Finite Group Databasebeta

← C4×#15689421 #26223880

Subgroups — C4×#15689421 #26223880

Order 12032.

948 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689421C211yesmaximal, characteristic
2C4×#13094015C211yesmaximal, characteristic, complemented
3C4×D4×Dic47C211yesmaximal, characteristic
4#20244344C211yesmaximal, characteristic, complemented
5#20244159C211yesmaximal, characteristic, complemented
6#20243979C211yesmaximal, characteristic, complemented
7#20243969C211yesmaximal, characteristic, complemented
8#20243647C211yesmaximal, characteristic
9#20243637C211yesmaximal, characteristic
10#20243396C211yesmaximal, characteristic
11C4×#13093958C211yesmaximal, characteristic, complemented
12C4×#13093955C211yesmaximal, characteristic, complemented
13C4×Dic47⋊C8C211yesmaximal, characteristic, complemented
14C4×#13093945C211yesmaximal, characteristic
15C4×#102×C47C211yesmaximal, characteristic
16C2×#13094015C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C22×C4×Dic47—21no—
20#15689421C418yesdirect product factor
21C2×#13093958C2211yescharacteristic
22C2×#13093958C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×Dic47⋊C8C2211yescharacteristic
26C2×Dic47⋊C8C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093945C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30#15689022C2211yescharacteristic
31#15689019C2211yescharacteristic
32#15689010—21no—
33#15689006—21no—
34#15688988C2211yescharacteristic
35C4⋊C4×Dic47C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4×Dic47⋊C4—21no—
38C4×C4⋊Dic47—21no—
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41C22⋊C4×Dic47—21no—
42C22⋊C4×Dic47—21no—
43C4×C22⋊Dic47—21no—
44C42×Dic47C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×C4×C8×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C23×Dic47—21no—
61C23×Dic47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094015—22no—
65#13094015C2×C414yescomplemented
66D4×Dic47—22no—
67D4×Dic47—22no—
68D4×Dic47—22no—
69D4×Dic47C2×C414yes—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—44no—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47D411yescharacteristic
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47C2311yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91#13093958—22no—
92#13093958C2×C414yescomplemented
93#13093955—22no—
94#13093955C2×C414yescomplemented
95Dic47⋊C8—22no—
96Dic47⋊C8C2×C414yescomplemented
97#13093945—22no—
98#13093945C2×C414yes—
99C4×C47⋊C8—22no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×D4×C47C2311yescharacteristic
109C23×C4×C47—21no—
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C4×D4×C47—22no—
113C4×D4×C47Q814yes—
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C42×C47—21no—
119C2×C42×C47C2311yescharacteristic
120#102×C47—22no—
121#102×C47C2×C414yes—
122C4×C8×C47—22no—
123#95×C47—21no—
124C24×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47C22×C412yes—
129C2×D4×C47C2×Q812yes—
130C2×D4×C47#4112yes—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—24no—
134C22×C4×C47—21no—
135C22×C4×C47—42no—
136C22×C4×C47—42no—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—24no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—24no—
145C22×Dic47—21no—
146C22×Dic47—42no—
147C22×Dic47—42no—
148C22×Dic47—42no—
149C22×Dic47—42no—
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47C22×C412yes—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47C22×C412yes—
165C4⋊C4×C47C2×Q812yes—
166C22⋊C4×C47—24no—
167C42×C47—42no—
168C42×C47C2×Q812yes—
169C42×C47#4112yes—
170C22⋊Dic47—24no—
171C22⋊Dic47—24no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—24no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—42no—
179C4⋊Dic47C22×C412yes—
180C4⋊Dic47C22×C412yes—
181C4⋊Dic47#4112yes—
182Dic47⋊C4—24no—
183Dic47⋊C4—42no—
184C4×Dic47—22no—
185C4×Dic47—22no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188C4×Dic47—22no—
189C4×Dic47—41no—
190C4×Dic47—42no—
191C4×Dic47—42no—
192C4×Dic47—42no—
193C4×Dic47—42no—
194C4×Dic47C22×C412yes—
195C4×Dic47#4112yes—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8C22×C412yes—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47C22×D411yescharacteristic
210D4×C47—22no—
211D4×C47—22no—
212D4×C47—22no—
213D4×C47C4×Q814yes—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—48no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47C22⋊Q812yes—
227C2×C4×C47C22⋊Q812yes—
228C2×C4×C47#13512yes—
229C2×C4×C47AGL(1,ℤ/8)12yes—
230C2×C4×C47C23×C411yescharacteristic
231C2×C4×C47C22×Q811yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—21no—
245C2×Dic47—22no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—48no—
249C2×Dic47—48no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47C4×D412yes—
260C2×Dic47C4×D412yes—
261C2×Dic47#13512yes—
262C8×C47—22no—
263C8×C47—41no—
264C47⋊C8—22no—
265C47⋊C8—41no—
266C4×#478—471nomaximal, Sylow
267C22×C47—24no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—42no—
274C22×C47C2×C4×D411yescharacteristic
275C22×C47C2×C22⋊Q811yescharacteristic
276C22×C47#52511yescharacteristic
277C22×C47C2×#13511yescharacteristic
278C22×C47C2×#13511yescharacteristic
279C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C47#57511yescharacteristic
281C4×C47—22no—
282C4×C47—42no—
283C4×C47—42no—
284C4×C47—81no—
285C4×C47#47814yescomplemented
286C4×C47C2×C4×Q811yescharacteristic
287C4×C47C4×#4111yescharacteristic
288C4×C47#52911yescharacteristic
289C4×C47#53311yescharacteristic
290Dic47—22no—
291Dic47—22no—
292Dic47—22no—
293Dic47—41no—
294Dic47—42no—
295Dic47—42no—
296Dic47—42no—
297Dic47—81no—
298C2×#478—471no—
299C4×#125—471no—
300C42×D4—471no—
301#1989—471no—
302#1949—471no—
303#1921—471no—
304#1919—471no—
305#1864—471no—
306#1862—471no—
307#1804—471no—
308C4×#107—471no—
309C4×#106—471no—
310C4×C4⋊C8—471no—
311C4×#102—471no—
312C4×#102D4711yescharacteristic
313C2×C47—44no—
314C2×C47C4×C22⋊Q811yescharacteristic
315C2×C47C4×#13511yescharacteristic
316C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C47C2×#47811yescharacteristic
318C2×C47#307611yescharacteristic
319C2×C47#310211yescharacteristic
320C2×C47#310511yescharacteristic
321C2×#125—471no—
322C2×C4×D4—471no—
323C2×C4×D4—471no—
324C2×C4×D4C2×D4711yescharacteristic
325C22×C42—941no—
326#478—478no—
327C2×#107—471no—
328C2×#107—471no—
329C2×#106—471no—
330C2×#106—471no—
331C2×C4⋊C8—471no—
332C2×C4⋊C8—471no—
333C2×#102—471no—
334C2×#102—471no—
335C2×#102C2×D4711yescharacteristic
336C2×#102C2×D4711yescharacteristic
337C2×C4×C8—471no—
338C2×C4×C8C2×D4711yescharacteristic
339#389—471no—
340#389—471no—
341#387—941no—
342#387—941no—
343#382—471no—
344C4×C4⋊C4—471no—
345C4×C4⋊C4—471no—
346C4×C4⋊C4—471no—
347C4×C4⋊C4—471no—
348C4×C4⋊C4—941no—
349C4×C4⋊C4—941no—
350C4×C4⋊C4C2×D4711yescharacteristic
351C4×C22⋊C4—941no—
352C4×C22⋊C4—941no—
353C4×C22⋊C4—941no—
354C43—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340C2×D4711yescharacteristic
362#340C2×D4711yescharacteristic
363#336—471no—
364#336—471no—
365C47C4×#47811yescharacteristic, complemented, Sylow
366C22×D4C22×D4711yescharacteristic
367C23×C4—21no—
368C23×C4—941no—
369C23×C4—941no—
370C22×C8—471no—
371C22×C8—471no—
372C22×C8C22×D4711yescharacteristic
373C22×C8C22×D4711yescharacteristic
374#125—474no—
375#125—942no—
376C4×D4—22no—
377C4×D4—474no—
378C4×D4—942no—
379C4×D4—942no—
380C4×D4—942no—
381C4×D4Dic9414yes—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C4⋊C4C22×D4711yescharacteristic
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—941no—
395C2×C22⋊C4—941no—
396C2×C42—21no—
397C2×C42—471no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—471no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—941no—
404C2×C42—941no—
405C2×C42—941no—
406C2×C42—941no—
407C2×C42—1884no—
408C2×C42C22×D4711yescharacteristic
409#107—474no—
410#107—942no—
411#106—474no—
412#106—942no—
413C4⋊C8—474no—
414C4⋊C8—942no—
415#102—22no—
416#102—474no—
417#102—942no—
418#102C4×D4714yes—
419C4×C8—22no—
420C4×C8—942no—
421#95—21no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427#95—941no—
428#95—941no—
429#95—941no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×Dic9412yes—
435C2×D4C2×C4×D4712yes—
436C2×D4#1062535012yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—471no—
444C22×C4—471no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4—941no—
450C22×C4—941no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—1882no—
454C22×C4—1882no—
455C22×C4—1882no—
456C22×C4—1882no—
457C22×C4D4×D4711yescharacteristic
458C22×C4D4×D4711yescharacteristic
459C22×C4C23×D4711yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—472no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8C2×C4×D4712yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—472no—
477C4⋊C4—472no—
478C4⋊C4—472no—
479C4⋊C4—944no—
480C4⋊C4—944no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C4⋊C4—941no—
486C4⋊C4—1882no—
487C4⋊C4—1882no—
488C4⋊C4C2×Dic9412yes—
489C4⋊C4C2×C4×D4712yes—
490C22⋊C4—24no—
491C22⋊C4—944no—
492C22⋊C4—944no—
493C42—42no—
494C42—472no—
495C42—472no—
496C42—942no—
497C42—942no—
498C42—942no—
499C42—941no—
500C42—941no—
501C42—1881no—
502C42—1882no—
503C42—1882no—
504C42—1882no—
505C42—1882no—
506C42C2×Dic9412yes—
507C42#1062535212yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D4711yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic9414yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—48no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—472no—
533C2×C4—472no—
534C2×C4—472no—
535C2×C4—941no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—941no—
540C2×C4—941no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—941no—
544C2×C4—942no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—1888no—
548C2×C4—1888no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4#1309400112yes—
559C2×C4#1309400112yes—
560C2×C4#1309404312yes—
561C2×C4#1309405112yes—
562C2×C4C22×C4×D4711yescharacteristic
563C2×C4C22×Dic9411yescharacteristic
564C2×C4#41×D4711yescharacteristic
565C2×C4#41×D4711yescharacteristic
566C2×C4C2×#1062535211yescharacteristic
567C2×C4C2×#1062535011yescharacteristic
568C8—22no—
569C8—41no—
570C8—942no—
571C8—1881no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C4×D4×D4711yescharacteristic
580C22C2×#1309400111yescharacteristic
581C22#1568963211yescharacteristic
582C22#135×D4711yescharacteristic
583C22C2×#1309404311yescharacteristic
584C22C2×#1309405111yescharacteristic
585C22#1568992011yescharacteristic
586C4—22no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—942no—
591C4—942no—
592C4—942no—
593C4—3761no—
594C4—1882no—
595C4—1882no—
596C4—1881no—
597C4—1882no—
598C4#1568942114yesdirect product factor
599C4C2×C4×Dic9411yescharacteristic
600C4C4×#1062535211yescharacteristic
601C4#1568965311yescharacteristic
602C4#1568967611yescharacteristic
603C2—44no—
604C2C4×#1309400111yescharacteristic
605C2C4×#1309404311yescharacteristic
606C2C4×#1309405111yescharacteristic
607C2C2×#1568942111yescharacteristic
608C2#2025317211yescharacteristic
609C2#2025340211yescharacteristic
610C2#2025343911yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#102×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC4×#102D47
hypocenterC47C4×#478
LCSG ▷ C2×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
2. C4×#13094015C2semidirect
4. #20244344C2semidirect
5. #20244159C2semidirect
6. #20243979C2semidirect
7. #20243969C2semidirect
11. C4×#13093958C2semidirect
12. C4×#13093955C2semidirect
13. C4×Dic47⋊C8C2semidirect
20. #15689421C4direct (N × H)
65. #13094015C2×C4semidirect
92. #13093958C2×C4semidirect
94. #13093955C2×C4semidirect
96. Dic47⋊C8C2×C4semidirect
285. C4×C47#478semidirect
365. C47C4×#478semidirect
598. C4#15689421direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894212
C4×#130940152
C4×D4×Dic472
#202443442
#202441592
#202439792
#202439692
#202436472
#202436372
#202433962
C4×#130939582
C4×#130939552
C4×Dic47⋊C82
C4×#130939452
C4×#102×C472
C4×#47847