Finite Group Databasebeta

← C4×#15689464 #26223446

Subgroups — C4×#15689464 #26223446

Order 12032.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689464C211yesmaximal, characteristic
2C4×#13094003C211yesmaximal, characteristic
3#20244029C211yesmaximal, characteristic, complemented
4#20244022C211yesmaximal, characteristic, complemented
5#20243966C211yesmaximal, characteristic, complemented
6#20243959C211yesmaximal, characteristic, complemented
7#20243700C211yesmaximal, characteristic
8#20243587C211yesmaximal, characteristic
9#20243483C211yesmaximal, characteristic
10C4×#13093957C211yesmaximal, characteristic, complemented
11C4×#13093955C211yesmaximal, characteristic, complemented
12C4×#13093947C211yesmaximal, characteristic, complemented
13C4×#13093947C211yesmaximal, characteristic, complemented
14C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic
15C4×#117×C47C211yesmaximal, characteristic
16C2×#13094003C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18#15689464C418yesdirect product factor
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093947C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093947C2211yescharacteristic
26C2×#13093947C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31#15689029C2211yescharacteristic
32#15689021C2211yescharacteristic
33#15689007—21no—
34#15688996—21no—
35#15688983C2211yescharacteristic
36C4×Dic47⋊C4—21no—
37C4×C4⋊Dic47C2211yescharacteristic
38C4×C22⋊Dic47—21no—
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688878C2211yescharacteristic
48#15688878C2211yescharacteristic
49C2×#117×C47C2211yescharacteristic
50C2×#117×C47C2211yescharacteristic
51C22×C42×C47C2211yescharacteristic
52C4×C4⋊C4×C47C2211yescharacteristic
53C4×C4⋊C4×C47C2211yescharacteristic
54C4×C4⋊C4×C47C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C4×C22⋊C4×C47—21no—
57C4×C22⋊C4×C47—21no—
58C47×#376—21no—
59C43×C47—21no—
60C22×Dic94C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094003—22no—
66#13094003C2×C414yes—
67C4×Dic94—22no—
68C4×Dic94—22no—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C4×Dic47—21no—
78C2×C4×Dic47—21no—
79#13093957—22no—
80#13093957C2×C414yescomplemented
81#13093955—22no—
82#13093955C2×C414yescomplemented
83#13093947—22no—
84#13093947—22no—
85#13093947C2×C414yescomplemented
86#13093947C2×C414yescomplemented
87C22⋊(C47⋊C8)—22no—
88C22⋊(C47⋊C8)C2×C414yes—
89C4×C47⋊C8—22no—
90C4×C47⋊C8—22no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C23×C4×C47D411yescharacteristic
97C23×C4×C47D411yescharacteristic
98C23×C4×C47C2311yescharacteristic
99#117×C47—22no—
100#117×C47—22no—
101#117×C47—22no—
102#117×C47C2×C414yes—
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C22⋊C4×C47—21no—
108C2×C22⋊C4×C47—21no—
109C2×C42×C47—21no—
110C2×C42×C47—21no—
111C2×C42×C47—22no—
112C2×C42×C47—22no—
113C2×C42×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47—21no—
116C2×C42×C47—21no—
117C2×C42×C47D411yescharacteristic
118C2×C42×C47D411yescharacteristic
119C2×C42×C47C2311yescharacteristic
120C2×C42×C47C2311yescharacteristic
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125C24×C47C2×D411yescharacteristic
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—41no—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47#4112yes—
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94C22×C412yes—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47C22×C412yes—
163C4⋊C4×C47C22×C412yes—
164C4⋊C4×C47#4112yes—
165C4⋊C4×C47#4112yes—
166C22⋊C4×C47—24no—
167C22⋊C4×C47—24no—
168C42×C47—24no—
169C42×C47—24no—
170C42×C47—22no—
171C42×C47—22no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—42no—
175C42×C47—42no—
176C42×C47—42no—
177C42×C47—42no—
178C42×C47#4112yes—
179C42×C47#4112yes—
180C22⋊Dic47—24no—
181C22⋊Dic47—42no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47C22×C412yes—
188Dic47⋊C4—24no—
189Dic47⋊C4—42no—
190Dic47⋊C4—42no—
191C4×Dic47—42no—
192C4×Dic47—42no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8C22×C412yes—
204C2×C47⋊C8C22×C412yes—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47C4×D412yes—
212C23×C47C22×D411yescharacteristic
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—21no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—44no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—44no—
240C2×C4×C47C4×D412yes—
241C2×C4×C47#12312yes—
242C2×C4×C47#12312yes—
243C2×C4×C47#13512yes—
244C2×C4×C47#13512yes—
245C2×C4×C47#13712yes—
246C2×C4×C47C23×C411yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264Dic94—22no—
265Dic94—22no—
266Dic94—41no—
267Dic94—41no—
268C47⋊C8—22no—
269C47⋊C8—22no—
270C47⋊C8—41no—
271C47⋊C8—41no—
272C4×#485—471nomaximal, Sylow
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47C2×C4×D411yescharacteristic
281C22×C47C2×#12311yescharacteristic
282C22×C47#52511yescharacteristic
283C22×C47C2×#13511yescharacteristic
284C22×C47C2×#13511yescharacteristic
285C22×C47C2×#13711yescharacteristic
286C22×C47#57511yescharacteristic
287C4×C47—22no—
288C4×C47—22no—
289C4×C47—22no—
290C4×C47—41no—
291C4×C47—42no—
292C4×C47—42no—
293C4×C47—42no—
294C4×C47—42no—
295C4×C47#48514yescomplemented
296C4×C47C4×#4111yescharacteristic
297C4×C47C4×#4111yescharacteristic
298C4×C47#52911yescharacteristic
299C4×C47#52911yescharacteristic
300Dic47—42no—
301Dic47—42no—
302Dic47—81no—
303Dic47—81no—
304C2×#485—471no—
305C4×C22⋊Q8—471no—
306C4×#117D4711yescharacteristic
307#1928—471no—
308#1927—471no—
309#1919—471no—
310#1918—471no—
311#1873—471no—
312#1854—471no—
313#1825—471no—
314C4×#107—471no—
315C4×#106—471no—
316C4×#103—471no—
317C4×#103—471no—
318C4×C22⋊C8—471no—
319C2×C47—42no—
320C2×C47C4×#12311yescharacteristic
321C2×C47C4×#13511yescharacteristic
322C2×C47C4×#13711yescharacteristic
323C2×C47C2×#48511yescharacteristic
324C2×C47#309111yescharacteristic
325C2×C47#310211yescharacteristic
326C2×C47#310711yescharacteristic
327C2×C22⋊Q8—471no—
328C2×C4×Q8—471no—
329C2×#117C2×D4711yescharacteristic
330C2×#117C2×D4711yescharacteristic
331C22×C42C2×D4711yescharacteristic
332#485—478no—
333C2×#107—471no—
334C2×#107—471no—
335C2×#106—471no—
336C2×#106—471no—
337C2×#103—471no—
338C2×#103—471no—
339C2×#103—471no—
340C2×#103—471no—
341C2×C22⋊C8—471no—
342C2×C22⋊C8—471no—
343C2×C4×C8—471no—
344C2×C4×C8—471no—
345#391—471no—
346#389—471no—
347#387—941no—
348#385—941no—
349#380—471no—
350C4×C4⋊C4—471no—
351C4×C4⋊C4—941no—
352C4×C4⋊C4C2×D4711yescharacteristic
353C4×C4⋊C4C2×D4711yescharacteristic
354C4×C4⋊C4C2×D4711yescharacteristic
355C4×C4⋊C4C2×D4711yescharacteristic
356C4×C22⋊C4—21no—
357C4×C22⋊C4—21no—
358C4×C22⋊C4—941no—
359#376—21no—
360C43—21no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#336—471no—
370#336—471no—
371C47C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8—471no—
373C23×C4C22⋊D4711yescharacteristic
374C23×C4C22⋊D4711yescharacteristic
375C23×C4C22×D4711yescharacteristic
376C22×C8—471no—
377C22×C8—471no—
378C22×C8—471no—
379C22×C8—471no—
380C22⋊Q8—474no—
381C22⋊Q8—942no—
382C4×Q8—942no—
383C4×Q8—942no—
384#117—22no—
385#117—22no—
386#117—22no—
387#117C4×D4714yes—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—941no—
411C2×C42—941no—
412C2×C42C22⋊D4711yescharacteristic
413C2×C42C22⋊D4711yescharacteristic
414C2×C42C22×D4711yescharacteristic
415C2×C42C22×D4711yescharacteristic
416#107—474no—
417#107—942no—
418#106—474no—
419#106—942no—
420#103—474no—
421#103—474no—
422#103—942no—
423#103—942no—
424C22⋊C8—474no—
425C22⋊C8—942no—
426C4×C8—942no—
427C4×C8—942no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436#95—941no—
437C24C2×C22⋊D4711yescharacteristic
438C2×Q8—472no—
439C2×Q8—941no—
440C2×Q8—941no—
441C2×Q8—941no—
442C2×Q8—941no—
443C2×Q8—941no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4C2×C4×D4712yes—
463C22×C4#1062535012yes—
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C2×C22⋊D4711yescharacteristic
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C2×C22⋊D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470C2×C8—472no—
471C2×C8—472no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—472no—
487C4⋊C4—944no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—1882no—
494C4⋊C4—1882no—
495C4⋊C4C2×C4×D4712yes—
496C4⋊C4C2×C4×D4712yes—
497C4⋊C4#1062535012yes—
498C4⋊C4#1062535012yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—944no—
502C22⋊C4—1882no—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—1882no—
514C42—1882no—
515C42#1062535012yes—
516C42#1062535012yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C4×C22⋊D4712yes—
524C23C22×C22⋊D4711yescharacteristic
525Q8—942no—
526Q8—942no—
527Q8—1881no—
528Q8—1881no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4C4×C22⋊D4712yes—
569C2×C4#1309400712yes—
570C2×C4#1309400712yes—
571C2×C4#1309404412yes—
572C2×C4#1309404412yes—
573C2×C4#1309405412yes—
574C2×C4C22×C4×D4711yescharacteristic
575C2×C4C2×#1062535011yescharacteristic
576C2×C4C2×#1062535011yescharacteristic
577C2×C4C2×#1062535011yescharacteristic
578C2×C4C2×#1062535011yescharacteristic
579C2×C4C2×#1062535011yescharacteristic
580C8—942no—
581C8—942no—
582C8—1881no—
583C8—1881no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×C22⋊D4711yescharacteristic
592C22C2×#1309400711yescharacteristic
593C22#1568963411yescharacteristic
594C22C2×#1309404411yescharacteristic
595C22C2×#1309404411yescharacteristic
596C22C2×#1309405411yescharacteristic
597C22#1568992111yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—42no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—3761no—
607C4—3761no—
608C4—1882no—
609C4—1882no—
610C4#1568946414yesdirect product factor
611C4C4×#1062535011yescharacteristic
612C4C4×#1062535011yescharacteristic
613C4#1568965711yescharacteristic
614C4#1568965711yescharacteristic
615C2—42no—
616C2C4×#1309400711yescharacteristic
617C2C4×#1309404411yescharacteristic
618C2C4×#1309405411yescharacteristic
619C2C2×#1568946411yescharacteristic
620C2#2025330111yescharacteristic
621C2#2025340611yescharacteristic
622C2#2025345911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#117×C47C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenterC4×#117D47
hypocenterC47C4×#485
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
3. #20244029C2semidirect
4. #20244022C2semidirect
5. #20243966C2semidirect
6. #20243959C2semidirect
10. C4×#13093957C2semidirect
11. C4×#13093955C2semidirect
12. C4×#13093947C2semidirect
13. C4×#13093947C2semidirect
18. #15689464C4direct (N × H)
80. #13093957C2×C4semidirect
82. #13093955C2×C4semidirect
85. #13093947C2×C4semidirect
86. #13093947C2×C4semidirect
295. C4×C47#485semidirect
371. C47C4×#485semidirect
610. C4#15689464direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894642
C4×#130940032
#202440292
#202440222
#202439662
#202439592
#202437002
#202435872
#202434832
C4×#130939572
C4×#130939552
C4×#130939472
C4×C22⋊(C47⋊C8)2
C4×#117×C472
C4×#48547