Finite Group Databasebeta

← C4×#15689259 #26221851

Subgroups — C4×#15689259 #26221851

Order 12032.

1964 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D188C211yesmaximal, characteristic, complemented
2C2×#15689259C211yesmaximal, characteristic
3C2×C4×D376C212yesmaximal, complemented
4C4×#13093998C211yesmaximal, characteristic, complemented
5#20243735C212yesmaximal, complemented
6#20243703C211yesmaximal, characteristic, complemented
7#20243638C212yesmaximal, complemented
8#20243578C211yesmaximal, characteristic, complemented
9#20243320C211yesmaximal, characteristic, complemented
10C4×#13093946C212yesmaximal, complemented
11C4×C22⋊C8×C47C211yesmaximal, characteristic, complemented
12C23×D188C2211yescharacteristic
13C23×C4×D47—21no—
14C22×D376C2212yes—
15C2×#13093998C2211yescharacteristic
16C2×C4×D188—28no—
17C2×C4×D188—21no—
18C2×C4×D188—21no—
19C2×C4×D188C2212yes—
20C2×C4×D188C2212yes—
21C2×C4×D188C2211yescharacteristic, complemented
22C2×C4×C22⋊D47—21no—
23C22×C4⋊Dic47C2211yescharacteristic
24C22×(C2×D47)⋊C4—21no—
25#15689259C418yesdirect product factor
26C4×D376—28no—
27C2×#13093958C2212yescomplemented
28C2×#13093946C2212yes—
29C2×#13093946C2212yes—
30C2×#13093946C2212yes—
31C2×#13093946C2212yescomplemented
32#15689025C2211yescharacteristic, complemented
33#15689021C2211yescharacteristic, complemented
34#15689014—21no—
35#15689002—21no—
36#15688983C2211yescharacteristic
37C4×C4⋊Dic47C2211yescharacteristic, complemented
38C4×(C2×D47)⋊C4—21no—
39#15688886C2212yescomplemented
40#15688886C2212yescomplemented
41C22×C42×C47C2211yescharacteristic
42C2×C22⋊C8×C47C2211yescharacteristic
43C2×C22⋊C8×C47C2211yescharacteristic
44C2×C4×C8×C47C2212yescomplemented
45C47×#336C2212yescomplemented
46C24×D47—21no—
47C22×D188—21no—
48C22×D188—22no—
49C22×D188—24no—
50C22×D188—24no—
51C22×D188C2×C414yescomplemented
52C22×D188C2311yescharacteristic
53C22×D188C2312yes—
54C22×C22⋊D47—21no—
55C23×Dic47—21no—
56C22×C4×D47—28no—
57C22×C4×D47—22no—
58C22×C4×D47—22no—
59C22×C4×D47—21no—
60C22×C4×D47—44no—
61C22×C4×D47—44no—
62C2×D376—24no—
63C2×D376—24no—
64C2×D376C2×C418yescomplemented
65#13093998—22no—
66#13093998C2×C414yescomplemented
67C4×D188—28no—
68C4×D188—28no—
69C4×D188—24no—
70C4×D188—24no—
71C4×D188—24no—
72C4×D188—24no—
73C4×D188D418yescomplemented
74C4×C22⋊D47—44no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47—24no—
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2312yes—
81C2×Dic47⋊C4—21no—
82C2×C22⋊Dic47—21no—
83C2×(C2×D47)⋊C4—28no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—22no—
86C2×(C2×D47)⋊C4—22no—
87C2×(C2×D47)⋊C4—21no—
88C2×(C2×D47)⋊C4—21no—
89C2×(C2×D47)⋊C4—21no—
90C2×C4×Dic47—21no—
91#13093958—24no—
92#13093946—24no—
93#13093946—24no—
94#13093946—24no—
95#13093946C2×C418yescomplemented
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C23×C4×C47D411yescharacteristic
100C23×C4×C47D411yescharacteristic
101C23×C4×C47C2311yescharacteristic
102C22×C8×C47C2312yes—
103C22×C8×C47C2312yes—
104C2×C42×C47—24no—
105C2×C42×C47D411yescharacteristic
106C2×C42×C47D411yescharacteristic
107C2×C42×C47C2311yescharacteristic
108C2×C42×C47C2311yescharacteristic
109C22⋊C8×C47—22no—
110C22⋊C8×C47C2×C414yescomplemented
111C4×C8×C47—24no—
112C24×C47C2×D411yescharacteristic
113C23×D47—24no—
114C23×D47—24no—
115C23×D47—24no—
116C23×D47—22no—
117C23×D47—21no—
118C23×D47—42no—
119C23×D47—42no—
120C23×D47—42no—
121C23×D47—42no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—41no—
129C22×C4×C47D814yescomplemented
130C22×C4×C47C22×C412yes—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47#4112yes—
137C22×C4×C47C2411yescharacteristic
138C2×C22⋊D47—24no—
139C2×C22⋊D47—42no—
140C2×C22⋊D47—42no—
141C22×Dic47—24no—
142C22×Dic47—21no—
143C22×Dic47—22no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—42no—
147C22×Dic47—42no—
148C2×D188—24no—
149C2×D188—24no—
150C2×D188—216no—
151C2×D188—24no—
152C2×D188—22no—
153C2×D188—22no—
154C2×D188—22no—
155C2×D188—216no—
156C2×D188—22no—
157C2×D188—22no—
158C2×D188—21no—
159C2×D188C22×C414yes—
160C2×D188C22×C412yescomplemented
161C2×D188C2×D414yes—
162C2×D188#4114yes—
163C2×C4×D47—28no—
164C2×C4×D47—28no—
165C2×C4×D47—44no—
166C2×C4×D47—432no—
167C2×C4×D47—44no—
168C2×C4×D47—44no—
169C2×C4×D47—44no—
170C2×C4×D47—42no—
171C2×C4×D47—42no—
172C2×C4×D47—42no—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C2×C8×C47—22no—
177C2×C8×C47—22no—
178C2×C8×C47C22×C414yescomplemented
179C42×C47—22no—
180C42×C47—22no—
181C42×C47—22no—
182C42×C47C2×D414yes—
183C22⋊Dic47—42no—
184(C2×D47)⋊C4—28no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—28no—
187(C2×D47)⋊C4—42no—
188(C2×D47)⋊C4—42no—
189(C2×D47)⋊C4—42no—
190C4⋊Dic47—24no—
191C4⋊Dic47—22no—
192C4⋊Dic47—21no—
193C4⋊Dic47—22no—
194C4⋊Dic47—22no—
195C4⋊Dic47C22×C412yescomplemented
196C4⋊Dic47C2×D414yes—
197Dic47⋊C4—42no—
198C4×Dic47—42no—
199D376—216no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47C4×D412yes—
205C23×C47C2×D812yes—
206C23×C47#13512yes—
207C23×C47C22×D411yescharacteristic
208C22×D47—216no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47—24no—
212C22×D47—22no—
213C22×D47—48no—
214C22×D47—48no—
215C22×D47—48no—
216C22×D47—48no—
217C22×D47—48no—
218C22×D47—42no—
219C22×D47—42no—
220C22×D47—42no—
221C22×D47—42no—
222C22×D47—48no—
223C22×D47—41no—
224C22×D47—41no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—41no—
228C22×D47—41no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—28no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—44no—
242C2×C4×C47C4×D412yes—
243C2×C4×C47#12012yes—
244C2×C4×C47#12012yes—
245C2×C4×C47C2×D812yes—
246C2×C4×C47C2×D812yes—
247C2×C4×C47AGL(1,ℤ/8)12yes—
248C2×C4×C47C23×C411yescharacteristic
249C2×C4×C47C22×D412yes—
250C2×C4×C47C2×#4112yes—
251C2×C4×C47C2×#4111yescharacteristic
252C22⋊D47—48no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—48no—
256C2×Dic47—41no—
257C2×Dic47—42no—
258C2×Dic47—41no—
259C2×Dic47—42no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266D188—216no—
267D188—24no—
268D188—24no—
269D188—24no—
270D188—24no—
271D188—42no—
272D188C4×D418yescomplemented
273C4×D47—416no—
274C4×D47—416no—
275C4×D47—84no—
276C8×C47—24no—
277C8×C47—42no—
278C4×#447—471nomaximal, Sylow
279C22×C47—22no—
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47C4×D814yescomplemented
286C22×C47C2×C4×D411yescharacteristic
287C22×C47C2×#12011yescharacteristic
288C22×C47#52511yescharacteristic
289C22×C47C22×D811yescharacteristic
290C22×C47C2×#13511yescharacteristic
291C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C47#57511yescharacteristic
293C2×D47—28no—
294C2×D47—432no—
295C2×D47—44no—
296C2×D47—44no—
297C2×D47—44no—
298C2×D47—44no—
299C2×D47—44no—
300C2×D47—44no—
301C2×D47—42no—
302C2×D47—42no—
303C2×D47—42no—
304C2×D47—81no—
305C2×D47—81no—
306C2×D47—81no—
307C2×D47—81no—
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—41no—
312C4×C47#44714yescomplemented
313C4×C47C2×C4×D412yes—
314C4×C47#52512yes—
315Dic47—42no—
316Dic47—44no—
317Dic47—81no—
318Dic47—81no—
319C22×C4×D4—471no—
320C2×#447—471no—
321C2×C4×D8—472no—
322C4×#121—471no—
323#1879—472no—
324#1874—471no—
325#1862—472no—
326#1852—471no—
327#1786—471no—
328C4×#102—472no—
329C4×C22⋊C8D4711yescharacteristic, complemented
330C2×C47—24no—
331C2×C47C4×#12011yescharacteristic
332C2×C47C2×C4×D811yescharacteristic
333C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
334C2×C47C2×#44711yescharacteristic
335C2×C47#300211yescharacteristic
336C2×C47#310011yescharacteristic
337C2×C47#310511yescharacteristic
338D47—48no—
339D47—82no—
340C23×D4—471no—
341C24×C4—941no—
342C22×D8—472no—
343C2×#121—471no—
344C2×C4×D4—472no—
345C2×C4×D4—472no—
346C2×C4×D4—471no—
347C2×C4×D4—948no—
348C2×C4×D4—941no—
349C2×C4×D4—941no—
350C2×C4×D4—941no—
351C22×C4⋊C4—471no—
352C22×C22⋊C4—941no—
353C22×C42C2×D4711yescharacteristic
354#447—478no—
355C4×D8—948no—
356C2×#107—472no—
357C2×#102—472no—
358C2×#102—472no—
359C2×#102—472no—
360C2×#102—472no—
361C2×C22⋊C8C2×D4711yescharacteristic
362C2×C22⋊C8C2×D4711yescharacteristic
363C2×C4×C8C2×D4712yescomplemented
364#390—471no—
365#389—471no—
366#388—941no—
367#386—941no—
368#380—471no—
369C4×C4⋊C4—471no—
370C4×C22⋊C4—941no—
371#340—472no—
372#340—472no—
373#336C2×D4712yescomplemented
374C47C4×#44711yescharacteristic, complemented, Sylow
375C25—941no—
376C22×D4—474no—
377C22×D4—472no—
378C22×D4—471no—
379C22×D4—944no—
380C22×D4—944no—
381C22×D4—941no—
382C22×D4—941no—
383C22×D4—942no—
384C23×C4—948no—
385C23×C4—941no—
386C23×C4—942no—
387C23×C4—942no—
388C23×C4—941no—
389C23×C4—1884no—
390C23×C4—1884no—
391C23×C4D18811yescharacteristic
392C23×C4D18811yescharacteristic
393C23×C4C22×D4711yescharacteristic
394C2×D8—478no—
395C2×D8—944no—
396C2×D8—944no—
397C22×C8C22×D4712yes—
398C22×C8C22×D4712yes—
399#121—474no—
400#121—942no—
401C4×D4—478no—
402C4×D4—944no—
403C4×D4—944no—
404C4×D4—948no—
405C4×D4—948no—
406C4×D4—944no—
407C4×D4—944no—
408C4×D4—1884no—
409C2×C4⋊C4—472no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—944no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C22⋊C4—948no—
417C2×C22⋊C4—942no—
418C2×C22⋊C4—942no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C42—24no—
425C2×C42—941no—
426C2×C42D18811yescharacteristic
427C2×C42D18811yescharacteristic
428C2×C42C22×D4711yescharacteristic
429C2×C42C22×D4711yescharacteristic
430#107—944no—
431#102—478no—
432#102—944no—
433#102—944no—
434#102—944no—
435C22⋊C8—22no—
436C22⋊C8C4×D4714yescomplemented
437C4×C8—24no—
438#95—941no—
439#95—941no—
440#95—941no—
441C24—944no—
442C24—944no—
443C24—944no—
444C24—942no—
445C24—941no—
446C24—1882no—
447C24—1882no—
448C24—1882no—
449C24—1882no—
450C24C2×D18811yescharacteristic
451C2×D4—474no—
452C2×D4—472no—
453C2×D4—474no—
454C2×D4—474no—
455C2×D4—9416no—
456C2×D4—944no—
457C2×D4—944no—
458C2×D4—9416no—
459C2×D4—944no—
460C2×D4—944no—
461C2×D4—942no—
462C2×D4—942no—
463C2×D4—942no—
464C2×D4—942no—
465C2×D4—942no—
466C2×D4—941no—
467C2×D4—1882no—
468C2×D4—1882no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—948no—
477C22×C4—944no—
478C22×C4—941no—
479C22×C4—942no—
480C22×C4—948no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—1884no—
484C22×C4—18832no—
485C22×C4—1882no—
486C22×C4—1882no—
487C22×C4—1882no—
488C22×C4—1882no—
489C22×C4—1882no—
490C22×C4—1884no—
491C22×C4—1884no—
492C22×C4—1884no—
493C22×C4D37614yescomplemented
494C22×C4C2×C4×D4712yes—
495C22×C4C2×D18811yescharacteristic
496C22×C4C2×D18811yescharacteristic
497C22×C4C2×D18811yescharacteristic
498C22×C4C2×D18811yescharacteristic
499C22×C4C2×D18811yescharacteristic
500C22×C4#1062535012yes—
501C22×C4C23×D4711yescharacteristic
502D8—9416no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8C2×C4×D4714yescomplemented
509C4⋊C4—472no—
510C4⋊C4—474no—
511C4⋊C4—944no—
512C4⋊C4—942no—
513C4⋊C4—941no—
514C4⋊C4—942no—
515C4⋊C4—942no—
516C4⋊C4—1882no—
517C22⋊C4—948no—
518C22⋊C4—944no—
519C22⋊C4—948no—
520C22⋊C4—1882no—
521C22⋊C4—1882no—
522C22⋊C4—1882no—
523C22⋊C4—1882no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—1882no—
528C42D4×D4714yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—9416no—
534C23—944no—
535C23—944no—
536C23—944no—
537C23—942no—
538C23—1888no—
539C23—1888no—
540C23—1888no—
541C23—1888no—
542C23—1888no—
543C23—1881no—
544C23—1881no—
545C23—1881no—
546C23—1881no—
547C23—1881no—
548C23—1881no—
549C23—1882no—
550C23—1882no—
551C23—1882no—
552C23—1882no—
553C23—1888no—
554C23C4×D18812yes—
555C23C2×D37612yes—
556C23#1309404612yes—
557C23C22×D18811yescharacteristic
558D4—478no—
559D4—9416no—
560D4—944no—
561D4—944no—
562D4—944no—
563D4—944no—
564D4—1888no—
565D4—1882no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—942no—
580C2×C4—944no—
581C2×C4—3764no—
582C2×C4—18816no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1888no—
590C2×C4—1881no—
591C2×C4—1882no—
592C2×C4—1881no—
593C2×C4—1882no—
594C2×C4—18816no—
595C2×C4C4×D18812yes—
596C2×C4#1309399412yes—
597C2×C4#1309399412yes—
598C2×C4C2×D37612yes—
599C2×C4C2×D37612yes—
600C2×C4#1309405212yes—
601C2×C4C22×C4×D4711yescharacteristic
602C2×C4C2×D4×D4712yes—
603C2×C4#41×D4712yes—
604C2×C4C2×#1062535011yescharacteristic
605C8—24no—
606C8—42no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—948no—
614C22—3761no—
615C22—3761no—
616C22—3761no—
617C22—3761no—
618C22—18832no—
619C22—1882no—
620C22—1882no—
621C22—1882no—
622C22—1884no—
623C22—1884no—
624C22—1884no—
625C22—1884no—
626C22—1884no—
627C22—1884no—
628C22C4×D37614yescomplemented
629C22C2×C4×D18811yescharacteristic
630C22C2×#1309399411yescharacteristic
631C22#1568963511yescharacteristic
632C22C22×D37611yescharacteristic
633C22C2×#1309404611yescharacteristic
634C22C2×#1309405211yescharacteristic
635C22#1568992311yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—3761no—
641C4—3761no—
642C4—1884no—
643C4—1882no—
644C4#1568925914yesdirect product factor
645C4C4×D4×D4712yes—
646C4#1568963212yes—
647C2—24no—
648C2—3762no—
649C2—1888no—
650C2C4×#1309399411yescharacteristic
651C2C2×C4×D37611yescharacteristic
652C2C4×#1309405211yescharacteristic
653C2C2×#1568925911yescharacteristic
654C2#2025261011yescharacteristic
655C2#2025338911yescharacteristic
656C2#2025343711yescharacteristic
subgroupG/H
centerC22×C4C2×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×C22⋊C8×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689259C2
cosocleC22×C4×C47C24
hypercenterC4×C22⋊C8D47
hypocenterC47C4×#447
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C4×D188C2semidirect
3. C2×C4×D376C2semidirect
4. C4×#13093998C2semidirect
5. #20243735C2semidirect
6. #20243703C2semidirect
7. #20243638C2semidirect
8. #20243578C2semidirect
9. #20243320C2semidirect
10. C4×#13093946C2semidirect
11. C4×C22⋊C8×C47C2semidirect
21. C2×C4×D188C22semidirect
25. #15689259C4direct (N × H)
27. C2×#13093958C22semidirect
31. C2×#13093946C22semidirect
32. #15689025C22semidirect
33. #15689021C22semidirect
37. C4×C4⋊Dic47C22semidirect
39. #15688886C22semidirect
40. #15688886C22semidirect
44. C2×C4×C8×C47C22semidirect
45. C47×#336C22semidirect
51. C22×D188C2×C4semidirect
64. C2×D376C2×C4semidirect
66. #13093998C2×C4semidirect
73. C4×D188D4semidirect
95. #13093946C2×C4semidirect
110. C22⋊C8×C47C2×C4semidirect
129. C22×C4×C47D8semidirect
160. C2×D188C22×C4semidirect
178. C2×C8×C47C22×C4semidirect
195. C4⋊Dic47C22×C4semidirect
272. D188C4×D4semidirect
285. C22×C47C4×D8semidirect
312. C4×C47#447semidirect
329. C4×C22⋊C8D47semidirect
363. C2×C4×C8C2×D47semidirect
373. #336C2×D47semidirect
374. C47C4×#447semidirect
436. C22⋊C8C4×D47semidirect
493. C22×C4D376semidirect
508. C2×C8C2×C4×D47semidirect
628. C22C4×D376semidirect
644. C4#15689259direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D1882
C2×#156892592
C2×C4×D3762
C4×#130939982
#202437352
#202437032
#202436382
#202435782
#202433202
C4×#130939462
C4×C22⋊C8×C472
C4×#44747