Finite Group Databasebeta

← C4×#16810912 #27391663

Subgroups — C4×#16810912 #27391663

Order 13568.

1000 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
1C2×#16810912C211yesmaximal, characteristic
2C4×#13133216C211yesmaximal, characteristic, complemented
3C4×#13133179C211yesmaximal, characteristic
4#20275661C211yesmaximal, characteristic, complemented
5#20275475C211yesmaximal, characteristic, complemented
6#20275302C211yesmaximal, characteristic, complemented
7#20275298C211yesmaximal, characteristic, complemented
8#20274977C211yesmaximal, characteristic
9#20274973C211yesmaximal, characteristic
10#20274714C211yesmaximal, characteristic
11C4×#13133148C211yesmaximal, characteristic, complemented
12C4×#13133146C211yesmaximal, characteristic, complemented
13C4×#13133137C211yesmaximal, characteristic
14C4×#13133139C211yesmaximal, characteristic
15C4×#106×C53C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×C4×Dic106—21no—
18C2×C4×D212C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×C42×D53—21no—
22#16810912C418yesdirect product factor
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810527—21no—
33#16810522C2211yescharacteristic
34#16810520—21no—
35#16810519C2211yescharacteristic
36#16810492C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×#11719919—21no—
40C4×C4⋊Dic53C2211yescharacteristic
41C4×#11719921—21no—
42#16810470—21no—
43#16810470—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106—21no—
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133216—22no—
68#13133216C2×C414yescomplemented
69C4×Dic106—44no—
70C4×D212—22no—
71C4×D212Q814yescomplemented
72#13133179—22no—
73#13133179—22no—
74#13133179—22no—
75#13133179C2×C414yes—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96C42×D53—44no—
97#13133148—22no—
98#13133148C2×C414yescomplemented
99#13133146—22no—
100#13133146C2×C414yescomplemented
101#13133139—22no—
102#13133139C2×C414yes—
103#13133137—22no—
104#13133137C2×C414yes—
105C4×#88322—22no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120#106×C53—22no—
121#106×C53C2×C414yescomplemented
122C4×C8×C53—22no—
123#95×C53—21no—
124#95×C53—21no—
125C23×D53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212C22×C412yes—
144C2×D212C2×Q812yes—
145C2×D212#4112yes—
146C2×C4×D53—24no—
147C2×C4×D53—24no—
148C2×C4×D53—24no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×Dic106—24no—
156C2×Dic106—42no—
157C2×Dic106—42no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C412yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C22×C412yes—
170C4⋊C4×C53C22×C412yes—
171C4⋊C4×C53#4112yes—
172C42×C53—42no—
173C42×C53—42no—
174C42×C53C2×Q812yes—
175C42×C53#4112yes—
176#11719921—24no—
177#11719921—24no—
178#11719921—24no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53C2×Q812yes—
186#11719919—24no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190C4×Dic53—22no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—22no—
194C4×Dic53—41no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53SD1614yes—
199C4×Dic53C22×C412yes—
200C4×Dic53#4112yes—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322C22×C412yes—
207C23×C53C22×D411yescharacteristic
208C22×D53—22no—
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53C22⋊Q812yes—
230C2×C4×C53C22⋊Q812yes—
231C2×C4×C53C2×SD1612yes—
232C2×C4×C53AGL(1,ℤ/8)12yes—
233C2×C4×C53C23×C411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53C4×D412yes—
256C2×Dic53C4×D412yes—
257C2×Dic53C2×SD1612yes—
258C2×Dic53C2×SD1612yes—
259C2×Dic53#13512yes—
260D212—22no—
261D212—22no—
262D212—22no—
263D212C4×Q814yescomplemented
264C4×D53—48no—
265C4×D53—48no—
266C4×D53—48no—
267Dic106—48no—
268C8×C53—22no—
269C8×C53—41no—
270#88322—22no—
271#88322—41no—
272C4×#476—531nomaximal, Sylow
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C2×C22⋊Q811yescharacteristic
275C22×C53#52511yescharacteristic
276C22×C53C22×SD1611yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—24no—
281C2×D53—42no—
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—42no—
285C2×D53—42no—
286C2×D53—42no—
287C4×C53—22no—
288C4×C53—42no—
289C4×C53—42no—
290C4×C53—42no—
291C4×C53—81no—
292C4×C53#47614yescomplemented
293C4×C53C2×C4×Q811yescharacteristic
294C4×C53C4×#4111yescharacteristic
295C4×C53#52911yescharacteristic
296C4×C53#53311yescharacteristic
297Dic53—22no—
298Dic53—22no—
299Dic53—41no—
300Dic53—42no—
301Dic53—42no—
302Dic53—81no—
303Dic53C4×SD1614yes—
304C2×#476—531no—
305C4×C4⋊Q8—531no—
306C42×D4—531no—
307#1987—531no—
308#1948—531no—
309#1920—531no—
310#1920—531no—
311#1864—531no—
312#1864—531no—
313#1802—531no—
314C4×#106—531no—
315C4×#106D5311yescharacteristic, complemented
316C4×C4⋊C8—531no—
317C4×#102—531no—
318C4×#102—531no—
319C2×C53C4×C22⋊Q811yescharacteristic
320C2×C53C2×C4×SD1611yescharacteristic
321C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
322C2×C53C2×#47611yescharacteristic
323C2×C53#307611yescharacteristic
324C2×C53#310111yescharacteristic
325C2×C53#310611yescharacteristic
326D53—44no—
327C2×C4⋊Q8—531no—
328C2×C4×Q8—1061no—
329C2×C4×D4—531no—
330C2×C4×D4—531no—
331C2×C4×D4—531no—
332C22×C42—1061no—
333#476—538no—
334C2×#106—531no—
335C2×#106—531no—
336C2×#106C2×D5311yescharacteristic
337C2×#106C2×D5311yescharacteristic
338C2×C4⋊C8—531no—
339C2×C4⋊C8—531no—
340C2×#102—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×C4×C8—531no—
345C2×C4×C8C2×D5311yescharacteristic
346#391—1061no—
347#389—531no—
348#389—531no—
349#389—1061no—
350#384—531no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—1061no—
355C4×C4⋊C4C2×D5311yescharacteristic
356C4×C4⋊C4C2×D5311yescharacteristic
357C4×C22⋊C4—1061no—
358C4×C22⋊C4—1061no—
359C4×C22⋊C4—1061no—
360C43—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371C53C4×#47611yescharacteristic, complemented, Sylow
372C22×Q8—1061no—
373C22×D4—531no—
374C23×C4—1061no—
375C23×C4—1061no—
376C23×C4—1061no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381C4⋊Q8—534no—
382C4⋊Q8—1062no—
383C4×Q8—2124no—
384C4×D4—534no—
385C4×D4—534no—
386C4×D4—1062no—
387C4×D4—1062no—
388C4×D4—1062no—
389C4×D4—1062no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—2124no—
417C2×C42C22×D5311yescharacteristic
418#106—22no—
419#106—534no—
420#106—1062no—
421#106C4×D5314yescomplemented
422C4⋊C8—534no—
423C4⋊C8—1062no—
424#102—534no—
425#102—534no—
426#102—1062no—
427#102—1062no—
428C4×C8—22no—
429C4×C8—1062no—
430#95—21no—
431#95—21no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24—1061no—
438C2×Q8—1064no—
439C2×Q8—2122no—
440C2×Q8—2122no—
441C2×D4—532no—
442C2×D4—532no—
443C2×D4—532no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—1064no—
456C22×C4—1064no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1064no—
460C22×C4—1061no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—532no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8C2×C4×D5312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—532no—
488C4⋊C4—532no—
489C4⋊C4—1064no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4C2×C4×D5312yes—
499C4⋊C4C2×C4×D5312yes—
500C4⋊C4#1171994512yes—
501C22⋊C4—1064no—
502C22⋊C4—1064no—
503C22⋊C4—1064no—
504C42—42no—
505C42—42no—
506C42—534no—
507C42—532no—
508C42—532no—
509C42—1062no—
510C42—1062no—
511C42—1061no—
512C42—1061no—
513C42—2121no—
514C42—2122no—
515C42—2122no—
516C42—2122no—
517C42#1171994512yes—
518C42Q8×D5312yes—
519C23—1062no—
520C23—1062no—
521C23—1062no—
522C23—2121no—
523C23—2121no—
524C23—2121no—
525C23—2121no—
526C23C2×D4×D5311yescharacteristic
527Q8—2128no—
528D4—534no—
529D4—1062no—
530D4—1062no—
531D4—1062no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—2128no—
561C2×C4—2128no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2128no—
570C2×C4#1313319512yes—
571C2×C4#1313319512yes—
572C2×C4SD16×D5312yes—
573C2×C4#1313324412yes—
574C2×C4C22×C4×D5311yescharacteristic
575C2×C4C2×Q8×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4C2×#1171994511yescharacteristic
579C2×C4C2×#1171994511yescharacteristic
580C8—22no—
581C8—41no—
582C8—1062no—
583C8—2121no—
584C22—1064no—
585C22—2122no—
586C22—2122no—
587C22—2122no—
588C22—2122no—
589C22—2122no—
590C22—2122no—
591C22C4×D4×D5311yescharacteristic
592C22C2×#1313319511yescharacteristic
593C22#1681113211yescharacteristic
594C22C2×SD16×D5311yescharacteristic
595C22#135×D5311yescharacteristic
596C22C2×#1313324411yescharacteristic
597C22#1681142011yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—534no—
604C4—1062no—
605C4—1062no—
606C4—4241no—
607C4—2122no—
608C4—2121no—
609C4—2122no—
610C4#1681091214yesdirect product factor
611C4C4×Q8×D5311yescharacteristic
612C4C4×#1171994511yescharacteristic
613C4#1681115111yescharacteristic
614C4#1681117711yescharacteristic
615C2—2124no—
616C2C4×#1313319511yescharacteristic
617C2C4×SD16×D5311yescharacteristic
618C2C4×#1313324411yescharacteristic
619C2C2×#1681091211yescharacteristic
620C2#2028450111yescharacteristic
621C2#2028472111yescharacteristic
622C2#2028477311yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#106×C53C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenterC4×#106D53
hypocenterC53C4×#476
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. C4×#13133216C2semidirect
4. #20275661C2semidirect
5. #20275475C2semidirect
6. #20275302C2semidirect
7. #20275298C2semidirect
11. C4×#13133148C2semidirect
12. C4×#13133146C2semidirect
15. C4×#106×C53C2semidirect
22. #16810912C4direct (N × H)
68. #13133216C2×C4semidirect
71. C4×D212Q8semidirect
98. #13133148C2×C4semidirect
100. #13133146C2×C4semidirect
121. #106×C53C2×C4semidirect
263. D212C4×Q8semidirect
292. C4×C53#476semidirect
315. C4×#106D53semidirect
371. C53C4×#476semidirect
421. #106C4×D53semidirect
610. C4#16810912direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109122
C4×#131332162
C4×#131331792
#202756612
#202754752
#202753022
#202752982
#202749772
#202749732
#202747142
C4×#131331482
C4×#131331462
C4×#131331372
C4×#131331392
C4×#106×C532
C4×#47653