Finite Group Databasebeta

← C4×#14583196 #25104197

Subgroups — C4×#14583196 #25104197

Order 11008.

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×#14583196C211yesmaximal, characteristic
2C42×D172C211yesmaximal, characteristic
3#19135553C211yesmaximal, characteristic, complemented
4#19135361C211yesmaximal, characteristic, complemented
5#19135184C211yesmaximal, characteristic, complemented
6#19135170C211yesmaximal, characteristic, complemented
7#19134855C211yesmaximal, characteristic
8#19134841C211yesmaximal, characteristic
9#19134602C211yesmaximal, characteristic
10C4×#13063258C211yesmaximal, characteristic, complemented
11C4×#13063256C211yesmaximal, characteristic, complemented
12C4×C4⋊(C43⋊C8)C211yesmaximal, characteristic, complemented
13C4×#13063245C211yesmaximal, characteristic
14C4×#13063245C211yesmaximal, characteristic
15C4×#125×C43C211yesmaximal, characteristic, complemented
16C2×C4×D172C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18C2×C4×D172C2211yescharacteristic
19C2×C42×D43—21no—
20#14583196C418yesdirect product factor
21C2×#13063258C2211yescharacteristic
22C2×#13063258C2211yescharacteristic
23C2×#13063256C2211yescharacteristic
24C2×#13063256C2211yescharacteristic
25C2×C4⋊(C43⋊C8)C2211yescharacteristic
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×#13063245C2211yescharacteristic
28C2×#13063245C2211yescharacteristic
29C2×#13063245C2211yescharacteristic
30C2×#13063245C2211yescharacteristic
31C2×C4×C43⋊C8C2211yescharacteristic
32C2×C4×C43⋊C8C2211yescharacteristic
33C4×C4⋊Dic43C2211yescharacteristic
34C4×C4⋊Dic43C2211yescharacteristic
35C4×C4⋊Dic43C2211yescharacteristic
36C4×(C2×D43)⋊C4—21no—
37C4×(C2×D43)⋊C4—21no—
38C4×(C2×D43)⋊C4—21no—
39#14582661C2211yescharacteristic
40#14582661C2211yescharacteristic
41#14582661C2211yescharacteristic
42#14582661C2211yescharacteristic
43#14582661C2211yescharacteristic
44#14582661C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582654C2211yescharacteristic
48#14582654C2211yescharacteristic
49C2×#125×C43C2211yescharacteristic
50C43×#389C2211yescharacteristic
51C43×#389C2211yescharacteristic
52C43×#387—21no—
53C43×#387—21no—
54C43×#382C2211yescharacteristic
55C4×C4⋊C4×C43—21no—
56C4×C4⋊C4×C43—21no—
57C4×C4⋊C4×C43C2211yescharacteristic
58C4×C4⋊C4×C43C2211yescharacteristic
59C43×C43C2211yescharacteristic
60C22×D172C2311yescharacteristic
61C22×C4×D43—21no—
62C22×C4×D43—21no—
63C22×C4×D43—21no—
64C22×C43⋊C8C2311yescharacteristic
65C22×C43⋊C8C2311yescharacteristic
66C22×C43⋊C8C2311yescharacteristic
67C22×C43⋊C8C2311yescharacteristic
68C4×D172—22no—
69C4×D172—22no—
70C4×D172—22no—
71C4×D172—22no—
72C4×D172C2×C414yes—
73C4×D172Q814yes—
74C2×C4⋊Dic43C2311yescharacteristic
75C2×C4⋊Dic43C2311yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×(C2×D43)⋊C4—21no—
78C2×(C2×D43)⋊C4—21no—
79C2×(C2×D43)⋊C4—21no—
80C2×C4×Dic43—21no—
81C2×C4×Dic43—21no—
82C2×C4×Dic43—21no—
83C42×D43—44no—
84#13063258—22no—
85#13063258C2×C414yescomplemented
86#13063256—22no—
87#13063256C2×C414yescomplemented
88C4⋊(C43⋊C8)—22no—
89C4⋊(C43⋊C8)C2×C414yescomplemented
90#13063245—22no—
91#13063245—22no—
92#13063245C2×C414yes—
93#13063245C2×C414yes—
94C4×C43⋊C8—22no—
95C4×C43⋊C8—22no—
96C43⋊#95—21no—
97C43⋊#95—21no—
98C43⋊#95—21no—
99#125×C43—22no—
100#125×C43C2×C414yescomplemented
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—21no—
105C2×C4⋊C4×C43C2311yescharacteristic
106C2×C4⋊C4×C43C2311yescharacteristic
107C2×C4⋊C4×C43C2311yescharacteristic
108C2×C4⋊C4×C43C2311yescharacteristic
109C2×C42×C43—21no—
110C2×C42×C43—21no—
111C2×C42×C43D411yescharacteristic
112C2×C42×C43D411yescharacteristic
113C2×C42×C43D411yescharacteristic
114C2×C42×C43D411yescharacteristic
115C2×C42×C43C2311yescharacteristic
116C2×C42×C43C2311yescharacteristic
117C2×C42×C43C2311yescharacteristic
118#95×C43—21no—
119#95×C43—21no—
120#95×C43—21no—
121#95×C43—21no—
122#95×C43—21no—
123#95×C43—21no—
124C23×D43—21no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43—21no—
128C22×C4×C43—21no—
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2411yescharacteristic
136C22×Dic43—21no—
137C22×Dic43—21no—
138C22×Dic43—21no—
139C2×D172—21no—
140C2×D172—21no—
141C2×D172—21no—
142C2×D172C22×C412yes—
143C2×D172C2×Q812yes—
144C2×D172#4112yes—
145C2×C4×D43—24no—
146C2×C4×D43—24no—
147C2×C4×D43—24no—
148C2×C4×D43—42no—
149C2×C4×D43—42no—
150C2×C4×D43—42no—
151C2×C4×D43—42no—
152C2×C4×D43—42no—
153C2×C4×D43—42no—
154C4⋊C4×C43—24no—
155C4⋊C4×C43—24no—
156C4⋊C4×C43—21no—
157C4⋊C4×C43—21no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43—21no—
160C4⋊C4×C43—21no—
161C4⋊C4×C43—21no—
162C4⋊C4×C43—42no—
163C4⋊C4×C43—42no—
164C4⋊C4×C43C22×C412yes—
165C4⋊C4×C43C22×C412yes—
166C42×C43—22no—
167C42×C43—22no—
168C42×C43—22no—
169C42×C43—21no—
170C42×C43—21no—
171C42×C43—41no—
172C42×C43—42no—
173C42×C43—42no—
174C42×C43C22×C412yes—
175C42×C43C2×Q812yes—
176C42×C43#4112yes—
177C42×C43#4112yes—
178(C2×D43)⋊C4—24no—
179(C2×D43)⋊C4—24no—
180(C2×D43)⋊C4—24no—
181C4⋊Dic43—21no—
182C4⋊Dic43—21no—
183C4⋊Dic43—21no—
184C4⋊Dic43C22×C412yes—
185C4⋊Dic43C2×Q812yes—
186C4⋊Dic43#4112yes—
187C4×Dic43—42no—
188C4×Dic43—42no—
189C4×Dic43—42no—
190C2×C43⋊C8—21no—
191C2×C43⋊C8—21no—
192C2×C43⋊C8—21no—
193C2×C43⋊C8—21no—
194C2×C43⋊C8—21no—
195C2×C43⋊C8—21no—
196C2×C43⋊C8—21no—
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8C22×C412yes—
201C2×C43⋊C8C22×C412yes—
202C23×C43C22×D411yescharacteristic
203C22×D43—22no—
204C22×D43—22no—
205C22×D43—22no—
206C22×D43—41no—
207C22×D43—41no—
208C22×D43—41no—
209C22×D43—41no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—22no—
219C2×C4×C43—22no—
220C2×C4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43—41no—
233C2×C4×C43C4×D412yes—
234C2×C4×C43C4×D412yes—
235C2×C4×C43C22⋊Q812yes—
236C2×C4×C43C22⋊Q812yes—
237C2×C4×C43#13512yes—
238C2×C4×C43#13512yes—
239C2×C4×C43AGL(1,ℤ/8)12yes—
240C2×C4×C43C23×C411yescharacteristic
241C2×C4×C43C22×Q811yescharacteristic
242C2×C4×C43C2×#4111yescharacteristic
243C2×C4×C43C2×#4111yescharacteristic
244C2×C4×C43C2×#4111yescharacteristic
245C2×C4×C43C2×#4111yescharacteristic
246C2×Dic43—22no—
247C2×Dic43—22no—
248C2×Dic43—22no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255D172—22no—
256D172—22no—
257D172—22no—
258D172C4×Q814yes—
259C4×D43—48no—
260C4×D43—48no—
261C4×D43—48no—
262C43⋊C8—22no—
263C43⋊C8—22no—
264C43⋊C8—41no—
265C43⋊C8—41no—
266C4×#478—431nomaximal, Sylow
267C22×C43C2×C4×D411yescharacteristic
268C22×C43C2×C22⋊Q811yescharacteristic
269C22×C43#52511yescharacteristic
270C22×C43C2×#13511yescharacteristic
271C22×C43C2×#13511yescharacteristic
272C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C43#57511yescharacteristic
274C2×D43—24no—
275C2×D43—42no—
276C2×D43—42no—
277C2×D43—42no—
278C2×D43—42no—
279C2×D43—42no—
280C2×D43—42no—
281C4×C43—22no—
282C4×C43—22no—
283C4×C43—22no—
284C4×C43—22no—
285C4×C43—41no—
286C4×C43—42no—
287C4×C43—42no—
288C4×C43—81no—
289C4×C43—81no—
290C4×C43#47814yescomplemented
291C4×C43C2×C4×Q811yescharacteristic
292C4×C43C4×#4111yescharacteristic
293C4×C43#52911yescharacteristic
294C4×C43#53311yescharacteristic
295Dic43—42no—
296Dic43—42no—
297Dic43—42no—
298C2×#478—431no—
299C4×#125D4311yescharacteristic, complemented
300C42×D4—431no—
301#1989—431no—
302#1949—431no—
303#1921—431no—
304#1919—431no—
305#1864—431no—
306#1862—431no—
307#1804—431no—
308C4×#107—431no—
309C4×#106—431no—
310C4×C4⋊C8—431no—
311C4×#102—431no—
312C4×#102—431no—
313C2×C43C4×C22⋊Q811yescharacteristic
314C2×C43C4×#13511yescharacteristic
315C2×C43C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C43C2×#47811yescharacteristic
317C2×C43#307611yescharacteristic
318C2×C43#310211yescharacteristic
319C2×C43#310511yescharacteristic
320D43—44no—
321C2×#125C2×D4311yescharacteristic
322C2×C4×D4—431no—
323C2×C4×D4—431no—
324C2×C4×D4—431no—
325C22×C42—861no—
326#478—438no—
327C2×#107—431no—
328C2×#107—431no—
329C2×#106—431no—
330C2×#106—431no—
331C2×C4⋊C8—431no—
332C2×C4⋊C8—431no—
333C2×#102—431no—
334C2×#102—431no—
335C2×#102—431no—
336C2×#102—431no—
337C2×C4×C8—431no—
338C2×C4×C8—431no—
339#389C2×D4311yescharacteristic
340#389C2×D4311yescharacteristic
341#387—21no—
342#387—21no—
343#382C2×D4311yescharacteristic
344C4×C4⋊C4—21no—
345C4×C4⋊C4—21no—
346C4×C4⋊C4—431no—
347C4×C4⋊C4—431no—
348C4×C4⋊C4—431no—
349C4×C4⋊C4C2×D4311yescharacteristic
350C4×C4⋊C4C2×D4311yescharacteristic
351C4×C22⋊C4—861no—
352C4×C22⋊C4—861no—
353C4×C22⋊C4—861no—
354C43C2×D4311yescharacteristic
355#340—431no—
356#340—431no—
357#340—431no—
358#340—431no—
359#340—431no—
360#340—431no—
361#340—431no—
362#340—431no—
363#336—431no—
364#336—431no—
365C43C4×#47811yescharacteristic, complemented, Sylow
366C22×D4—431no—
367C23×C4—861no—
368C23×C4—861no—
369C23×C4—861no—
370C22×C8—431no—
371C22×C8—431no—
372C22×C8—431no—
373C22×C8—431no—
374#125—22no—
375#125C4×D4314yescomplemented
376C4×D4—434no—
377C4×D4—434no—
378C4×D4—862no—
379C4×D4—862no—
380C4×D4—862no—
381C4×D4—862no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—431no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—431no—
389C2×C4⋊C4C22×D4311yescharacteristic
390C2×C4⋊C4C22×D4311yescharacteristic
391C2×C4⋊C4C22×D4311yescharacteristic
392C2×C4⋊C4C22×D4311yescharacteristic
393C2×C22⋊C4—861no—
394C2×C22⋊C4—861no—
395C2×C22⋊C4—861no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—861no—
399C2×C42—861no—
400C2×C42—861no—
401C2×C42—1724no—
402C2×C42C22⋊D4311yescharacteristic
403C2×C42C22⋊D4311yescharacteristic
404C2×C42C22⋊D4311yescharacteristic
405C2×C42C22⋊D4311yescharacteristic
406C2×C42C22×D4311yescharacteristic
407C2×C42C22×D4311yescharacteristic
408C2×C42C22×D4311yescharacteristic
409#107—434no—
410#107—862no—
411#106—434no—
412#106—862no—
413C4⋊C8—434no—
414C4⋊C8—862no—
415#102—434no—
416#102—434no—
417#102—862no—
418#102—862no—
419C4×C8—862no—
420C4×C8—862no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—861no—
428#95—861no—
429#95—861no—
430C24—861no—
431C2×D4—432no—
432C2×D4—432no—
433C2×D4—432no—
434C2×D4—861no—
435C2×D4—861no—
436C2×D4—861no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—864no—
442C22×C4—861no—
443C22×C4—861no—
444C22×C4—861no—
445C22×C4—864no—
446C22×C4—864no—
447C22×C4—1722no—
448C22×C4—1722no—
449C22×C4—1722no—
450C22×C4—1722no—
451C22×C4—1722no—
452C22×C4—1722no—
453C22×C4C2×C22⋊D4311yescharacteristic
454C22×C4C2×C22⋊D4311yescharacteristic
455C22×C4C2×C22⋊D4311yescharacteristic
456C22×C4C2×C22⋊D4311yescharacteristic
457C22×C4C2×C22⋊D4311yescharacteristic
458C22×C4C2×C22⋊D4311yescharacteristic
459C22×C4C23×D4311yescharacteristic
460C2×C8—432no—
461C2×C8—432no—
462C2×C8—861no—
463C2×C8—861no—
464C2×C8—861no—
465C2×C8—861no—
466C2×C8—861no—
467C2×C8—861no—
468C2×C8—861no—
469C2×C8—861no—
470C2×C8—861no—
471C2×C8—861no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—432no—
483C4⋊C4—432no—
484C4⋊C4—432no—
485C4⋊C4—861no—
486C4⋊C4—861no—
487C4⋊C4—861no—
488C4⋊C4C2×C4×D4312yes—
489C4⋊C4C2×C4×D4312yes—
490C22⋊C4—864no—
491C22⋊C4—864no—
492C22⋊C4—864no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—1722no—
502C42—1722no—
503C42—1722no—
504C42C2×C4×D4312yes—
505C42#1062189812yes—
506C42Q8×D4312yes—
507C42#1062190212yes—
508C23—862no—
509C23—862no—
510C23—862no—
511C23—1721no—
512C23—1721no—
513C23—1721no—
514C23—1721no—
515C23C22×C22⋊D4311yescharacteristic
516D4—434no—
517D4—862no—
518D4—862no—
519D4—862no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—862no—
544C2×C4—862no—
545C2×C4—862no—
546C2×C4—1728no—
547C2×C4—1728no—
548C2×C4—1721no—
549C2×C4—1721no—
550C2×C4—1721no—
551C2×C4—1721no—
552C2×C4—1721no—
553C2×C4—1721no—
554C2×C4—1728no—
555C2×C4C4×C22⋊D4312yes—
556C2×C4C4×C22⋊D4312yes—
557C2×C4#1306330112yes—
558C2×C4#1306330112yes—
559C2×C4#1306334512yes—
560C2×C4#1306334512yes—
561C2×C4#1306335012yes—
562C2×C4C22×C4×D4311yescharacteristic
563C2×C4C2×Q8×D4311yescharacteristic
564C2×C4#41×D4311yescharacteristic
565C2×C4#41×D4311yescharacteristic
566C2×C4C2×#1062189811yescharacteristic
567C2×C4C2×#1062190211yescharacteristic
568C8—862no—
569C8—862no—
570C8—1721no—
571C8—1721no—
572C22—864no—
573C22—1722no—
574C22—1722no—
575C22—1722no—
576C22—1722no—
577C22—1722no—
578C22—1722no—
579C22C2×C4×C22⋊D4311yescharacteristic
580C22C2×#1306330111yescharacteristic
581C22#1458340611yescharacteristic
582C22C2×#1306334511yescharacteristic
583C22C2×#1306334511yescharacteristic
584C22C2×#1306335011yescharacteristic
585C22#1458369711yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—1722no—
596C4—1722no—
597C4—1722no—
598C4#1458319614yesdirect product factor
599C4C4×Q8×D4311yescharacteristic
600C4C4×#1062190211yescharacteristic
601C4#1458343611yescharacteristic
602C4#1458345311yescharacteristic
603C2—1724no—
604C2C4×#1306330111yescharacteristic
605C2C4×#1306334511yescharacteristic
606C2C4×#1306335011yescharacteristic
607C2C2×#1458319611yescharacteristic
608C2#1914437711yescharacteristic
609C2#1914460711yescharacteristic
610C2#1914463811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC4×#125×C43C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenterC4×#125D43
hypocenterC43C4×#478
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
3. #19135553C2semidirect
4. #19135361C2semidirect
5. #19135184C2semidirect
6. #19135170C2semidirect
10. C4×#13063258C2semidirect
11. C4×#13063256C2semidirect
12. C4×C4⋊(C43⋊C8)C2semidirect
15. C4×#125×C43C2semidirect
20. #14583196C4direct (N × H)
85. #13063258C2×C4semidirect
87. #13063256C2×C4semidirect
89. C4⋊(C43⋊C8)C2×C4semidirect
100. #125×C43C2×C4semidirect
290. C4×C43#478semidirect
299. C4×#125D43semidirect
365. C43C4×#478semidirect
375. #125C4×D43semidirect
598. C4#14583196direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145831962
C42×D1722
#191355532
#191353612
#191351842
#191351702
#191348552
#191348412
#191346022
C4×#130632582
C4×#130632562
C4×C4⋊(C43⋊C8)2
C4×#130632452
C4×#125×C432
C4×#47843