Finite Group Databasebeta

← #29647738

Subgroups — #29647738

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459658C211yesmaximal, characteristic
2#21448759C211yesmaximal, characteristic, complemented
3#21444397C212yesmaximal, complemented
4#21444392C211yesmaximal, characteristic, complemented
5#21444388C211yesmaximal, characteristic, complemented
6#21444376C212yesmaximal, complemented
7#21444311C212yesmaximal, complemented
8#21444306C212yesmaximal, complemented
9C2×#16839500C211yesmaximal, characteristic, complemented
10C2×#16839497C211yesmaximal, characteristic, complemented
11C61×#2972C211yesmaximal, characteristic, complemented
12#16840461—22no—
13#16840458C412yescomplemented
14#16840452C412yescomplemented
15#16840441—22no—
16#16840421C412yescomplemented
17#16840408C412yescomplemented
18#16840359—21no—
19#16840350—21no—
20#16840321—21no—
21#16840296C2212yes—
22#16840273C2212yes—
23C2×#14519756C2211yescharacteristic
24C2×#14519751C2211yescharacteristic
25#16840235—21no—
26#16839655—21no—
27#16839648—21no—
28#16839642—21no—
29#16839630C2211yescharacteristic, complemented
30#16839625C2211yescharacteristic, complemented
31#16839622C2211yescharacteristic
32C22⋊C4×Dic61—21no—
33C4×C22⋊Dic61C2211yescharacteristic, complemented
34#16839608C2211yescharacteristic, complemented
35C2×#14519684C2211yescharacteristic
36#16839500C2212yescomplemented
37#16839500C2212yescomplemented
38#16839500C2212yescomplemented
39#16839500C2212yescomplemented
40#16839497C2212yes—
41#16839497C2212yes—
42#16839497C2212yes—
43#16839497C2212yes—
44C2×C4×D4×C61C2211yescharacteristic
45C8×D4×C61—22no—
46C8×D4×C61—22no—
47C61×#431C2212yescomplemented
48C2×C22⋊C8×C61C2211yescharacteristic
49C2×C22⋊C8×C61C2211yescharacteristic
50C2×C22⋊C8×C61C2212yescomplemented
51C2×#11758222C2×C412yescomplemented
52C22×Dic122C2×C411yescharacteristic
53C22×D244C2×C411yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56#14519772—22no—
57#14519771—22no—
58#14519770—22no—
59#14519769C2×C412yescomplemented
60#14519767—22no—
61#14519765C2×C412yescomplemented
62#14519763—22no—
63#14519762—22no—
64#14519760—22no—
65#14519760—22no—
66#14519759—22no—
67#14519758—22no—
68#14519756—21no—
69#14519756—21no—
70#14519756C2×C412yes—
71#14519756C2×C412yescomplemented
72#14519756C2312yes—
73#14519755—22no—
74#14519751—21no—
75#14519751—21no—
76#14519751C2×C412yes—
77#14519751C2×C412yescomplemented
78#14519751C2312yes—
79#14519750—22no—
80#14519749—22no—
81#14519749—22no—
82#14519746—22no—
83#14519745—22no—
84C4×Dic122C2×C412yescomplemented
85C4×D244C2×C412yescomplemented
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88C4×C22⋊D61—22no—
89#14519737—22no—
90#14519737—22no—
91#14519736C2×C412yescomplemented
92#14519734—22no—
93#14519733—22no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61C2×C411yescharacteristic
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758196—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61—21no—
102C2×#11758198—21no—
103C22⋊C4×D61—22no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684C2311yescharacteristic
112#14519684C2311yescharacteristic
113#14519684C2311yescharacteristic
114#14519684C2311yescharacteristic
115C22×D4×C61D411yescharacteristic
116C23×C4×C61C2311yescharacteristic
117C23×C4×C61C2311yescharacteristic
118C22×C8×C61—21no—
119C22×C8×C61—21no—
120C22×C8×C61—22no—
121C4×D4×C61C2×C412yes—
122C4×D4×C61C2×C412yes—
123C4×D4×C61D412yes—
124C4×D4×C61D412yes—
125C2×C4⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130C22⋊C8×C61—21no—
131C22⋊C8×C61—21no—
132C22⋊C8×C61—22no—
133C22⋊C8×C61C2312yes—
134C22⋊C8×C61C2312yes—
135C4×C8×C61—22no—
136C24×C61C2×D411yescharacteristic
137C24×C61C2×D411yescharacteristic
138C23×D61—21no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61C22⋊C412yes—
142C2×D4×C61C22⋊C412yes—
143C22×C4×C61—81no—
144C22×C4×C61—81no—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—42no—
161C2×C22⋊D61—42no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—42no—
170C22×Dic61—42no—
171C22×Dic61—81no—
172#11758222—82no—
173C2×D244—21no—
174C2×D244—81no—
175C2×D244C22×C412yes—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×Dic122—21no—
179C2×Dic122—81no—
180C2×Dic122C22×C412yes—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C2×C8×C61—81no—
184C2×C8×C61—81no—
185C2×C8×C61—82no—
186C4⋊C4×C61C22⋊C412yes—
187C4⋊C4×C61C22⋊C412yes—
188C22⋊C4×C61C22⋊C412yes—
189C22⋊C4×C61C22⋊C412yes—
190C22⋊C4×C61C22⋊C412yes—
191C22⋊C4×C61C22⋊C412yes—
192C42×C61C22×C412yes—
193C42×C61C2×D412yes—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61—22no—
198C22⋊Dic61—42no—
199C22⋊Dic61—42no—
200C22⋊Dic61—81no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—81no—
210C4⋊Dic61C22×C412yes—
211C4⋊Dic61C22×C412yes—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217#11758196—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—42no—
221C4×Dic61—81no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C2×C22⋊C412yes—
229C23×C61C2×C22⋊C411yescharacteristic
230C23×C61C2×C22⋊C411yescharacteristic
231C23×C61#13512yes—
232C23×C61#13512yes—
233C23×C61C22×D411yescharacteristic
234C22×D61—22no—
235C22×D61—42no—
236C22×D61—81no—
237C22×D61—81no—
238D4×C61—24no—
239D4×C61—24no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—41no—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C22⋊D61—42no—
260C22⋊D61—42no—
261C22⋊D61—44no—
262C22⋊D61—82no—
263C22⋊D61—82no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—42no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—81no—
281C2×Dic61—81no—
282C2×Dic61—84no—
283D244—42no—
284D244—81no—
285C4×D61—82no—
286C4×D61—82no—
287Dic122—42no—
288Dic122—81no—
289C8×C61—42no—
290C8×C61—42no—
291#17615—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61#41612yes—
306C22×C61#41612yes—
307C22×C61C22×C22⋊C411yescharacteristic
308C22×C61C2×#13511yescharacteristic
309C22×C61C2×#13511yescharacteristic
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—162no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—22no—
318C4×C61—21no—
319C4×C61—22no—
320C4×C61—22no—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
324Dic61—42no—
325Dic61—81no—
326Dic61—81no—
327Dic61—81no—
328#3505—611no—
329#2972D6111yescharacteristic, complemented
330#2353—611no—
331#1521—612no—
332#1520—611no—
333#1519—611no—
334#1516—612no—
335#1501—612no—
336#1500—612no—
337C2×#328—611no—
338C2×#327—611no—
339C2×C61—21no—
340C2×C61—21no—
341C2×C61—22no—
342C2×C61—22no—
343C2×C61—42no—
344C2×C61#288311yescharacteristic
345C2×C61C2×#41611yescharacteristic
346C2×C61#290111yescharacteristic
347D61—82no—
348#553—612no—
349#553—1222no—
350#552—612no—
351#551—1222no—
352#548—612no—
353#546—612no—
354#539—1221no—
355#538—1221no—
356#534—1221no—
357#529—612no—
358#525—612no—
359C2×C22⋊Q8—611no—
360C2×#121—611no—
361#518—1221no—
362C2×C4×D4C2×D6111yescharacteristic
363C8×D4—22no—
364C8×D4—22no—
365#431C2×D6112yescomplemented
366C2×C22⋊C8C2×D6111yescharacteristic
367C2×C22⋊C8C2×D6111yescharacteristic
368C2×C22⋊C8C2×D6112yescomplemented
369#388—1221no—
370#387—1221no—
371#386—1221no—
372#383—611no—
373#381—611no—
374#380—611no—
375C4×C22⋊C4—611no—
376C4×C22⋊C4—1221no—
377#376—611no—
378C2×#95—611no—
379#328—612no—
380#328—612no—
381#328—612no—
382#328—612no—
383#327—612no—
384#327—612no—
385#327—612no—
386#327—612no—
387C61#1761511yescharacteristic, complemented, Sylow
388C2×#41—612no—
389C22×Q8—611no—
390C22×D4—611no—
391C22×D4—1221no—
392C22×D4D24411yescharacteristic
393C23×C4—1221no—
394C23×C4C22×D6111yescharacteristic
395C23×C4C22×D6111yescharacteristic
396C22×C8—21no—
397C22×C8—21no—
398C22×C8—22no—
399#126—1222no—
400#126—1222no—
401#126—1222no—
402#125—612no—
403#125—1222no—
404#124—612no—
405#124—1222no—
406#123—1222no—
407#123—1222no—
408#123—1222no—
409#123—1222no—
410#123—1222no—
411C22⋊Q8—612no—
412C22⋊Q8—612no—
413C22⋊Q8—612no—
414C22⋊Q8—1221no—
415C22⋊Q8—1221no—
416C22⋊Q8—1222no—
417#121—612no—
418#121—612no—
419#121—612no—
420#121—1222no—
421#121—1222no—
422#121—1221no—
423#121—1221no—
424#121—1222no—
425#120—1222no—
426#120—1222no—
427C4×Q8—612no—
428C4×D4—612no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4C4×D6112yes—
435C4×D4C4×D6112yes—
436C4×D4C22⋊D6112yes—
437C4×D4C22⋊D6112yes—
438#117—612no—
439#117—1222no—
440#117—1222no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4D24411yescharacteristic
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4D24411yescharacteristic
453C2×C22⋊C4D24411yescharacteristic
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457C4⋊C8—22no—
458C22⋊C8—21no—
459C22⋊C8—21no—
460C22⋊C8—22no—
461C22⋊C8C22×D6112yes—
462C22⋊C8C22×D6112yes—
463C4×C8—22no—
464#95—611no—
465#95—611no—
466#95—611no—
467#95—611no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473C24—1221no—
474C24C2×D24411yescharacteristic
475C24C2×D24411yescharacteristic
476#41—4882no—
477C2×Q8—612no—
478C2×Q8—1221no—
479C2×Q8—4881no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—612no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—1221no—
488C2×D4—4881no—
489C2×D4—2442no—
490C2×D4—2442no—
491C2×D4#1175819812yes—
492C2×D4#1175819812yes—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—4881no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4C2×C4×D6111yescharacteristic
508C22×C4C2×C4×D6111yescharacteristic
509C22×C4C2×C4×D6112yes—
510C22×C4C2×D24411yescharacteristic
511C22×C4C2×D24411yescharacteristic
512C22×C4C2×D24411yescharacteristic
513C22×C4C2×D24411yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6112yes—
517C22×C4C23×D6111yescharacteristic
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—81no—
521C2×C8—81no—
522C2×C8—82no—
523C4⋊C4—612no—
524C4⋊C4—612no—
525C4⋊C4—1222no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—4881no—
534C4⋊C4#1175819812yes—
535C4⋊C4#1175819812yes—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—4881no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4#1175819812yes—
550C22⋊C4#1175819812yes—
551C22⋊C4#1175819812yes—
552C22⋊C4#1175819812yes—
553C42—1222no—
554C42—1222no—
555C42—4881no—
556C42—2442no—
557C42C2×C4×D6112yes—
558C42C2×C22⋊D6112yes—
559C23—21no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—1222no—
566C23—4881no—
567C23—4881no—
568C23—2442no—
569C23C2×#1175819812yes—
570C23C2×#1175819811yescharacteristic
571C23C2×#1175819811yescharacteristic
572C23#1451979912yes—
573C23#1451979912yes—
574C23C22×D24411yescharacteristic
575Q8—4881no—
576Q8—2442no—
577D4—24no—
578D4—24no—
579D4—4881no—
580D4—4882no—
581D4—4882no—
582D4—2442no—
583D4—2444no—
584D4—2442no—
585D4—2442no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—4882no—
603C2×C4—4882no—
604C2×C4—4884no—
605C2×C4—4881no—
606C2×C4—4881no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2442no—
618C2×C4—2441no—
619C2×C4C2×#1175819812yes—
620C2×C4C2×#1175819812yes—
621C2×C4C2×#1175819812yes—
622C2×C4C2×#1175819811yescharacteristic
623C2×C4C2×#1175819811yescharacteristic
624C2×C4C22×C4×D6111yescharacteristic
625C2×C4C22×C22⋊D6111yescharacteristic
626C8—42no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—9762no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22#1683975912yes—
646C22#1683975912yes—
647C22C22×#1175819811yescharacteristic
648C22C2×#1451979911yescharacteristic
649C22C2×#1451979911yescharacteristic
650C4—22no—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—21no—
655C4—21no—
656C4—22no—
657C4—4881no—
658C4—4881no—
659C4—4881no—
660C4—2442no—
661C4—2442no—
662C4—2442no—
663C4—2442no—
664C2—21no—
665C2—21no—
666C2—22no—
667C2—22no—
668C2—42no—
669C2—4882no—
670C2#2145427911yescharacteristic
671C2C2×#1683975911yescharacteristic
672C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2972C2
socleC22×C61C22×C22⋊C4
prime radical#21459658C2
cosocleC22×C4×C61C24
hypercenter#2972D61
hypocenterC61#17615
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21448759C2semidirect
3. #21444397C2semidirect
4. #21444392C2semidirect
5. #21444388C2semidirect
6. #21444376C2semidirect
7. #21444311C2semidirect
8. #21444306C2semidirect
9. C2×#16839500C2semidirect
10. C2×#16839497C2semidirect
11. C61×#2972C2semidirect
13. #16840458C4semidirect
14. #16840452C4semidirect
16. #16840421C4semidirect
17. #16840408C4semidirect
29. #16839630C22semidirect
30. #16839625C22semidirect
33. C4×C22⋊Dic61C22semidirect
34. #16839608C22semidirect
36. #16839500C22semidirect
37. #16839500C22semidirect
38. #16839500C22semidirect
39. #16839500C22semidirect
47. C61×#431C22semidirect
50. C2×C22⋊C8×C61C22semidirect
51. C2×#11758222C2×C4semidirect
59. #14519769C2×C4semidirect
61. #14519765C2×C4semidirect
71. #14519756C2×C4semidirect
77. #14519751C2×C4semidirect
84. C4×Dic122C2×C4semidirect
85. C4×D244C2×C4semidirect
91. #14519736C2×C4semidirect
329. #2972D61semidirect
365. #431C2×D61semidirect
368. C2×C22⋊C8C2×D61semidirect
387. C61#17615semidirect

Maximal subgroups

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

subgroupindex
#214596582
#214487592
#214443972
#214443922
#214443882
#214443762
#214443112
#214443062
C2×#168395002
C2×#168394972
C61×#29722
#1761561