Finite Group Databasebeta

← C4×#15689126 #26214052

Subgroups — C4×#15689126 #26214052

Order 12032.

2018 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D47C211yesmaximal, characteristic
2C2×#15689126C212yesmaximal
3C2×C4×#10625324C211yesmaximal, characteristic, complemented
4C4×#13093982C211yesmaximal, characteristic, complemented
5#20243215C212yesmaximal, complemented
6#20243212C212yesmaximal, complemented
7C4×#13093944C211yesmaximal, characteristic, complemented
8C4×#13093946C211yesmaximal, characteristic, complemented
9C4×#13093945C211yesmaximal, characteristic, complemented
10#20243106C212yesmaximal, complemented
11C4×#102×C47C211yesmaximal, characteristic, complemented
12C22×D4×D47C2211yescharacteristic
13C23×C4×D47—21no—
14C22×#10625324C2212yes—
15C2×C4×D188C2211yescharacteristic
16C2×D4×Dic47C2211yescharacteristic
17C2×#13093986C2211yescharacteristic
18C2×C4×C22⋊D47—21no—
19C2×(C22⋊D47)⋊C4—21no—
20C4×D4×D47—24no—
21C4×D4×D47C418yescomplemented
22C2×#13093982C2212yes—
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C22⋊C4×D47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689126C4116yesdirect product factor
27C2×#13093946C2212yes—
28C2×#13093945C2212yes—
29C2×#13093944C2212yes—
30C4×#10625324—24no—
31C2×#13093934C2212yescomplemented
32C2×C4×C47⋊C8C2211yescharacteristic, complemented
33C4⋊C4×Dic47C2212yescomplemented
34C4×C4⋊Dic47C2211yescharacteristic, complemented
35C4×(C2×D47)⋊C4—21no—
36#15688970—21no—
37#15688968—21no—
38C42×Dic47—21no—
39#15688886C2212yescomplemented
40#15688885C2212yescomplemented
41#15688885C2212yescomplemented
42C2×C4×D4×C47C2211yescharacteristic
43C2×#102×C47C2212yes—
44C2×C4×C8×C47C2211yescharacteristic, complemented
45C4×C4⋊C4×C47C2211yescharacteristic, complemented
46C47×#340C2212yescomplemented
47C24×D47—21no—
48C22×D188C2311yescharacteristic
49C22×C22⋊D47—21no—
50C2×D4×D47—22no—
51C2×D4×D47—24no—
52C2×D4×D47C2×C414yes—
53C2×D4×D47C2×C418yes—
54C23×Dic47—21no—
55C22×C4×D47—21no—
56C22×C4×D47—21no—
57C22×C4×D47—21no—
58C22×C4×D47—28no—
59C22×C4×D47—44no—
60C22×C4×D47—44no—
61C22×C4×D47D411yescharacteristic
62C22×C4×D47D411yescharacteristic
63C22×C4×D47C2311yescharacteristic
64C2×#10625324—24no—
65C2×#10625324—24no—
66C2×#10625324C2×C418yescomplemented
67C22×C47⋊C8C2312yes—
68C4×D188—22no—
69C4×D188C2×C414yes—
70D4×Dic47—22no—
71D4×Dic47C2×C414yes—
72#13093986—22no—
73#13093986C2×C414yes—
74C4×C22⋊D47—28no—
75(C22⋊D47)⋊C4—28no—
76#13093982—24no—
77#13093982C2×C418yescomplemented
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2312yes—
80C2×Dic47⋊C4—21no—
81C4⋊C4×D47—22no—
82C4⋊C4×D47C2×C414yes—
83C2×C22⋊Dic47—21no—
84C2×(C2×D47)⋊C4—22no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C22⋊C4×D47—28no—
88C2×C4×Dic47—22no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—22no—
91C2×C4×Dic47D411yescharacteristic
92C2×C4×Dic47D411yescharacteristic
93C2×C4×Dic47C2311yescharacteristic
94C42×D47C2×C414yes—
95C42×D47D414yes—
96#13093946C2×C418yescomplemented
97#13093945C2×C418yescomplemented
98#13093944C2×C418yescomplemented
99#13093934—24no—
100C4×C47⋊C8—22no—
101C47⋊#95—21no—
102C47⋊#95—22no—
103C22×D4×C47C2311yescharacteristic
104C23×C4×C47—21no—
105C22×C8×C47C2312yes—
106C4×D4×C47—22no—
107C4×D4×C47C2×C414yes—
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2312yes—
110C2×C22⋊C4×C47—21no—
111C2×C42×C47—21no—
112C2×C42×C47C2311yescharacteristic
113#102×C47C2×C418yescomplemented
114C4×C8×C47—22no—
115#95×C47—21no—
116C24×C47—21no—
117C23×D47—21no—
118C23×D47—24no—
119C23×D47—28no—
120C23×D47—42no—
121C23×D47—42no—
122C23×D47—44no—
123C23×D47C2×D411yescharacteristic
124C2×D4×C47—21no—
125C2×D4×C47—22no—
126C2×D4×C47C22×C412yes—
127C2×D4×C47C22×C414yes—
128C22×C4×C47—22no—
129C22×C4×C47—21no—
130C22×C4×C47—24no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2411yescharacteristic
136C2×C22⋊D47—24no—
137C2×C22⋊D47—28no—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—24no—
141C22×Dic47—42no—
142C22×Dic47—42no—
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146D4×D47—28no—
147D4×D47—216no—
148D4×D47C42116yescomplemented
149C2×D188—21no—
150C2×D188—22no—
151C2×D188C22×C414yes—
152C2×D188C22×C412yes—
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—24no—
156C2×C4×D47—24no—
157C2×C4×D47—24no—
158C2×C4×D47—432no—
159C2×C4×D47—42no—
160C2×C4×D47—42no—
161C2×C4×D47—42no—
162C2×C4×D47—42no—
163C2×C4×D47—42no—
164C2×C4×D47—42no—
165C2×C4×D47—42no—
166C2×C4×D47C22⋊C414yes—
167C2×C4×D47C22⋊C414yes—
168C2×C4×D47C22×C412yes—
169C2×C4×D47C22×C414yes—
170C2×C4×D47C2×D412yes—
171C2×C4×D47#4114yes—
172C2×C8×C47—22no—
173C2×C8×C47—22no—
174C2×C8×C47C22×C414yescomplemented
175C4⋊C4×C47—21no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47C22×C412yes—
178C4⋊C4×C47C22×C414yescomplemented
179C22⋊C4×C47—24no—
180C42×C47—42no—
181C42×C47C22×C412yes—
182C42×C47C2×D412yes—
183C22⋊Dic47—24no—
184(C2×D47)⋊C4—28no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—24no—
187C4⋊Dic47—21no—
188C4⋊Dic47—22no—
189C4⋊Dic47C22×C412yes—
190C4⋊Dic47C22×C414yescomplemented
191Dic47⋊C4—24no—
192C4×Dic47—28no—
193C4×Dic47—44no—
194C4×Dic47—42no—
195C4×Dic47C22⋊C414yes—
196C4×Dic47C22⋊C414yes—
197C4×Dic47C22×C412yes—
198C4×Dic47C2×D412yes—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—22no—
201C2×C47⋊C8C22×C414yescomplemented
202#10625324—216no—
203C23×C47—22no—
204C23×C47—24no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—42no—
208C23×C47C22×D411yescharacteristic
209C22×D47—21no—
210C22×D47—21no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—24no—
214C22×D47—216no—
215C22×D47—48no—
216C22×D47—48no—
217C22×D47—416no—
218C22×D47—416no—
219C22×D47—41no—
220C22×D47—41no—
221C22×D47—42no—
222C22×D47C2×C22⋊C412yes—
223C22×D47C4×D414yes—
224D4×C47—22no—
225D4×C47—24no—
226D4×C47C2×C4214yes—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—24no—
230C2×C4×C47—42no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—42no—
234C2×C4×C47—48no—
235C2×C4×C47C2×C22⋊C412yes—
236C2×C4×C47C2×C22⋊C412yes—
237C2×C4×C47AGL(1,ℤ/8)12yes—
238C2×C4×C47AGL(1,ℤ/8)12yes—
239C2×C4×C47C23×C411yescharacteristic
240C2×C4×C47C23×C412yes—
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C2×#4112yes—
243C22⋊D47—216no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—24no—
249C2×Dic47—48no—
250C2×Dic47—42no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—42no—
254C2×Dic47C2×C22⋊C412yes—
255C2×Dic47C2×C22⋊C412yes—
256C2×Dic47C2×C22⋊C412yes—
257C2×Dic47C4×D414yes—
258D188—22no—
259D188—24no—
260D188C2×C4214yes—
261C4×D47—28no—
262C4×D47—44no—
263C4×D47—48no—
264C4×D47—48no—
265C4×D47—48no—
266C4×D47C2×C4214yes—
267C4×D47#11714yes—
268C4×D47C4×D418yes—
269C8×C47—24no—
270C47⋊C8—24no—
271C4×#418—471nomaximal, Sylow
272C22×C47—24no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47C22×C22⋊C411yescharacteristic
278C22×C47C2×C4×D412yes—
279C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C47#57512yes—
282C2×D47—22no—
283C2×D47—22no—
284C2×D47—24no—
285C2×D47—24no—
286C2×D47—24no—
287C2×D47—432no—
288C2×D47—42no—
289C2×D47—42no—
290C2×D47—44no—
291C2×D47—44no—
292C2×D47C4×C22⋊C414yes—
293C4×C47—22no—
294C4×C47—42no—
295C4×C47—44no—
296C4×C47#41814yescomplemented
297C4×C47C22×C4211yescharacteristic
298C4×C47C2×#11711yescharacteristic
299C4×C47C2×C4×D412yes—
300Dic47—22no—
301Dic47—22no—
302Dic47—42no—
303Dic47—44no—
304Dic47C4×C22⋊C414yes—
305C22×C4×D4—471no—
306C2×#418—472no—
307C2×C4×M16—471no—
308C4×#117—471no—
309#1762—472no—
310#1762—472no—
311C4×#102—471no—
312C4×#102—471no—
313C4×#102—471no—
314C4×#102D4711yescharacteristic, complemented
315#1736—472no—
316C2×C47—44no—
317C2×C47C2×C4×C22⋊C411yescharacteristic
318C2×C47C2×#41811yescharacteristic
319C2×C47#289611yescharacteristic
320C2×C47C4×AGL(1,ℤ/8)12yes—
321C2×C47C4×AGL(1,ℤ/8)12yes—
322D47—24no—
323D47—44no—
324C23×D4—471no—
325C24×C4—941no—
326C22×M16—472no—
327C2×C4×D4—471no—
328C2×C4×D4—478no—
329C2×C4×D4—471no—
330C2×C4×D4—471no—
331C2×C4×D4—941no—
332C2×C4×D4—944no—
333C2×C4×D4—941no—
334C2×C4×D4C2×D4711yescharacteristic
335C2×#117—472no—
336C22×C4⋊C4—471no—
337C22×C22⋊C4—941no—
338C22×C42—471no—
339#418—4716no—
340C2×#102—472no—
341C2×#102—472no—
342C2×#102—472no—
343C2×#102C2×D4712yes—
344C4×M16—944no—
345C2×#97—472no—
346C2×C4×C8—471no—
347C2×C4×C8C2×D4711yescharacteristic, complemented
348C4×C4⋊C4—472no—
349C4×C4⋊C4—471no—
350C4×C4⋊C4C2×D4711yescharacteristic, complemented
351C4×C22⋊C4—941no—
352C4×C22⋊C4—941no—
353#376—941no—
354C43—941no—
355#340—472no—
356#340—472no—
357#340—472no—
358#340C2×D4712yescomplemented
359C47C4×#41811yescharacteristic, complemented, Sylow
360C25—941no—
361C22×D4—474no—
362C22×D4—478no—
363C22×D4—471no—
364C22×D4—942no—
365C22×D4—944no—
366C22×D4—941no—
367C22×D4C22×D4711yescharacteristic
368C23×C4—21no—
369C23×C4—471no—
370C23×C4—471no—
371C23×C4—471no—
372C23×C4—941no—
373C23×C4—948no—
374C23×C4—941no—
375C23×C4—941no—
376C23×C4—941no—
377C23×C4—1884no—
378C23×C4—1884no—
379C2×M16—478no—
380C2×M16—944no—
381C2×M16—944no—
382C22×C8—472no—
383C22×C8C22×D4712yes—
384C4×D4—22no—
385C4×D4—474no—
386C4×D4—474no—
387C4×D4—474no—
388C4×D4—948no—
389C4×D4—942no—
390C4×D4—948no—
391C4×D4—942no—
392C4×D4—942no—
393C4×D4C4×D4714yes—
394#117—478no—
395#117—944no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—472no—
398C2×C4⋊C4—474no—
399C2×C4⋊C4—942no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C4⋊C4C22×D4712yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—948no—
405C2×C22⋊C4—941no—
406C2×C22⋊C4—942no—
407C2×C22⋊C4—941no—
408C2×C22⋊C4—941no—
409C2×C42—21no—
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—471no—
413C2×C42—474no—
414C2×C42—474no—
415C2×C42—942no—
416C2×C42—942no—
417C2×C42—941no—
418C2×C42C22×D4711yescharacteristic
419#102—478no—
420#102—478no—
421#102—478no—
422#102C4×D4718yescomplemented
423#97—944no—
424C4×C8—22no—
425C4×C8—942no—
426#95—21no—
427#95—942no—
428#95—941no—
429C24—21no—
430C24—471no—
431C24—944no—
432C24—948no—
433C24—941no—
434C24—1882no—
435C24—1882no—
436C24—1884no—
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—4716no—
440C2×D4—474no—
441C2×D4—472no—
442C2×D4—948no—
443C2×D4—9416no—
444C2×D4—944no—
445C2×D4—948no—
446C2×D4—941no—
447C2×D4—942no—
448C2×D4C2×C4×D4714yes—
449C2×D4C2×C4×D4712yes—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—471no—
456C22×C4—472no—
457C22×C4—474no—
458C22×C4—472no—
459C22×C4—474no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—474no—
463C22×C4—474no—
464C22×C4—944no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—941no—
471C22×C4—944no—
472C22×C4—1882no—
473C22×C4—1882no—
474C22×C4—1882no—
475C22×C4—18832no—
476C22×C4—1882no—
477C22×C4—1882no—
478C22×C4—1882no—
479C22×C4—1882no—
480C22×C4—1882no—
481C22×C4—1882no—
482C22×C4D4×D4711yescharacteristic
483C22×C4D4×D4711yescharacteristic
484C22×C4C23×D4711yescharacteristic
485M16—9416no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—474no—
489C2×C8—942no—
490C2×C8—942no—
491C2×C8C2×C4×D4714yescomplemented
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—472no—
495C4⋊C4—474no—
496C4⋊C4—944no—
497C4⋊C4—941no—
498C4⋊C4—942no—
499C4⋊C4C2×C4×D4712yes—
500C4⋊C4C2×C4×D4714yescomplemented
501C22⋊C4—24no—
502C22⋊C4—944no—
503C22⋊C4—948no—
504C22⋊C4—944no—
505C22⋊C4—944no—
506C42—42no—
507C42—474no—
508C42—474no—
509C42—472no—
510C42—472no—
511C42—948no—
512C42—1884no—
513C42—1882no—
514C42C2×C4×D4712yes—
515C42D4×D4712yes—
516C23—22no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—472no—
522C23—474no—
523C23—9416no—
524C23—941no—
525C23—941no—
526C23—942no—
527C23—942no—
528C23—944no—
529C23—1888no—
530C23—18816no—
531C23—18816no—
532C23—1888no—
533C23—1881no—
534C23—1881no—
535C23—1882no—
536C23C2×D4×D4711yescharacteristic
537D4—22no—
538D4—24no—
539D4—474no—
540D4—9416no—
541D4—942no—
542D4—944no—
543D4C42×D4714yes—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—48no—
552C2×C4—472no—
553C2×C4—474no—
554C2×C4—474no—
555C2×C4—474no—
556C2×C4—478no—
557C2×C4—472no—
558C2×C4—472no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—942no—
562C2×C4—948no—
563C2×C4—942no—
564C2×C4—944no—
565C2×C4—1884no—
566C2×C4—1888no—
567C2×C4—1888no—
568C2×C4—1888no—
569C2×C4—1881no—
570C2×C4—1882no—
571C2×C4—1881no—
572C2×C4—1882no—
573C2×C4—1888no—
574C2×C4C22⋊C4×D4712yes—
575C2×C4C22⋊C4×D4712yes—
576C2×C4#1309404812yes—
577C2×C4#1309405112yes—
578C2×C4C22×C4×D4711yescharacteristic
579C2×C4C22×C4×D4712yes—
580C2×C4C2×D4×D4711yescharacteristic
581C2×C4#41×D4712yes—
582C8—24no—
583C8—944no—
584C22—24no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—474no—
590C22—944no—
591C22—944no—
592C22—942no—
593C22—942no—
594C22—944no—
595C22—18832no—
596C22—1884no—
597C22—1884no—
598C22—1882no—
599C22—1882no—
600C22C2×C22⋊C4×D4711yescharacteristic
601C22C4×D4×D4712yes—
602C22C2×#1309404811yescharacteristic
603C22C2×#1309405111yescharacteristic
604C22#1568992012yes—
605C4—22no—
606C4—44no—
607C4—42no—
608C4—474no—
609C4—942no—
610C4—942no—
611C4—1884no—
612C4—1882no—
613C4#1568912614yesdirect product factor
614C4C2×C42×D4711yescharacteristic
615C4#117×D4711yescharacteristic
616C4C4×D4×D4712yes—
617C2—44no—
618C2—944no—
619C2—1884no—
620C2C4×C22⋊C4×D4711yescharacteristic
621C2C2×#1568912611yescharacteristic
622C2#2025190311yescharacteristic
623C2C4×#1309404812yes—
624C2C4×#1309405112yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC4×#102×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×D47C22
cosocleC22×C4×C47C24
hypercenterC4×#102D47
hypocenterC47C4×#418
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. C2×C4×#10625324C2semidirect
4. C4×#13093982C2semidirect
5. #20243215C2semidirect
6. #20243212C2semidirect
7. C4×#13093944C2semidirect
8. C4×#13093946C2semidirect
9. C4×#13093945C2semidirect
10. #20243106C2semidirect
11. C4×#102×C47C2semidirect
21. C4×D4×D47C4semidirect
26. #15689126C4direct (N × H)
31. C2×#13093934C22semidirect
32. C2×C4×C47⋊C8C22semidirect
33. C4⋊C4×Dic47C22semidirect
34. C4×C4⋊Dic47C22semidirect
39. #15688886C22semidirect
40. #15688885C22semidirect
41. #15688885C22semidirect
44. C2×C4×C8×C47C22semidirect
45. C4×C4⋊C4×C47C22semidirect
46. C47×#340C22semidirect
66. C2×#10625324C2×C4semidirect
77. #13093982C2×C4semidirect
96. #13093946C2×C4semidirect
97. #13093945C2×C4semidirect
98. #13093944C2×C4semidirect
113. #102×C47C2×C4semidirect
148. D4×D47C42semidirect
174. C2×C8×C47C22×C4semidirect
178. C4⋊C4×C47C22×C4semidirect
190. C4⋊Dic47C22×C4semidirect
201. C2×C47⋊C8C22×C4semidirect
296. C4×C47#418semidirect
314. C4×#102D47semidirect
347. C2×C4×C8C2×D47semidirect
350. C4×C4⋊C4C2×D47semidirect
358. #340C2×D47semidirect
359. C47C4×#418semidirect
422. #102C4×D47semidirect
491. C2×C8C2×C4×D47semidirect
500. C4⋊C4C2×C4×D47semidirect
613. C4#15689126direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D4×D472
C2×#156891262
C2×C4×#106253242
C4×#130939822
#202432152
#202432122
C4×#130939442
C4×#130939462
C4×#130939452
#202431062
C4×#102×C472
C4×#41847