Finite Group Databasebeta

← C4×#15689003 #26137425

Subgroups — C4×#15689003 #26137425

Order 12032.

3366 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
1C23×C4×Dic47C211yesmaximal, characteristic, complemented
2C2×#15689003C212yesmaximal
3C2×C4×C22⋊Dic47C212yesmaximal, complemented
4C2×C4×C22⋊Dic47C211yesmaximal, characteristic, complemented
5#20241625C212yesmaximal, complemented
6#20241620C212yesmaximal, complemented
7#20241610C212yesmaximal, complemented
8C4×C47⋊#95C212yesmaximal, complemented
9C22×C4×D4×C47C211yesmaximal, characteristic
10C24×Dic47C2212yes—
11C22×C22⋊Dic47C2212yes—
12C22×C22⋊Dic47C2214yes—
13C22×C4×Dic47—28no—
14C22×C4×Dic47—24no—
15C22×C4×Dic47C2212yes—
16C22×C4×Dic47C2212yes—
17C22×C4×Dic47C2212yes—
18#15689003C4116yesdirect product factor
19C4×C22⋊Dic47—28no—
20C4×C22⋊Dic47—24no—
21C2×C47⋊#95C2214yes—
22C2×C47⋊#95C2214yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2214yes—
25C23×D4×C47C2211yescharacteristic
26C24×C4×C47C2211yescharacteristic
27C24×C4×C47C2211yescharacteristic
28C2×C4×D4×C47—24no—
29C2×C4×D4×C47—24no—
30C2×C4×D4×C47C418yescomplemented
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×C22⋊C4×C47C2211yescharacteristic
33C22×C22⋊C4×C47C2211yescharacteristic
34C22×C42×C47C2211yescharacteristic
35C23×Dic47—24no—
36C23×Dic47—24no—
37C23×Dic47—28no—
38C23×Dic47—28no—
39C23×Dic47C2×C418yescomplemented
40C23×Dic47C2314yes—
41C23×Dic47C2314yes—
42C2×C22⋊Dic47—24no—
43C2×C22⋊Dic47—24no—
44C2×C22⋊Dic47—28no—
45C2×C22⋊Dic47—28no—
46C2×C22⋊Dic47C2×C418yescomplemented
47C2×C22⋊Dic47C2×C4116yescomplemented
48C2×C4×Dic47—232no—
49C2×C4×Dic47—24no—
50C2×C4×Dic47—24no—
51C2×C4×Dic47—24no—
52C2×C4×Dic47—28no—
53C2×C4×Dic47—24no—
54C2×C4×Dic47—24no—
55C2×C4×Dic47—24no—
56C2×C4×Dic47D418yescomplemented
57C47⋊#95—24no—
58C47⋊#95—28no—
59C47⋊#95—28no—
60C47⋊#95C2×C4116yescomplemented
61C25×C47C2311yescharacteristic
62C25×C47C2311yescharacteristic
63C22×D4×C47—22no—
64C22×D4×C47—22no—
65C22×D4×C47—24no—
66C22×D4×C47C2×C418yes—
67C22×D4×C47C2×C414yes—
68C23×C4×C47—24no—
69C23×C4×C47—24no—
70C23×C4×C47—24no—
71C23×C4×C47—22no—
72C23×C4×C47—22no—
73C23×C4×C47—24no—
74C23×C4×C47C2×C414yes—
75C23×C4×C47C2×C414yescomplemented
76C23×C4×C47D414yes—
77C23×C4×C47D414yes—
78C23×C4×C47C2311yescharacteristic
79C23×C4×C47C2311yescharacteristic
80C23×C4×C47C2311yescharacteristic
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C4×D4×C47—232no—
84C2×C4⋊C4×C47—22no—
85C2×C4⋊C4×C47—22no—
86C2×C4⋊C4×C47C2×C414yescomplemented
87C2×C22⋊C4×C47—22no—
88C2×C22⋊C4×C47—22no—
89C2×C22⋊C4×C47—22no—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47C2×C414yes—
92C2×C22⋊C4×C47C2×C414yescomplemented
93C2×C42×C47—22no—
94C2×C42×C47—22no—
95C2×C42×C47C2×C414yescomplemented
96C24×C47—22no—
97C24×C47—22no—
98C24×C47—24no—
99C24×C47—21no—
100C24×C47—21no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—24no—
105C24×C47C22×C414yes—
106C24×C47C22×C412yes—
107C24×C47C22×C412yes—
108C24×C47C22×C414yes—
109C24×C47C2×D412yes—
110C24×C47C2×D412yes—
111C24×C47#4114yes—
112C24×C47C2411yescharacteristic
113C2×D4×C47—28no—
114C2×D4×C47—28no—
115C2×D4×C47—216no—
116C2×D4×C47—216no—
117C2×D4×C47C42116yescomplemented
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—216no—
124C22×C4×C47—28no—
125C22×C4×C47—216no—
126C22×C4×C47—28no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—24no—
134C22×C4×C47—28no—
135C22×C4×C47—216no—
136C22×C4×C47—44no—
137C22×C4×C47—48no—
138C22×C4×C47—44no—
139C22×C4×C47C22⋊C4116yescomplemented
140C22×C4×C47C22×C412yes—
141C22×C4×C47C22×C412yes—
142C22×C4×C47C22×C414yes—
143C22×C4×C47C22×C412yes—
144C22×C4×C47C22×C412yes—
145C22×C4×C47C22×C412yes—
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D414yes—
151C22×C4×C47#4114yes—
152C22×Dic47—216no—
153C22×Dic47—232no—
154C22×Dic47—216no—
155C22×Dic47—216no—
156C22×Dic47—24no—
157C22×Dic47—24no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—232no—
161C22×Dic47—24no—
162C22×Dic47—24no—
163C22×Dic47—24no—
164C22×Dic47—24no—
165C22×Dic47—24no—
166C22×Dic47—24no—
167C22×Dic47C22×C418yes—
168C22×Dic47C22×C418yes—
169C22×Dic47C2×D418yes—
170C22×Dic47#4118yes—
171C4⋊C4×C47—28no—
172C22⋊C4×C47—28no—
173C22⋊C4×C47—28no—
174C42×C47—28no—
175C22⋊Dic47—216no—
176C22⋊Dic47—232no—
177C4×Dic47—216no—
178C4×Dic47—216no—
179C4×Dic47—48no—
180C23×C47—28no—
181C23×C47—24no—
182C23×C47—28no—
183C23×C47—24no—
184C23×C47—22no—
185C23×C47—24no—
186C23×C47—22no—
187C23×C47—24no—
188C23×C47—28no—
189C23×C47—28no—
190C23×C47—24no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—28no—
194C23×C47—24no—
195C23×C47—24no—
196C23×C47—24no—
197C23×C47—28no—
198C23×C47—24no—
199C23×C47—41no—
200C23×C47—42no—
201C23×C47—41no—
202C23×C47—42no—
203C23×C47—44no—
204C23×C47—44no—
205C23×C47—42no—
206C23×C47C2×C4214yes—
207C23×C47C2×C4214yescomplemented
208C23×C47C2×C22⋊C414yes—
209C23×C47#11714yes—
210C23×C47C4×D418yes—
211C23×C47C4×D418yes—
212C23×C47C23×C412yes—
213C23×C47C23×C411yescharacteristic
214C23×C47C22×D412yes—
215C23×C47C22×D411yescharacteristic
216C23×C47C22×D411yescharacteristic
217C23×C47C2×#4114yes—
218C23×C47C2×#4112yes—
219C23×C47C2×#4112yes—
220D4×C47—232no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—232no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—41no—
235C2×C4×C47—42no—
236C2×C4×C47—41no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—416no—
240C2×C4×C47C2×C4214yescomplemented
241C2×C4×C47C2×C22⋊C414yes—
242C2×C4×C47C2×C22⋊C414yes—
243C2×C4×C47C4×D418yes—
244C2×C4×C47#12014yes—
245C2×C4×C47#12118yes—
246C2×C4×C47#12314yes—
247C2×Dic47—264no—
248C2×Dic47—28no—
249C2×Dic47—28no—
250C2×Dic47—28no—
251C2×Dic47—28no—
252C2×Dic47—28no—
253C2×Dic47—28no—
254C2×Dic47—28no—
255C2×Dic47—28no—
256C2×Dic47—28no—
257C2×Dic47—44no—
258C2×Dic47—44no—
259C2×Dic47—44no—
260C2×Dic47—44no—
261C2×Dic47C4×D4116yescomplemented
262C4×#386—471nomaximal, Sylow
263C22×C47—28no—
264C22×C47—24no—
265C22×C47—28no—
266C22×C47—24no—
267C22×C47—28no—
268C22×C47—216no—
269C22×C47—28no—
270C22×C47—24no—
271C22×C47—28no—
272C22×C47—42no—
273C22×C47—44no—
274C22×C47—42no—
275C22×C47—44no—
276C22×C47—48no—
277C22×C47—44no—
278C22×C47C4×C22⋊C418yescomplemented
279C22×C47C22×C4211yescharacteristic
280C22×C47C22×C22⋊C411yescharacteristic
281C22×C47C2×#11711yescharacteristic
282C22×C47C2×C4×D414yes—
283C22×C47C2×C4×D412yes—
284C22×C47C2×C4×D412yes—
285C22×C47C2×C4×D412yes—
286C22×C47C4×#4114yes—
287C22×C47C4×#4112yes—
288C22×C47C2×#12011yescharacteristic
289C22×C47C2×#12112yes—
290C22×C47C2×#12311yescharacteristic
291C22×C47#52512yes—
292C22×C47#52514yes—
293C22×C47#52914yes—
294C22×C47#53212yes—
295C4×C47—24no—
296C4×C47—24no—
297C4×C47—42no—
298C4×C47—42no—
299C4×C47#38618yescomplemented
300Dic47—216no—
301Dic47—48no—
302C22×C4×D4D4711yescharacteristic
303C23×C42—471no—
304C2×#386—472no—
305C2×C4×C22⋊C4—471no—
306C2×C4×C22⋊C4—472no—
307#1437—472no—
308#1436—472no—
309C22⋊#95—472no—
310C4×#95—472no—
311C2×C47—28no—
312C2×C47—44no—
313C2×C47C2×C4×C22⋊C411yescharacteristic
314C2×C47C42×D412yes—
315C2×C47C2×#38611yescharacteristic
316C2×C47#229012yes—
317C2×C47#229711yescharacteristic
318C2×C47C4×#12012yes—
319C2×C47C4×#12114yes—
320C2×C47C4×#12312yes—
321C23×D4C2×D4711yescharacteristic
322C24×C4—472no—
323C24×C4C2×D4711yescharacteristic
324C24×C4C2×D4711yescharacteristic
325C2×C4×D4—24no—
326C2×C4×D4—24no—
327C2×C4×D4Dic4718yescomplemented
328C22×C4⋊C4C2×D4711yescharacteristic
329C22×C22⋊C4—472no—
330C22×C22⋊C4—474no—
331C22×C22⋊C4C2×D4711yescharacteristic
332C22×C22⋊C4C2×D4711yescharacteristic
333C22×C42—472no—
334C22×C42—472no—
335C22×C42—472no—
336C22×C42—948no—
337C22×C42—944no—
338C22×C42C2×D4711yescharacteristic
339#386—4716no—
340C4×C22⋊C4—948no—
341C4×C22⋊C4—944no—
342C2×#95—474no—
343C2×#95—474no—
344C2×#95—474no—
345C2×#95—472no—
346C47C4×#38611yescharacteristic, complemented, Sylow
347C25C22×D4711yescharacteristic
348C25C22×D4711yescharacteristic
349C22×D4—24no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4C4×D4718yes—
353C22×D4C2×Dic4714yes—
354C23×C4—24no—
355C23×C4—24no—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—22no—
359C23×C4—22no—
360C23×C4—474no—
361C23×C4—478no—
362C23×C4—474no—
363C23×C4—948no—
364C23×C4—948no—
365C23×C4—944no—
366C23×C4—944no—
367C23×C4C2×Dic4714yes—
368C23×C4C2×Dic4714yescomplemented
369C23×C4C22⋊D4714yes—
370C23×C4C22⋊D4714yes—
371C23×C4C22×D4711yescharacteristic
372C23×C4C22×D4711yescharacteristic
373C23×C4C22×D4711yescharacteristic
374C23×C4C22×D4711yescharacteristic
375C23×C4C22×D4711yescharacteristic
376C4×D4—232no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4C2×Dic4714yescomplemented
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—4716no—
385C2×C22⋊C4—478no—
386C2×C22⋊C4—948no—
387C2×C22⋊C4—948no—
388C2×C22⋊C4—944no—
389C2×C22⋊C4—944no—
390C2×C22⋊C4C2×Dic4714yes—
391C2×C22⋊C4C2×Dic4714yescomplemented
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—478no—
395C2×C42—948no—
396C2×C42—9432no—
397C2×C42—944no—
398C2×C42—944no—
399C2×C42—944no—
400C2×C42—944no—
401C2×C42—944no—
402C2×C42—944no—
403C2×C42C2×Dic4714yescomplemented
404#95—4716no—
405#95—948no—
406#95—948no—
407#95—944no—
408C24—22no—
409C24—24no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—21no—
414C24—24no—
415C24—22no—
416C24—22no—
417C24C2×C4×D4714yes—
418C24C2×C4×D4714yes—
419C24#1062535014yes—
420C24C22×Dic4712yes—
421C24C22×Dic4712yes—
422C24C2×C22⋊D4712yes—
423C24C2×C22⋊D4712yes—
424C24C23×D4711yescharacteristic
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—216no—
428C2×D4—216no—
429C2×D4C4×Dic47116yescomplemented
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—216no—
436C22×C4—28no—
437C22×C4—216no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—216no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—478no—
452C22×C4—478no—
453C22×C4—478no—
454C22×C4—478no—
455C22×C4—9432no—
456C22×C4—9416no—
457C22×C4—9416no—
458C22×C4—9432no—
459C22×C4—9416no—
460C22×C4—944no—
461C22×C4—944no—
462C22×C4—944no—
463C22×C4—944no—
464C22×C4—944no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—944no—
468C22×C4—944no—
469C22×C4—944no—
470C22×C4C22⋊Dic47116yescomplemented
471C22×C4C2×C4×D4714yes—
472C22×C4D4×D4714yes—
473C22×C4#1062535214yes—
474C22×C4C22×Dic4712yes—
475C22×C4C22×Dic4712yes—
476C22×C4C22×Dic4712yes—
477C22×C4C22×Dic4712yes—
478C22×C4C22×Dic4712yes—
479C22×C4C2×C22⋊D4712yes—
480C22×C4C2×C22⋊D4712yes—
481C22×C4C2×C22⋊D4712yes—
482C22×C4C2×C22⋊D4712yes—
483C4⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—9432no—
487C22⋊C4—9416no—
488C42—28no—
489C42—9416no—
490C42—9416no—
491C42—1888no—
492C23—28no—
493C23—24no—
494C23—28no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—24no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23C2×C4×Dic4714yes—
519C23C2×C4×Dic4714yescomplemented
520C23C2×C22⋊Dic4714yes—
521C23#1309398314yes—
522C23C4×C22⋊D4718yes—
523C23C4×C22⋊D4718yes—
524C23C22×C4×D4712yes—
525C23C23×Dic4711yescharacteristic
526C23C2×D4×D4712yes—
527C23C22×C22⋊D4711yescharacteristic
528C23C22×C22⋊D4711yescharacteristic
529C23#41×D4714yes—
530C23C2×#1062535212yes—
531C23C2×#1062535012yes—
532D4—232no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—232no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—416no—
552C2×C4—4716no—
553C2×C4—9464no—
554C2×C4—948no—
555C2×C4—948no—
556C2×C4—948no—
557C2×C4—948no—
558C2×C4—948no—
559C2×C4—948no—
560C2×C4—948no—
561C2×C4—948no—
562C2×C4—948no—
563C2×C4—1884no—
564C2×C4—1884no—
565C2×C4—1884no—
566C2×C4—1884no—
567C2×C4C2×C4×Dic4714yescomplemented
568C2×C4C2×C22⋊Dic4714yes—
569C2×C4C2×C22⋊Dic4714yes—
570C2×C4D4×Dic4718yes—
571C2×C4#1309399214yes—
572C2×C4#1309399618yes—
573C2×C4#1309400514yes—
574C22—24no—
575C22—28no—
576C22—24no—
577C22—28no—
578C22—216no—
579C22—28no—
580C22—28no—
581C22—24no—
582C22—28no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—48no—
588C22—44no—
589C22C4×C22⋊Dic4718yescomplemented
590C22C22×C4×Dic4711yescharacteristic
591C22C22×C22⋊Dic4711yescharacteristic
592C22C2×#1309398311yescharacteristic
593C22C4×D4×D4714yes—
594C22C2×C4×C22⋊D4712yes—
595C22C2×C4×C22⋊D4712yes—
596C22C2×D4×Dic4712yes—
597C22C4×#1062535214yes—
598C22#41×Dic4712yes—
599C22C2×#1309399211yescharacteristic
600C22C2×#1309399612yes—
601C22C2×#1309400511yescharacteristic
602C22#1568963012yes—
603C22#1568963214yes—
604C22#1568965314yes—
605C22#1568966912yes—
606C4—24no—
607C4—24no—
608C4—42no—
609C4—42no—
610C4—9416no—
611C4—1888no—
612C4#1568900318yesdirect product factor
613C2—28no—
614C2—44no—
615C2C2×C4×C22⋊Dic4711yescharacteristic
616C2C4×D4×Dic4712yes—
617C2C2×#1568900311yescharacteristic
618C2#2024583312yes—
619C2#2024587911yescharacteristic
620C2C4×#1309399212yes—
621C2C4×#1309399614yes—
622C2C4×#1309400512yes—
subgroupG/H
centerC23×C4C22×D47
commutatorC22×C47C22×C42
FrattiniC24C23×D47
FittingC22×C4×D4×C47C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC22×C4×D4D47
hypocenterC47C4×#386
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C23×C4×Dic47C2semidirect
3. C2×C4×C22⋊Dic47C2semidirect
4. C2×C4×C22⋊Dic47C2semidirect
5. #20241625C2semidirect
6. #20241620C2semidirect
7. #20241610C2semidirect
8. C4×C47⋊#95C2semidirect
18. #15689003C4direct (N × H)
30. C2×C4×D4×C47C4semidirect
39. C23×Dic47C2×C4semidirect
46. C2×C22⋊Dic47C2×C4semidirect
47. C2×C22⋊Dic47C2×C4semidirect
56. C2×C4×Dic47D4semidirect
60. C47⋊#95C2×C4semidirect
75. C23×C4×C47C2×C4semidirect
86. C2×C4⋊C4×C47C2×C4semidirect
92. C2×C22⋊C4×C47C2×C4semidirect
95. C2×C42×C47C2×C4semidirect
117. C2×D4×C47C42semidirect
139. C22×C4×C47C22⋊C4semidirect
207. C23×C47C2×C42semidirect
240. C2×C4×C47C2×C42semidirect
261. C2×Dic47C4×D4semidirect
278. C22×C47C4×C22⋊C4semidirect
299. C4×C47#386semidirect
327. C2×C4×D4Dic47semidirect
346. C47C4×#386semidirect
368. C23×C4C2×Dic47semidirect
379. C2×C4⋊C4C2×Dic47semidirect
391. C2×C22⋊C4C2×Dic47semidirect
403. C2×C42C2×Dic47semidirect
429. C2×D4C4×Dic47semidirect
470. C22×C4C22⋊Dic47semidirect
519. C23C2×C4×Dic47semidirect
567. C2×C4C2×C4×Dic47semidirect
589. C22C4×C22⋊Dic47semidirect
612. C4#15689003direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×Dic472
C2×#156890032
C2×C4×C22⋊Dic472
#202416252
#202416202
#202416102
C4×C47⋊#952
C22×C4×D4×C472
C4×#38647