Finite Group Databasebeta

← C2×#20243830 #26218030

Subgroups — C2×#20243830 #26218030

Order 12032.

2514 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689635C211yesmaximal, characteristic
2C22×C42×D47C211yesmaximal, characteristic, complemented
3C22×#13093952C211yesmaximal, characteristic, complemented
4C22×#13093950C211yesmaximal, characteristic, complemented
5C2×#15689086C211yesmaximal, characteristic, complemented
6#20243830C218yesmaximal, direct product factor
7C2×#15688882C211yesmaximal, characteristic, complemented
8C2×C47×#339C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C23×C4×D47—21no—
11C23×C4×D47C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13#15689635—22no—
14#15689635—22no—
15#15689635C414yescomplemented
16#15689635C2214yes—
17C2×#13094008—21no—
18C2×#13094004C412yescomplemented
19C2×#13093999C412yescomplemented
20C2×#13093994—21no—
21C2×C4×D188—22no—
22C22×C4×Dic47—21no—
23C22×C4×Dic47C2211yescharacteristic
24C2×C42×D47—22no—
25C2×C42×D47—24no—
26C2×C42×D47—24no—
27C2×C42×D47C2214yescomplemented
28C2×#13093952—24no—
29C2×#13093952C2214yescomplemented
30C2×#13093950—24no—
31C2×#13093950C2214yescomplemented
32#15689086—22no—
33#15689086C2214yescomplemented
34C2×(C2×D47)⋊C8—22no—
35#15688882C2214yescomplemented
36C22×M16×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C22×C42×C47C2211yescharacteristic
39C47×#339C2214yescomplemented
40C24×D47D411yescharacteristic
41C2×#10625350—21no—
42C2×#10625350—21no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350C2×C412yescomplemented
48C2×#10625350C2312yes—
49C22×Dic94C2×C411yescharacteristic, complemented
50C22×D188—22no—
51C22×D188C2×C411yescharacteristic, complemented
52C22×C22⋊D47—21no—
53C23×Dic47—21no—
54C23×Dic47D411yescharacteristic
55C22×C4×D47—28no—
56C22×C4×D47—216no—
57C22×C4×D47—24no—
58C22×C4×D47—22no—
59C22×C4×D47—22no—
60C22×C4×D47—22no—
61C22×C4×D47—22no—
62C22×C4×D47—21no—
63C22×C4×D47C2×C412yes—
64C22×C4×D47C2×C412yes—
65C22×C4×D47D412yes—
66C22×C4×D47D412yes—
67C22×C4×D47D412yes—
68C22×C4×D47C2312yes—
69C2×#10625329—21no—
70C2×#10625329—22no—
71C2×#10625329—22no—
72C2×#10625329C2312yes—
73C22×C47⋊C8—21no—
74#13094008—24no—
75#13094008—24no—
76#13094004—24no—
77#13094004C2×C418yescomplemented
78#13093999—24no—
79#13093999C2×C418yescomplemented
80#13093994—24no—
81#13093994—24no—
82C4×D188—28no—
83C4×D188—28no—
84C2×C4⋊Dic47—22no—
85C2×(C2×D47)⋊C4—22no—
86C2×(C2×D47)⋊C4—22no—
87C2×C4×Dic47—28no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—22no—
90C2×C4×Dic47—24no—
91C2×C4×Dic47D414yescomplemented
92C2×C4×Dic47C2312yes—
93C42×D47—216no—
94C42×D47—28no—
95C42×D47—28no—
96#13093952—28no—
97#13093950—28no—
98(C2×D47)⋊C8—28no—
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C2×M16×C47—21no—
102C2×M16×C47—22no—
103C2×M16×C47—22no—
104C2×M16×C47C2312yes—
105C22×C8×C47—21no—
106#117×C47—21no—
107#117×C47—21no—
108#117×C47C2×C412yescomplemented
109#117×C47C2312yes—
110C2×C4⋊C4×C47C2×C412yescomplemented
111C2×C22⋊C4×C47—21no—
112C2×C42×C47—21no—
113C2×C42×C47—22no—
114C2×C42×C47—21no—
115C2×C42×C47D414yescomplemented
116C2×C42×C47C2312yes—
117C24×C47C2×D411yescharacteristic
118C23×D47—22no—
119C23×D47—22no—
120C23×D47—28no—
121C23×D47—21no—
122C23×D47C22⋊C412yes—
123C23×D47C22⋊C412yes—
124C23×D47C2×D412yes—
125C22×C4×C47—28no—
126C22×C4×C47—24no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47C22×C411yescharacteristic
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47#4112yes—
137C22×C4×C47#4112yes—
138C22×C4×C47C2411yescharacteristic
139C2×C22⋊D47—22no—
140C2×C22⋊D47—22no—
141C2×C22⋊D47—42no—
142C2×C22⋊D47—42no—
143C22×Dic47—28no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—21no—
149C22×Dic47C22⋊C412yes—
150C22×Dic47C22⋊C412yes—
151C22×Dic47C2×D412yes—
152#10625350—24no—
153#10625350—24no—
154#10625350—22no—
155#10625350—22no—
156#10625350—22no—
157#10625350—22no—
158#10625350—42no—
159#10625350—42no—
160#10625350—42no—
161#10625350—42no—
162C2×D188—22no—
163C2×D188—22no—
164C2×D188—28no—
165C2×D188—21no—
166C2×D188—48no—
167C2×D188C22×C412yescomplemented
168C2×C4×D47—216no—
169C2×C4×D47—264no—
170C2×C4×D47—232no—
171C2×C4×D47—24no—
172C2×C4×D47—24no—
173C2×C4×D47—24no—
174C2×C4×D47—24no—
175C2×C4×D47—24no—
176C2×C4×D47—24no—
177C2×C4×D47—24no—
178C2×C4×D47—24no—
179C2×C4×D47—24no—
180C2×C4×D47—24no—
181C2×C4×D47—22no—
182C2×C4×D47—22no—
183C2×C4×D47—22no—
184C2×C4×D47—22no—
185C2×C4×D47—22no—
186C2×C4×D47—22no—
187C2×C4×D47—42no—
188C2×C4×D47—42no—
189C2×C4×D47C22⋊C414yes—
190C2×C4×D47C22⋊C414yes—
191C2×C4×D47C22×C414yes—
192C2×C4×D47C2×D414yes—
193C2×Dic94—22no—
194C2×Dic94—22no—
195C2×Dic94—21no—
196C2×Dic94C22×C412yescomplemented
197M16×C47—22no—
198M16×C47—22no—
199M16×C47—42no—
200C2×C8×C47—22no—
201C2×C8×C47—42no—
202C4⋊C4×C47—22no—
203C4⋊C4×C47C22×C414yescomplemented
204C22⋊C4×C47—22no—
205C22⋊C4×C47—22no—
206C42×C47—22no—
207C42×C47—24no—
208C42×C47—22no—
209C42×C47—22no—
210C42×C47C2×D414yescomplemented
211(C2×D47)⋊C4—28no—
212(C2×D47)⋊C4—28no—
213C4⋊Dic47—28no—
214C4×Dic47—216no—
215C4×Dic47—24no—
216C4×Dic47—22no—
217C4×Dic47C2×D414yescomplemented
218#10625329—22no—
219#10625329—22no—
220#10625329—42no—
221C2×C47⋊C8—22no—
222C2×C47⋊C8—42no—
223C23×C47—24no—
224C23×C47C2×C22⋊C411yescharacteristic
225C23×C47C2×C22⋊C411yescharacteristic
226C23×C47#12012yes—
227C23×C47#12312yes—
228C23×C47C22×D411yescharacteristic
229C22×D47—216no—
230C22×D47—216no—
231C22×D47—22no—
232C22×D47—22no—
233C22×D47—24no—
234C22×D47—24no—
235C22×D47—24no—
236C22×D47—28no—
237C22×D47—28no—
238C22×D47—22no—
239C22×D47—42no—
240C22×D47—44no—
241C22×D47#10418yescomplemented
242C22×D47C2×C22⋊C414yes—
243C2×C4×C47—28no—
244C2×C4×C47—216no—
245C2×C4×C47—24no—
246C2×C4×C47—24no—
247C2×C4×C47—44no—
248C2×C4×C47C2×C22⋊C411yescharacteristic
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C4×D412yes—
251C2×C4×C47C4×D412yes—
252C2×C4×C47C4×D412yes—
253C2×C4×C47C4×D412yes—
254C2×C4×C47#12012yes—
255C2×C4×C47#12112yes—
256C2×C4×C47#12112yes—
257C2×C4×C47#12112yes—
258C2×C4×C47#12112yes—
259C2×C4×C47#12312yes—
260C2×C4×C47C23×C411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C22⋊D47—42no—
267C22⋊D47—42no—
268C22⋊D47—44no—
269C2×Dic47—28no—
270C2×Dic47—216no—
271C2×Dic47—22no—
272C2×Dic47—24no—
273C2×Dic47—22no—
274C2×Dic47—24no—
275C2×Dic47—24no—
276C2×Dic47—22no—
277C2×Dic47—42no—
278C2×Dic47C2×C22⋊C414yes—
279D188—22no—
280D188—22no—
281D188—416no—
282D188—42no—
283C4×D47—264no—
284C4×D47—24no—
285C4×D47—24no—
286C4×D47—24no—
287C4×D47—24no—
288C4×D47—28no—
289C4×D47—28no—
290C4×D47—42no—
291C4×D47—42no—
292C4×D47—44no—
293Dic94—22no—
294Dic94—22no—
295Dic94—42no—
296C8×C47—42no—
297C47⋊C8—42no—
298C2×#1896—471nomaximal, Sylow
299C22×C47—22no—
300C22×C47—22no—
301C22×C47—24no—
302C22×C47—24no—
303C22×C47#38612yes—
304C22×C47#38612yes—
305C22×C47C2×#10414yes—
306C22×C47C22×C22⋊C411yescharacteristic
307C22×C47C2×#12011yescharacteristic
308C22×C47C2×#12311yescharacteristic
309C2×D47—28no—
310C2×D47—28no—
311C2×D47—216no—
312C2×D47—24no—
313C2×D47—216no—
314C2×D47—24no—
315C2×D47—44no—
316C2×D47—42no—
317C2×D47—84no—
318C2×D47C2×#10418yescomplemented
319C4×C47—28no—
320C4×C47—44no—
321C4×C47#38612yes—
322C4×C47#38612yes—
323C4×C47C2×C4×D411yescharacteristic
324C4×C47C2×C4×D411yescharacteristic
325C4×C47C2×#12111yescharacteristic
326C4×C47C2×#12111yescharacteristic
327Dic47—28no—
328Dic47—24no—
329Dic47—42no—
330C2×#525—471no—
331C23×C42—471no—
332C22×#104—471no—
333C22×#104—471no—
334C2×#407—471no—
335#1896—478no—
336C2×#339—471no—
337C2×#339D4711yescharacteristic, complemented
338C2×C47—21no—
339C2×C47—21no—
340C2×C47—22no—
341C2×C47#189614yescomplemented
342C2×C47C2×#38611yescharacteristic
343C2×C47C22×#10412yes—
344D47—28no—
345D47—82no—
346C22×#41—471no—
347C24×C4—471no—
348C24×C4—941no—
349C22×M16—471no—
350C22×M16C2×D4711yescharacteristic
351#525—474no—
352#525—474no—
353#525—942no—
354#525—942no—
355C2×#123—941no—
356C2×C22⋊Q8—472no—
357C2×#121—472no—
358C2×#120—941no—
359C2×C4×D4—942no—
360C2×#117C2×D4711yescharacteristic
361C22×C42—471no—
362C22×C42—474no—
363C22×C42—944no—
364C22×C42—942no—
365C22×C42—941no—
366C22×C42—944no—
367C22×C42C2×D4711yescharacteristic
368C2×#104—474no—
369C2×#104—474no—
370C2×#104—944no—
371C2×#104—944no—
372#407—474no—
373#407—942no—
374C2×C22⋊C8—942no—
375#339—474no—
376#339C2×D4714yescomplemented
377C47C2×#189611yescharacteristic, complemented, Sylow
378C25—471no—
379C2×#41—472no—
380C2×#41—472no—
381C2×#41—941no—
382C2×#41—941no—
383C2×#41—942no—
384C2×#41—942no—
385C2×#41—942no—
386C2×#41—942no—
387C22×Q8—471no—
388C22×D4—471no—
389C22×D4—942no—
390C22×D4—941no—
391C23×C4—21no—
392C23×C4—471no—
393C23×C4—472no—
394C23×C4—472no—
395C23×C4—472no—
396C23×C4—472no—
397C23×C4—472no—
398C23×C4—472no—
399C23×C4—944no—
400C23×C4—9416no—
401C23×C4—948no—
402C23×C4—941no—
403C23×C4—942no—
404C23×C4—942no—
405C23×C4—942no—
406C23×C4—942no—
407C23×C4—941no—
408C23×C4C22×D4711yescharacteristic
409C2×M16—21no—
410C2×M16—22no—
411C2×M16—22no—
412C2×M16—472no—
413C2×M16—941no—
414C2×M16—942no—
415C2×M16—942no—
416C2×M16C22×D4712yes—
417C22×C8—21no—
418C22×C8—941no—
419#123—944no—
420#123—944no—
421C22⋊Q8—478no—
422C22⋊Q8—944no—
423#121—478no—
424#121—944no—
425#120—944no—
426#120—944no—
427C4×D4—948no—
428C4×D4—948no—
429#117—21no—
430#117—21no—
431#117C4×D4712yescomplemented
432#117C22×D4712yes—
433C2×C4⋊C4—942no—
434C2×C4⋊C4C4×D4712yescomplemented
435C2×C22⋊C4—21no—
436C2×C22⋊C4—942no—
437C2×C22⋊C4—942no—
438C2×C42—21no—
439C2×C42—22no—
440C2×C42—21no—
441C2×C42—474no—
442C2×C42—472no—
443C2×C42—948no—
444C2×C42—9416no—
445C2×C42—944no—
446C2×C42—948no—
447C2×C42—942no—
448C2×C42—941no—
449C2×C42—948no—
450C2×C42D18814yescomplemented
451C2×C42C22×D4712yes—
452#104—948no—
453#104—948no—
454C22⋊C8—948no—
455C24—472no—
456C24—472no—
457C24—472no—
458C24—948no—
459C24—942no—
460C24—942no—
461C24—941no—
462C24D4×D4711yescharacteristic
463#41—944no—
464#41—944no—
465#41—942no—
466#41—942no—
467#41—942no—
468#41—942no—
469#41—1882no—
470#41—1882no—
471#41—1882no—
472#41—1882no—
473C2×Q8—472no—
474C2×Q8—941no—
475C2×Q8—942no—
476C2×Q8—942no—
477C2×D4—472no—
478C2×D4—948no—
479C2×D4—942no—
480C2×D4—942no—
481C2×D4—941no—
482C2×D4—942no—
483C2×D4—942no—
484C2×D4—1888no—
485C2×D4—1882no—
486C2×D4—1882no—
487C22×C4—28no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—472no—
492C22×C4—472no—
493C22×C4—472no—
494C22×C4—474no—
495C22×C4—474no—
496C22×C4—474no—
497C22×C4—474no—
498C22×C4—9416no—
499C22×C4—9432no—
500C22×C4—9464no—
501C22×C4—948no—
502C22×C4—944no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—944no—
507C22×C4—944no—
508C22×C4—944no—
509C22×C4—944no—
510C22×C4—944no—
511C22×C4—944no—
512C22×C4—944no—
513C22×C4—944no—
514C22×C4—944no—
515C22×C4—944no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4—942no—
519C22×C4—942no—
520C22×C4—941no—
521C22×C4—942no—
522C22×C4—942no—
523C22×C4—1882no—
524C22×C4—1882no—
525C22×C4C2×C4×D4711yescharacteristic
526C22×C4C2×C4×D4711yescharacteristic
527C22×C4C2×D18812yes—
528C22×C4#1062535012yes—
529C22×C4D4×D4711yescharacteristic
530C22×C4D4×D4711yescharacteristic
531C22×C4D4×D4712yes—
532C22×C4D4×D4711yescharacteristic
533C22×C4#1062535412yes—
534C22×C4C23×D4711yescharacteristic
535M16—22no—
536M16—22no—
537M16—42no—
538M16—942no—
539M16—942no—
540M16—1882no—
541C2×C8—22no—
542C2×C8—42no—
543C2×C8—942no—
544C2×C8—1882no—
545C4⋊C4—22no—
546C4⋊C4—948no—
547C4⋊C4C2×C4×D4714yescomplemented
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—948no—
551C22⋊C4—948no—
552C42—22no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—474no—
557C42—9416no—
558C42—944no—
559C42—942no—
560C42C2×D18814yescomplemented
561C23—24no—
562C23—474no—
563C23—478no—
564C23—948no—
565C23—948no—
566C23—9416no—
567C23—9416no—
568C23—944no—
569C23—944no—
570C23—942no—
571C23—942no—
572C23—944no—
573C23—942no—
574C23—1882no—
575C23—1884no—
576C23C22⋊C4×D4711yescharacteristic
577C23C22⋊C4×D4711yescharacteristic
578C23#1309399412yes—
579C23#1309400912yes—
580C23C2×D4×D4711yescharacteristic
581Q8—942no—
582Q8—942no—
583Q8—1882no—
584D4—942no—
585D4—942no—
586D4—18816no—
587D4—1884no—
588D4—1882no—
589D4—1882no—
590D4—1882no—
591C2×C4—28no—
592C2×C4—216no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—44no—
596C2×C4—474no—
597C2×C4—9464no—
598C2×C4—948no—
599C2×C4—9416no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—948no—
606C2×C4—948no—
607C2×C4—944no—
608C2×C4—944no—
609C2×C4—944no—
610C2×C4—944no—
611C2×C4—942no—
612C2×C4—1882no—
613C2×C4—1884no—
614C2×C4—1882no—
615C2×C4—1882no—
616C2×C4C22⋊C4×D4711yescharacteristic
617C2×C4C22⋊C4×D4711yescharacteristic
618C2×C4#1309398612yes—
619C2×C4#1309398612yes—
620C2×C4C4×D18812yes—
621C2×C4C4×D18812yes—
622C2×C4#1309399412yes—
623C2×C4#1309399712yes—
624C2×C4#1309399712yes—
625C2×C4#1309399912yes—
626C2×C4#1309399912yes—
627C2×C4#1309400912yes—
628C2×C4C22×C4×D4711yescharacteristic
629C2×C4C2×D4×D4711yescharacteristic
630C2×C4C2×D4×D4711yescharacteristic
631C2×C4C22×D18811yescharacteristic
632C2×C4C2×#1062535011yescharacteristic
633C2×C4C2×#1062535411yescharacteristic
634C8—42no—
635C8—1882no—
636C22—22no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—478no—
641C22—948no—
642C22—9416no—
643C22—948no—
644C22—944no—
645C22—944no—
646C22—9416no—
647C22—3764no—
648C22—1884no—
649C22—1882no—
650C22#1568900112yes—
651C22#1568900112yes—
652C22#104×D4714yes—
653C22C2×C22⋊C4×D4711yescharacteristic
654C22C2×#1309399411yescharacteristic
655C22C2×#1309400911yescharacteristic
656C4—28no—
657C4—44no—
658C4—948no—
659C4—944no—
660C4—1882no—
661C4#1568900112yes—
662C4#1568900112yes—
663C4C2×#1309398611yescharacteristic
664C4C2×C4×D18811yescharacteristic
665C4C2×#1309399711yescharacteristic
666C4C2×#1309399911yescharacteristic
667C2—21no—
668C2—22no—
669C2—21no—
670C2—948no—
671C2—3762no—
672C2#2024383014yesdirect product factor
673C2C2×#1568900111yescharacteristic
674C2C2×#104×D4712yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#339C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689635C2
cosocleC22×C4×C47C24
hypercenterC2×#339D47
hypocenterC47C2×#1896
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
2. C22×C42×D47C2semidirect
3. C22×#13093952C2semidirect
4. C22×#13093950C2semidirect
5. C2×#15689086C2semidirect
6. #20243830C2direct (N × H)
7. C2×#15688882C2semidirect
8. C2×C47×#339C2semidirect
15. #15689635C4semidirect
18. C2×#13094004C4semidirect
19. C2×#13093999C4semidirect
27. C2×C42×D47C22semidirect
29. C2×#13093952C22semidirect
31. C2×#13093950C22semidirect
33. #15689086C22semidirect
35. #15688882C22semidirect
39. C47×#339C22semidirect
47. C2×#10625350C2×C4semidirect
49. C22×Dic94C2×C4semidirect
51. C22×D188C2×C4semidirect
77. #13094004C2×C4semidirect
79. #13093999C2×C4semidirect
91. C2×C4×Dic47D4semidirect
108. #117×C47C2×C4semidirect
110. C2×C4⋊C4×C47C2×C4semidirect
115. C2×C42×C47D4semidirect
167. C2×D188C22×C4semidirect
196. C2×Dic94C22×C4semidirect
203. C4⋊C4×C47C22×C4semidirect
210. C42×C47C2×D4semidirect
217. C4×Dic47C2×D4semidirect
241. C22×D47#104semidirect
318. C2×D47C2×#104semidirect
337. C2×#339D47semidirect
341. C2×C47#1896semidirect
376. #339C2×D47semidirect
377. C47C2×#1896semidirect
431. #117C4×D47semidirect
434. C2×C4⋊C4C4×D47semidirect
450. C2×C42D188semidirect
547. C4⋊C4C2×C4×D47semidirect
560. C42C2×D188semidirect
672. C2#20243830direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896352
C22×C42×D472
C22×#130939522
C22×#130939502
C2×#156890862
#202438302
C2×#156888822
C2×C47×#3392
C2×#189647