Finite Group Databasebeta

← #562×#6923 #30624908

Subgroups — #562×#6923 #30624908

Order 15616.

2248 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462365C211yesmaximal, characteristic, complemented
2#21462362C212yesmaximal, complemented
3#21462361C214yesmaximal, complemented
4#21462360C214yesmaximal, complemented
5#21462359C214yesmaximal, complemented
6#125×#6923C211yesmaximal, characteristic, complemented
7#124×#6923C212yesmaximal, complemented
8C22⋊Q8×#6923C214yesmaximal, complemented
9#123×#6923C214yesmaximal, complemented
10#121×#6923C214yesmaximal, complemented
11#562×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2212yes—
13C2×D4×#6923C2214yes—
14C2×D4×#6923C2212yes—
15C22×C4×#6923C2214yes—
16(C2×#6923)⋊Q8C2214yes—
17(C2×#6923)⋊Q8C2212yes—
18#16840714C2214yes—
19#16840714C2212yes—
20#16840713C2214yes—
21#16840713C2214yes—
22#16840713C2212yes—
23#16840712C2214yes—
24#16840712C2214yes—
25#16840712C2214yes—
26#16840712C2214yes—
27#16840712C2214yes—
28#16840712C2214yes—
29#16840711C2214yes—
30#16840711C2214yes—
31#16840711C2214yes—
32#16840711C2214yes—
33#16840711C2214yes—
34#16840710C2214yes—
35#16840710C2214yes—
36#16840710C2214yes—
37#16840709C2214yes—
38#16840709C2214yes—
39#16840709C2214yes—
40#16840706C2212yes—
41#16840706C2211yescharacteristic
42#16840705C2214yes—
43#16840704C2214yes—
44#16840704C2214yes—
45C4⋊C4×#6923C2214yes—
46C4⋊C4×#6923C2212yes—
47C4⋊C4×#6923C2214yes—
48C22⋊C4×#6923C2214yes—
49C22⋊C4×#6923C2214yes—
50C22⋊C4×#6923C2214yes—
51C42×#6923C2211yescharacteristic
52#16840510C411yescharacteristic, complemented
53#16840509C414yescomplemented
54#16840508C414yescomplemented
55#16840507C414yescomplemented
56#16840506C412yescomplemented
57#125×D61C2211yescharacteristic
58#124×D61C2212yes—
59#123×D61C2214yes—
60C22⋊Q8×D61C2214yes—
61#121×D61C2214yes—
62C61×#562C411yescharacteristic, complemented
63C23×#6923C2314yes—
64C23×#6923C2314yes—
65Q8×#6923—44no—
66D4×#6923—48no—
67D4×#6923—44no—
68C2×C4⋊#6923C2314yes—
69C2×C4⋊#6923C2314yes—
70C2×C4⋊#6923C2314yes—
71C2×C4⋊#6923C2314yes—
72C2×C4⋊#6923C2312yes—
73C2×C4⋊#6923C2312yes—
74C2×C4⋊#6923C2312yes—
75C2×C22⋊#6923C2314yes—
76C2×C22⋊#6923C2314yes—
77C2×C22⋊#6923C2314yes—
78C2×C22⋊#6923C2314yes—
79C2×C22⋊#6923C2314yes—
80C2×C22⋊#6923C2314yes—
81C2×C4×#6923—48no—
82C2×C4×#6923C2314yes—
83C2×C4×#6923C2314yes—
84C2×C4×#6923C2312yes—
85C2×C4×#6923C2311yescharacteristic
86C2×C4×#6923C2314yes—
87C2×C4×#6923C2312yes—
88C2×C4×#6923C2314yes—
89C2×C4×#6923C2311yescharacteristic
90(C2×#6923)⋊C4C2314yes—
91(C2×#6923)⋊C4C2314yes—
92(C2×#6923)⋊C4C2314yes—
93(C2×#6923)⋊C4C2314yes—
94(C2×#6923)⋊C4C2314yes—
95(C2×#6923)⋊C4C2314yes—
96(C2×#6923)⋊C4C2314yes—
97(C2×#6923)⋊C4C2314yes—
98(C2×#6923)⋊C4C2314yes—
99(C2×#6923)⋊C4C2314yes—
100(C2×#6923)⋊C4C2314yes—
101C2×Q8×D61C2312yes—
102C2×D4×D61C2314yes—
103C2×D4×D61C2312yes—
104C22×C4×D61C2314yes—
105#14519769C2×C411yescharacteristic, complemented
106#14519768C2×C412yescomplemented
107#14519767C2×C414yescomplemented
108#14519766C2×C412yescomplemented
109#14519765C2×C412yescomplemented
110#14519764C2×C412yescomplemented
111#14519763C2×C414yescomplemented
112#14519763C2×C414yescomplemented
113#14519762C2×C414yescomplemented
114#14519762C2×C414yescomplemented
115#14519761C2×C414yescomplemented
116#14519760C2×C414yescomplemented
117#14519759C2×C414yescomplemented
118#14519759C2×C414yescomplemented
119#14519758C2×C414yescomplemented
120#14519757C2×C414yescomplemented
121#14519756C2×C414yescomplemented
122#14519755C2×C414yescomplemented
123#14519755C2×C414yescomplemented
124#14519754C2×C414yescomplemented
125#14519754C2×C414yescomplemented
126#14519753C2×C414yescomplemented
127#14519752C2×C414yescomplemented
128#14519751C2×C414yescomplemented
129#14519750C2×C414yescomplemented
130#14519750C2×C414yescomplemented
131#14519749C2×C414yescomplemented
132#14519749C2×C414yescomplemented
133#14519748C2×C414yescomplemented
134C4⋊C4×D61C2314yes—
135C4⋊C4×D61C2314yes—
136C4⋊C4×D61C2312yes—
137C22⋊C4×D61C2314yes—
138C22⋊C4×D61C2314yes—
139C22⋊C4×D61C2314yes—
140C42×D61C2311yescharacteristic
141#125×C61C2×C411yescharacteristic, complemented
142#124×C61C2×C412yescomplemented
143#123×C61C2×C414yescomplemented
144C22⋊Q8×C61C2×C414yescomplemented
145#121×C61C2×C414yescomplemented
146C23×D61C2414yes—
147C22×#6923—28no—
148C22×#6923—28no—
149C22×#6923—28no—
150C22×#6923—44no—
151C22×#6923—44no—
152C22×#6923—44no—
153C22×#6923C2411yescharacteristic
154C22×#6923C2414yes—
155C22×#6923C2414yes—
156C22×#6923C2414yes—
157C22×#6923C2412yes—
158C22×#6923C2411yescharacteristic
159C2×Q8×C61C22×C412yes—
160C2×D4×C61C22×C414yes—
161C2×D4×C61C22×C412yes—
162C22×C4×C61C22×C414yes—
163C2×C22⋊D61C22×C414yes—
164C2×C22⋊D61C22×C414yes—
165C2×C22⋊D61C22×C414yes—
166C22×Dic61C22×C414yes—
167Q8×D61—42no—
168Q8×D61—42no—
169D4×D61—42no—
170D4×D61—44no—
171D4×D61—44no—
172D4×D61—42no—
173C2×D244C22×C412yes—
174C2×D244C22×C414yes—
175C2×C4×D61—44no—
176C2×C4×D61—44no—
177C2×C4×D61C22×C414yes—
178C2×C4×D61C22×C414yes—
179C2×C4×D61C2414yes—
180C2×C4×D61C2414yes—
181C2×C4×D61C2412yes—
182C2×C4×D61C2411yescharacteristic
183C2×Dic122C22×C412yes—
184C2×Dic122C22×C414yes—
185C22⋊#6923—44no—
186C22⋊#6923—44no—
187C22⋊#6923—44no—
188C22⋊#6923—44no—
189C22⋊#6923—44no—
190C22⋊#6923—44no—
191C4⋊#6923—42no—
192C4⋊#6923—44no—
193C4⋊#6923—44no—
194C4⋊#6923—44no—
195C4⋊#6923—42no—
196C4⋊#6923—44no—
197C4⋊#6923—42no—
198C4×#6923—28no—
199C4×#6923—28no—
200C4×#6923—24no—
201C4×#6923—22no—
202C4×#6923—44no—
203C4×#6923—44no—
204C4×#6923—41no—
205C4×#6923—42no—
206C4⋊C4×C61C22×C414yes—
207C4⋊C4×C61C22×C414yes—
208C4⋊C4×C61C22×C412yes—
209C22⋊C4×C61C22×C414yes—
210C22⋊C4×C61C22×C414yes—
211C22⋊C4×C61C22×C414yes—
212C42×C61C22×C411yescharacteristic
213C22⋊Dic61C22×C414yes—
214C22⋊Dic61C22×C414yes—
215C22⋊Dic61C22×C414yes—
216#11758198C22×C414yes—
217#11758198C22×C414yes—
218#11758198C22×C414yes—
219#11758198C22×C414yes—
220#11758198C22×C414yes—
221#11758198C22×C414yes—
222C4⋊Dic61C22×C414yes—
223C4⋊Dic61C22×C414yes—
224C4⋊Dic61C22×C412yes—
225#11758196C22×C414yes—
226#11758196C22×C414yes—
227#11758196C22×C414yes—
228#11758196C22×C414yes—
229#11758196C22×C414yes—
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C22×C412yes—
232C23×C61C23×C414yes—
233C22×D61—24no—
234C22×D61—24no—
235C22×D61—24no—
236C22×D61—24no—
237C22×D61—24no—
238C22×D61—24no—
239C22×D61C23×C414yes—
240C22×D61C2511yescharacteristic
241C2×#6923—24no—
242C2×#6923—24no—
243C2×#6923—22no—
244C2×#6923—24no—
245C2×#6923—24no—
246C2×#6923—21no—
247C2×#6923—24no—
248C2×#6923—22no—
249C2×#6923—22no—
250C2×#6923—22no—
251C2×#6923—24no—
252C2×#6923—24no—
253C2×#6923—24no—
254C2×#6923—24no—
255C2×#6923—416no—
256C2×#6923—416no—
257C2×#6923H(5,𝔽2)14yes—
258C2×#69232−1+412yes—
259Q8×C61—42no—
260D4×C61—42no—
261D4×C61—44no—
262C2×C4×C61—44no—
263C2×C4×C61C23×C412yes—
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C23×C414yes—
266C2×C4×C61C23×C414yes—
267C22⋊D61—44no—
268C22⋊D61—44no—
269C22⋊D61—44no—
270C2×Dic61—44no—
271C2×Dic61C23×C411yescharacteristic
272C2×Dic61C23×C412yes—
273C2×Dic61C23×C414yes—
274C2×Dic61C23×C414yes—
275D244—42no—
276D244—44no—
277C4×D61—24no—
278C4×D61—24no—
279C4×D61—22no—
280C4×D61—21no—
281C4×D61—21no—
282C4×D61—22no—
283C4×D61—24no—
284C4×D61—24no—
285C4×D61—44no—
286C4×D61—44no—
287Dic122—42no—
288Dic122—44no—
289C4×#562—611nomaximal, Sylow
290C22×C61—24no—
291C22×C61—24no—
292C22×C61—24no—
293C22×C61C24×C411yescharacteristic
294C2×D61—24no—
295C2×D61—24no—
296C2×D61—24no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—44no—
301C2×D61C2×H(5,𝔽2)12yes—
302C2×D61C2×2−1+411yescharacteristic
303C2×D61#58511yescharacteristic
304C2×D61#58512yes—
305#6923—44no—
306#6923—42no—
307#6923—44no—
308#6923—44no—
309#6923—41no—
310#6923#56214yesdirect product factor
311C4×C61—24no—
312C4×C61—22no—
313C4×C61—24no—
314C4×C61—21no—
315Dic61—24no—
316Dic61—22no—
317Dic61—24no—
318Dic61—21no—
319C2×#562—611no—
320#2382—611no—
321#2379—612no—
322#2378—614no—
323#2377—614no—
324#2376—614no—
325C4×#125—611no—
326C4×#124—612no—
327C4×C22⋊Q8—614no—
328C4×#123—614no—
329C4×#121—614no—
330C2×C61—44no—
331C2×C61C4×H(5,𝔽2)12yes—
332C2×C61C4×2−1+411yescharacteristic
333D61—44no—
334D61C2×#56211yescharacteristic
335D61#352912yes—
336D61#353111yescharacteristic
337#562—611no—
338#562—612no—
339#562—614no—
340#562—614no—
341#562—614no—
342#562#692311yescharacteristic, direct product factor
343C2×#125—611no—
344C2×#124—612no—
345C2×#123—614no—
346C2×C22⋊Q8—614no—
347C2×#121—614no—
348C2×C4×Q8—612no—
349C2×C4×D4—614no—
350C2×C4×D4—612no—
351C22×C42—614no—
352#391—612no—
353#391—614no—
354#390—614no—
355#390—612no—
356#389—612no—
357#389—614no—
358#389—614no—
359#388—614no—
360#388—614no—
361#388—614no—
362#388—614no—
363#388—614no—
364#388—614no—
365#387—614no—
366#387—614no—
367#387—614no—
368#387—614no—
369#387—614no—
370#386—614no—
371#386—614no—
372#386—614no—
373#385—614no—
374#385—614no—
375#385—614no—
376#382—611no—
377#382—612no—
378#381—614no—
379#380—614no—
380#380—614no—
381C4×C4⋊C4—614no—
382C4×C4⋊C4—612no—
383C4×C4⋊C4—614no—
384C4×C22⋊C4—614no—
385C4×C22⋊C4—614no—
386C4×C22⋊C4—614no—
387C43—611no—
388C61C4×#56211yescharacteristic, complemented, Sylow
389C22×Q8—612no—
390C22×D4—614no—
391C22×D4—612no—
392C23×C4—614no—
393C23×C4—614no—
394C23×C4—614no—
395#125—614no—
396#125—612no—
397#125—611no—
398#125C2×#692311yescharacteristic, complemented
399#124—612no—
400#124—612no—
401#124—614no—
402#124—614no—
403#124—612no—
404#124C2×#692312yescomplemented
405#123—614no—
406#123—614no—
407#123—614no—
408#123—614no—
409#123—614no—
410#123—614no—
411#123—614no—
412#123C2×#692314yescomplemented
413C22⋊Q8—614no—
414C22⋊Q8—614no—
415C22⋊Q8—614no—
416C22⋊Q8—614no—
417C22⋊Q8—614no—
418C22⋊Q8—614no—
419C22⋊Q8—614no—
420C22⋊Q8C2×#692314yescomplemented
421#121—614no—
422#121—614no—
423#121—614no—
424#121—614no—
425#121—614no—
426#121—614no—
427#121—614no—
428#121C2×#692314yescomplemented
429C4×Q8—2444no—
430C4×D4—2448no—
431C4×D4—2444no—
432C2×C4⋊C4—614no—
433C2×C4⋊C4—614no—
434C2×C4⋊C4—612no—
435C2×C4⋊C4—614no—
436C2×C4⋊C4—612no—
437C2×C4⋊C4—614no—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—614no—
440C2×C4⋊C4—614no—
441C2×C4⋊C4—612no—
442C2×C22⋊C4—614no—
443C2×C22⋊C4—614no—
444C2×C22⋊C4—614no—
445C2×C22⋊C4—614no—
446C2×C22⋊C4—614no—
447C2×C22⋊C4—614no—
448C2×C22⋊C4—614no—
449C2×C22⋊C4—614no—
450C2×C22⋊C4—614no—
451C2×C42—611no—
452C2×C42—612no—
453C2×C42—614no—
454C2×C42—614no—
455C2×C42—611no—
456C2×C42—612no—
457C2×C42—611no—
458C2×C42—614no—
459C2×C42—614no—
460C2×C42—2448no—
461#95—614no—
462#95—614no—
463#95—614no—
464#95—614no—
465#95—614no—
466#95—614no—
467#95—614no—
468#95—614no—
469#95—614no—
470#95—614no—
471#95—614no—
472C24—614no—
473C2×Q8—614no—
474C2×Q8—612no—
475C2×Q8—2442no—
476C2×Q8—2442no—
477C2×Q8C22×#692312yes—
478C2×D4—614no—
479C2×D4—614no—
480C2×D4—614no—
481C2×D4—614no—
482C2×D4—612no—
483C2×D4—2444no—
484C2×D4—2442no—
485C2×D4—2442no—
486C2×D4—2444no—
487C2×D4C22×#692314yes—
488C2×D4C22×#692312yes—
489C22×C4—611no—
490C22×C4—614no—
491C22×C4—611no—
492C22×C4—612no—
493C22×C4—614no—
494C22×C4—614no—
495C22×C4—614no—
496C22×C4—614no—
497C22×C4—611no—
498C22×C4—614no—
499C22×C4—614no—
500C22×C4—614no—
501C22×C4—612no—
502C22×C4—1228no—
503C22×C4—1228no—
504C22×C4—1228no—
505C22×C4—2444no—
506C22×C4—2444no—
507C22×C4—2444no—
508C22×C4—2444no—
509C22×C4—2444no—
510C22×C4C22×#692314yes—
511C4⋊C4—614no—
512C4⋊C4—614no—
513C4⋊C4—614no—
514C4⋊C4—614no—
515C4⋊C4—614no—
516C4⋊C4—614no—
517C4⋊C4—614no—
518C4⋊C4—612no—
519C4⋊C4—2444no—
520C4⋊C4—2442no—
521C4⋊C4—2444no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C4⋊C4—2444no—
525C4⋊C4—2444no—
526C4⋊C4C22×#692314yes—
527C4⋊C4C22×#692314yes—
528C4⋊C4C22×#692312yes—
529C22⋊C4—614no—
530C22⋊C4—614no—
531C22⋊C4—614no—
532C22⋊C4—614no—
533C22⋊C4—614no—
534C22⋊C4—614no—
535C22⋊C4—614no—
536C22⋊C4—614no—
537C22⋊C4—614no—
538C22⋊C4—2444no—
539C22⋊C4—2444no—
540C22⋊C4—2444no—
541C22⋊C4—2444no—
542C22⋊C4—2444no—
543C22⋊C4—2444no—
544C22⋊C4C22×#692314yes—
545C22⋊C4C22×#692314yes—
546C22⋊C4C22×#692314yes—
547C42—612no—
548C42—611no—
549C42—1228no—
550C42—1228no—
551C42—1224no—
552C42—1222no—
553C42—2444no—
554C42—2444no—
555C42—2441no—
556C42—2442no—
557C42C22×#692311yescharacteristic
558C23—611no—
559C23—614no—
560C23—1224no—
561C23—1224no—
562C23—1224no—
563C23—1224no—
564C23—1224no—
565C23—1224no—
566C23C23×#692314yes—
567Q8—42no—
568Q8—2444no—
569Q8—2442no—
570D4—44no—
571D4—42no—
572D4—2444no—
573D4—2442no—
574D4—2444no—
575D4—2444no—
576D4—2444no—
577C2×C4—44no—
578C2×C4—614no—
579C2×C4—612no—
580C2×C4—611no—
581C2×C4—612no—
582C2×C4—614no—
583C2×C4—614no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1222no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1222no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1222no—
595C2×C4—1221no—
596C2×C4—1224no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1224no—
601C2×C4—1224no—
602C2×C4—1224no—
603C2×C4—1224no—
604C2×C4—1224no—
605C2×C4—1224no—
606C2×C4—2444no—
607C2×C4—2444no—
608C2×C4—24416no—
609C2×C4—2444no—
610C2×C4—24416no—
611C2×C4C23×#692311yescharacteristic
612C2×C4C23×#692312yes—
613C2×C4C23×#692314yes—
614C2×C4C23×#692314yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—612no—
619C22—612no—
620C22—611no—
621C22—611no—
622C22—1224no—
623C22—1224no—
624C22—1224no—
625C22—2444no—
626C22—2444no—
627C22—2444no—
628C22—2444no—
629C22C24×#692311yescharacteristic
630C4—24no—
631C4—24no—
632C4—22no—
633C4—21no—
634C4—614no—
635C4—1224no—
636C4—1224no—
637C4—1222no—
638C4—1221no—
639C4—2444no—
640C4—2444no—
641C4—2444no—
642C4—2442no—
643C4—2441no—
644C2—44no—
645C2—611no—
646C2—612no—
647C2—611no—
648C2—2444no—
649C2H(5,𝔽2)×#692312yes—
650C22−1+4×#692311yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#562C4
prime radical#562×D61C2
cosocleC22×D61C25
hypercenter#562#6923
hypocenterC61C4×#562
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #21462365C2semidirect
2. #21462362C2semidirect
3. #21462361C2semidirect
4. #21462360C2semidirect
5. #21462359C2semidirect
6. #125×#6923C2semidirect
7. #124×#6923C2semidirect
8. C22⋊Q8×#6923C2semidirect
9. #123×#6923C2semidirect
10. #121×#6923C2semidirect
52. #16840510C4semidirect
53. #16840509C4semidirect
54. #16840508C4semidirect
55. #16840507C4semidirect
56. #16840506C4semidirect
62. C61×#562C4semidirect
105. #14519769C2×C4semidirect
106. #14519768C2×C4semidirect
107. #14519767C2×C4semidirect
108. #14519766C2×C4semidirect
109. #14519765C2×C4semidirect
110. #14519764C2×C4semidirect
111. #14519763C2×C4semidirect
112. #14519763C2×C4semidirect
113. #14519762C2×C4semidirect
114. #14519762C2×C4semidirect
115. #14519761C2×C4semidirect
116. #14519760C2×C4semidirect
117. #14519759C2×C4semidirect
118. #14519759C2×C4semidirect
119. #14519758C2×C4semidirect
120. #14519757C2×C4semidirect
121. #14519756C2×C4semidirect
122. #14519755C2×C4semidirect
123. #14519755C2×C4semidirect
124. #14519754C2×C4semidirect
125. #14519754C2×C4semidirect
126. #14519753C2×C4semidirect
127. #14519752C2×C4semidirect
128. #14519751C2×C4semidirect
129. #14519750C2×C4semidirect
130. #14519750C2×C4semidirect
131. #14519749C2×C4semidirect
132. #14519749C2×C4semidirect
133. #14519748C2×C4semidirect
141. #125×C61C2×C4semidirect
142. #124×C61C2×C4semidirect
143. #123×C61C2×C4semidirect
144. C22⋊Q8×C61C2×C4semidirect
145. #121×C61C2×C4semidirect
310. #6923#562direct (N × H)
342. #562#6923direct (N × H)
388. C61C4×#562semidirect
398. #125C2×#6923semidirect
404. #124C2×#6923semidirect
412. #123C2×#6923semidirect
420. C22⋊Q8C2×#6923semidirect
428. #121C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623652
#214623622
#214623612
#214623602
#214623592
#125×#69232
#124×#69232
C22⋊Q8×#69232
#123×#69232
#121×#69232
#562×D612
C4×#56261