Dic47C2211yescharacteristic
27C22×C22⋊Dic47C2211yescharacteristic
28C22×(C2×D47)⋊C4C2212yes—
29C22×(C2×D47)⋊C4C2212yes—
30C22×(C2×D47)⋊C4C2212yes—
31C22×(C2×D47)⋊C4C2212yes—
32C22×C4×Dic47C2211yescharacteristic
33C22×C4×Dic47C2211yescharacteristic
34#15689052C22116yescomplemented
35#15689038C22116yescomplemented
36#15689024C2218yescomplemented
37#15689022C2218yescomplemented
38#15688997C22116yescomplemented
39#15688992C2218yescomplemented
40C4×C22⋊Dic47C2218yescomplemented
41C2×C47⋊#95C2211yescharacteristic
42C2×C47⋊#95C2212yes—
43C2×C47⋊#95C2211yescharacteristic
44C24×C4×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C22×C4⋊C4×C47C2211yescharacteristic
47C22×C22⋊C4×C47C2212yes—
48C47×#380C2218yescomplemented
49C2×#95×C47C2212yes—
50C24×D47C2312yes—
51C22×D188—24no—
52C22×D188—24no—
53C22×D188D418yes—
54C22×D188C2314yes—
55C22×C22⋊D47—28no—
56C22×C22⋊D47—28no—
57C22×C22⋊D47—44no—
58C22×C22⋊D47C2318yes—
59C23×Dic47C2311yescharacteristic
60C23×Dic47C2312yes—
61C23×Dic47C2311yescharacteristic
62C23×Dic47C2311yescharacteristic
63C23×Dic47C2311yescharacteristic
64C22×C4×D47—28no—
65C22×C4×D47—28no—
66C22×C4×D47—44no—
67C22×C4×D47C2318yes—
68#13094006—264no—
69#13093998—232no—
70#13093986—232no—
71C2×C4⋊Dic47—24no—
72C2×C4⋊Dic47—28no—
73C2×C4⋊Dic47C2314yes—
74C2×C4⋊Dic47C2318yes—
75C2×Dic47⋊C4—28no—
76C2×Dic47⋊C4—44no—
77C2×Dic47⋊C4C2318yes—
78C2×Dic47⋊C4C2318yes—
79C2×C22⋊Dic47—24no—
80C2×C22⋊Dic47D418yescomplemented
81C2×C22⋊Dic47C2314yes—
82C2×C22⋊Dic47C2314yes—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—28no—
85C2×(C2×D47)⋊C4—28no—
86C2×(C2×D47)⋊C4—28no—
87C2×(C2×D47)⋊C4C2318yes—
88C2×(C2×D47)⋊C4C2318yes—
89C2×(C2×D47)⋊C4C2318yes—
90C2×(C2×D47)⋊C4C2318yes—
91C2×C4×Dic47—24no—
92C2×C4×Dic47—24no—
93C2×C4×Dic47C2314yes—
94C2×C4×Dic47C2314yes—
95C47⋊#95C2314yes—
96C47⋊#95C2314yes—
97C47⋊#95C2318yes—
98C25×C47C2311yescharacteristic
99C23×C4×C47—24no—
100C23×C4×C47—24no—
101C23×C4×C47—24no—
102C23×C4×C47C2311yescharacteristic
103C23×C4×C47C2312yes—
104C23×C4×C47C2314yes—
105C23×C4×C47C2312yes—
106C23×C4×C47C2311yescharacteristic
107C2×C4⋊C4×C47—24no—
108C2×C4⋊C4×C47—24no—
109C2×C4⋊C4×C47C2314yes—
110C2×C4⋊C4×C47C2314yes—
111C2×C22⋊C4×C47—28no—
112C2×C22⋊C4×C47C2318yes—
113#95×C47C2318yes—
114C24×C47—22no—
115C24×C47—22no—
116C24×C47—22no—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47—22no—
120C24×C47C2×D414yes—
121C24×C47C2412yes—
122C24×C47C2411yescharacteristic
123C23×D47—24no—
124C23×D47—24no—
125C23×D47—24no—
126C23×D47—24no—
127C23×D47—24no—
128C23×D47—24no—
129C23×D47C2×D418yes—
130C23×D47C2414yes—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47—24no—
134C22×C4×C47—22no—
135C22×C4×C47—24no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—24no—
139C22×C4×C47—24no—
140C22×C4×C47—28no—
141C22×C4×C47—28no—
142C22×C4×C47—28no—
143C22×C4×C47—416no—
144C22×C4×C47C2×D414yes—
145C22×C4×C47#4114yes—
146C22×C4×C47#4114yes—
147C22×C4×C47C2414yes—
148C22×C4×C47C2414yes—
149C22×C4×C47C2412yes—
150C22×C4×C47C2412yes—
151C2×C22⋊D47—216no—
152C2×C22⋊D47—216no—
153C2×C22⋊D47—48no—
154C2×C22⋊D47—432no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—24no—
163C22×Dic47—24no—
164C22×Dic47—22no—
165C22×Dic47—24no—
166C22×Dic47—22no—
167C22×Dic47C2×D414yes—
168C22×Dic47C2×D414yes—
169C22×Dic47#4114yes—
170C22×Dic47C2412yes—
171C22×Dic47C2414yes—
172C22×Dic47C2412yes—
173C22×Dic47C2412yes—
174C22×Dic47C2412yes—
175C2×D188—232no—
176C2×D188—28no—
177C2×D188—28no—
178C2×D188C2×D4116yes—
179C2×C4×D47—216no—
180C2×C4×D47—216no—
181C2×C4×D47—48no—
182C2×C4×D47—432no—
183C4⋊C4×C47—28no—
184C4⋊C4×C47—28no—
185C22⋊C4×C47—216no—
186C22⋊Dic47—28no—
187C22⋊Dic47C2×D4116yescomplemented
188(C2×D47)⋊C4—216no—
189(C2×D47)⋊C4—216no—
190(C2×D47)⋊C4—216no—
191(C2×D47)⋊C4—216no—
192C4⋊Dic47—28no—
193C4⋊Dic47—216no—
194Dic47⋊C4—216no—
195Dic47⋊C4—48no—
196C4×Dic47—28no—
197C4×Dic47—28no—
198C23×C47—28no—
199C23×C47—28no—
200C23×C47—28no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—44no—
208C23×C47—44no—
209C23×C47—44no—
210C23×C47—44no—
211C23×C47C22×D414yes—
212C23×C47C22×D412yes—
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—216no—
221C22×D47—216no—
222C22×D47—216no—
223C22×D47—24no—
224C22×D47—24no—
225C22×D47—24no—
226C22×D47—24no—
227C22×D47—24no—
228C22×D47—24no—
229C22×D47—48no—
230C22×D47—48no—
231C22×D47—48no—
232C22×D47—48no—
233C22×D47C22×D418yes—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—22no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—24no—
243C2×C4×C47—44no—
244C2×C4×C47—48no—
245C2×C4×C47—48no—
246C2×C4×C47—416no—
247C2×C4×C47#123116yescomplemented
248C2×C4×C47C22×D414yes—
249C2×C4×C47C2×#4114yes—
250C2×C4×C47C2×#4114yes—
251C22⋊D47—432no—
252C2×Dic47—28no—
253C2×Dic47—28no—
254C2×Dic47—28no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—22no—
265C2×Dic47—24no—
266C2×Dic47—22no—
267C2×Dic47—48no—
268C2×Dic47—44no—
269C2×Dic47C22×D414yes—
270C2×Dic47C22×D414yes—
271C2×Dic47C2×#4114yes—
272D188—232no—
273C4×D47—432no—
274C2×#2422—471nomaximal, Sylow
275C22×C47—24no—
276C22×C47—24no—
277C22×C47—24no—
278C22×C47—42no—
279C22×C47—416no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—42no—
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—28no—
298C2×D47—28no—
299C2×D47—28no—
300C2×D47—44no—
301C2×D47—432no—
302C2×D47—44no—
303C2×D47—44no—
304C2×D47—44no—
305C4×C47—42no—
306C4×C47—44no—
307C4×C47—44no—
308C4×C47C2×#12318yescomplemented
309Dic47—24no—
310Dic47—24no—
311Dic47—24no—
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×#384—471no—
324C2×#380D4711yescharacteristic, complemented
325C2×C4×C22⋊C4—471no—
326C2×C47—44no—
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—48no—
336C23×D4—472no—
337C23×D4—471no—
338C24×C4—472no—
339C24×C4C2×D4711yescharacteristic
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⋊C4—472no—
349C22×C4⋊C4—472no—
350C22×C4⋊C4C2×D4711yescharacteristic
351C22×C4⋊C4C2×D4711yescharacteristic
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⋊C4C2×D4712yes—
359C22×C42—471no—
360C22×C42—471no—
361#397—4716no—
362#394—4716no—
363#390—478no—
364#389—478no—
365#385—4716no—
366#384—478no—
367#380C2×D4718yescomplemented
368C4×C22⋊C4—478no—
369C2×#95—472no—
370C2×#95—471no—
371C2×#95—471no—
372C2×#95C2×D4712yes—
373C47C2×#242211yescharacteristic, complemented, Sylow
374C25—472no—
375C25C22×D4711yescharacteristic
376C22×D4—478no—
377C22×D4—478no—
378C22×D4—474no—
379C22×D4—948no—
380C22×D4—948no—
381C22×D4—944no—
382C22×D4—944no—
383C22×D4—1884no—
384C23×C4—24no—
385C23×C4—24no—
386C23×C4—24no—
387C23×C4—471no—
388C23×C4—471no—
389C23×C4—471no—
390C23×C4—471no—
391C23×C4—472no—
392C23×C4—478no—
393C23×C4—948no—
394C23×C4—948no—
395C23×C4—1884no—
396C23×C4C22×D4714yes—
397C23×C4C22×D4712yes—
398C23×C4C22×D4711yescharacteristic
399C23×C4C22×D4712yes—
400C23×C4C22×D4711yescharacteristic
401#123—9464no—
402#121—9432no—
403C4×D4—9432no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—474no—
407C2×C4⋊C4—478no—
408C2×C4⋊C4—478no—
409C2×C4⋊C4—478no—
410C2×C4⋊C4—948no—
411C2×C4⋊C4—948no—
412C2×C4⋊C4—944no—
413C2×C4⋊C4—1884no—
414C2×C4⋊C4C22×D4714yes—
415C2×C4⋊C4C22×D4714yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—478no—
418C2×C22⋊C4—474no—
419C2×C22⋊C4—474no—
420C2×C22⋊C4—478no—
421C2×C22⋊C4—478no—
422C2×C22⋊C4—478no—
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⋊C4C22×D4718yes—
430C2×C42—474no—
431C2×C42—474no—
432C2×C42—944no—
433C2×C42—944no—
434#95—474no—
435#95—474no—
436#95—478no—
437#95C22×D4718yes—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—478no—
445C24—474no—
446C24—944no—
447C24—944no—
448C24—944no—
449C24—944no—
450C24—944no—
451C24—944no—
452C24D4×D4714yes—
453C24C23×D4712yes—
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—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—416no—
476C22×C4—474no—
477C22×C4—474no—
478C22×C4—474no—
479C22×C4—472no—
480C22×C4—472no—
481C22×C4—472no—
482C22×C4—472no—
483C22×C4—474no—
484C22×C4—9416no—
485C22×C4—9416no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—944no—
494C22×C4—944no—
495C22×C4—942no—
496C22×C4—944no—
497C22×C4—942no—
498C22×C4—1888no—
499C22×C4—18832no—
500C22×C4#1062535014yes—
501C22×C4D4×D4714yes—
502C22×C4#1062535214yes—
503C22×C4C23×D4712yes—
504C22×C4C23×D4714yes—
505C22×C4C23×D4714yes—
506C22×C4C23×D4712yes—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—9416no—
510C4⋊C4—9416no—
511C4⋊C4—948no—
512C4⋊C4—1888no—
513C22⋊C4—216no—
514C22⋊C4—4716no—
515C22⋊C4—9416no—
516C22⋊C4—9416no—
517C22⋊C4—9416no—
518C22⋊C4—9416no—
519C22⋊C4—948no—
520C42—948no—
521C42—948no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—478no—
536C23—9416no—
537C23—9416no—
538C23—9416no—
539C23—944no—
540C23—944no—
541C23—944no—
542C23—944no—
543C23—944no—
544C23—944no—
545C23—1888no—
546C23—1888no—
547C23—1888no—
548C23—1888no—
549C23C2×D4×D4714yes—
550C23C2×D4×D4712yes—
551C23C2×D4×D4712yes—
552C23#41×D4712yes—
553C23C2×#1062535212yes—
554C23C2×#1062535012yes—
555C23#1309407012yes—
556C23#1309407212yes—
557C23C24×D4711yescharacteristic
558D4—9432no—
559D4—18832no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—416no—
573C2×C4—474no—
574C2×C4—474no—
575C2×C4—474no—
576C2×C4—948no—
577C2×C4—948no—
578C2×C4—948no—
579C2×C4—942no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—942no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—944no—
587C2×C4—944no—
588C2×C4—942no—
589C2×C4—944no—
590C2×C4—942no—
591C2×C4—18832no—
592C2×C4—1888no—
593C2×C4—1884no—
594C2×C4#13094006116yescomplemented
595C2×C4C2×D4×D4714yes—
596C2×C4C2×#1062535214yes—
597C2×C4C2×#1062535014yes—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—416no—
605C22—42no—
606C22—948no—
607C22—948no—
608C22—948no—
609C22—1884no—
610C22—1884no—
611C22—18832no—
612C22—1884no—
613C22—1884no—
614C22C2×#1309400614yes—
615C22#1568966814yes—
616C22#1568974614yes—
617C22#1568975714yes—
618C22#1568977414yes—
619C22#1568977514yes—
620C22#1568981714yes—
621C22C22×D4×D4711yescharacteristic
622C22C22×D4×D4711yescharacteristic
623C22C2×#41×D4711yescharacteristic
624C22C22×#1062535211yescharacteristic
625C22C22×#1062535011yescharacteristic
626C22C2×#1309407011yescharacteristic
627C22C2×#1309407211yescharacteristic
628C4—44no—
629C4—42no—
630C4—44no—
631C4—944no—
632C4—944no—
633C4—944no—
634C4—1884no—
635C4—1882no—
636C4C2×#1309400618yescomplemented
637C2—44no—
638C2—1888no—
639C2#2024690418yesdirect product factor
640C2C22×#1309400611yescharacteristic
641C2C2×#1568966811yescharacteristic
642C2C2×#1568974611yescharacteristic
643C2C2×#1568975711yescharacteristic
644C2C2×#1568977411yescharacteristic
645C2C2×#1568977511yescharacteristic
646C2C2×#1568981711yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#380C2
socleC24×C47C24
prime radicalC22×#13093998C2
hypercenterC2×#380D47
hypocenterC47C2×#2422
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#13094006C2semidirect
3. C22×#13093986C2semidirect
4. #20246904C2direct (N × H)
5. C2×#15689052C2semidirect
6. C2×#15689038C2semidirect
7. C2×#15689024C2semidirect
8. C2×#15689022C2semidirect
9. C2×#15688997C2semidirect
10. C2×#15688992C2semidirect
11. C2×C4×C22⋊Dic47C2semidirect
12. C2×C47×#380C2semidirect
17. C2×#13094006C22semidirect
21. C2×#13093986C22semidirect
34. #15689052C22semidirect
35. #15689038C22semidirect
36. #15689024C22semidirect
37. #15689022C22semidirect
38. #15688997C22semidirect
39. #15688992C22semidirect
40. C4×C22⋊Dic47C22semidirect
48. C47×#380C22semidirect
80. C2×C22⋊Dic47D4semidirect
187. C22⋊Dic47C2×D4semidirect
247. C2×C4×C47#123semidirect
308. C4×C47C2×#123semidirect
324. C2×#380D47semidirect
327. C2×C47#2422semidirect
367. #380C2×D47semidirect
373. C47C2×#2422semidirect
594. C2×C4#13094006semidirect
636. C4C2×#13094006semidirect
639. C2#20246904direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940062
C22×#130939982
C22×#130939862
#202469042
C2×#156890522
C2×#156890382
C2×#156890242
C2×#156890222
C2×#156889972
C2×#156889922
C2×C4×C22⋊Dic472
C2×C47×#3802
C2×#242247