Finite Group Databasebeta

← #29647801

Subgroups — #29647801

Order 15616.

1014 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#21459660C211yesmaximal, characteristic
2#21454858C211yesmaximal, characteristic, complemented
3#21448762C211yesmaximal, characteristic, complemented
4#21444395C212yesmaximal, complemented
5#21444361C212yesmaximal, complemented
6#21444352C211yesmaximal, characteristic, complemented
7#21444349C211yesmaximal, characteristic, complemented
8#21444334C211yesmaximal, characteristic, complemented
9#21444322C212yesmaximal, complemented
10#21444304C212yesmaximal, complemented
11C61×#1508C211yesmaximal, characteristic, complemented
12#16840460—22no—
13#16840460C412yescomplemented
14#16840454C412yescomplemented
15#16840439—22no—
16#16840423C412yescomplemented
17#16840403C412yescomplemented
18#16840361—21no—
19#16840352—21no—
20#16840324—21no—
21#16840298C2212yes—
22#16840275C2212yes—
23C2×#14519751C2211yescharacteristic
24#16840235—21no—
25C2×C4×C22⋊D61C2211yescharacteristic
26#16839838—22no—
27#16839837—22no—
28#16839812C2212yescomplemented
29#16839726C2211yescharacteristic
30#16839724C2212yescomplemented
31#16839650—21no—
32#16839646—21no—
33#16839638—21no—
34#16839629C2211yescharacteristic, complemented
35#16839625C2211yescharacteristic
36#16839621C2211yescharacteristic, complemented
37C4×#11758198—21no—
38#16839607C2211yescharacteristic, complemented
39#95×D61C2211yescharacteristic, complemented
40#16839499C2212yescomplemented
41#16839499C2212yescomplemented
42#16839497C2212yes—
43#16839497C2212yes—
44#16839495C2212yes—
45#16839495C2212yes—
46C2×C22⋊Q8×C61C2211yescharacteristic
47C61×#407C2211yescharacteristic
48C4×C22⋊C4×C61C2211yescharacteristic
49C61×#328C2212yescomplemented
50C61×#328C2212yescomplemented
51C2×#11758226C2×C412yes—
52C22×D244C2×C411yescharacteristic, complemented
53C22×C22⋊D61—21no—
54C22×C22⋊D61D411yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61C2311yescharacteristic
57C2×#11758194—21no—
58C2×#11758189—22no—
59#14519772—22no—
60#14519772—22no—
61#14519772—22no—
62#14519768—22no—
63#14519768C2×C412yescomplemented
64#14519766—22no—
65#14519766C2×C412yescomplemented
66#14519762—22no—
67#14519762—22no—
68#14519762—22no—
69#14519761—22no—
70#14519761—22no—
71#14519753—22no—
72#14519753C2×C412yes—
73#14519752—22no—
74#14519752C2×C412yes—
75#14519751—21no—
76#14519751—21no—
77#14519751—22no—
78#14519751—22no—
79#14519751C2×C412yescomplemented
80#14519751C2312yes—
81#14519747—22no—
82#14519747—22no—
83Q8×Dic61C2×C412yescomplemented
84#14519739—22no—
85#14519739C2×C412yescomplemented
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88C4×C22⋊D61C2×C412yes—
89C4×C22⋊D61C2×C412yes—
90C4×C22⋊D61D412yes—
91C4×C22⋊D61D412yes—
92#14519737—22no—
93#14519737—22no—
94#14519735—22no—
95#14519735C2×C412yes—
96#14519733—22no—
97C2×C4⋊Dic61C2×C411yescharacteristic, complemented
98C2×#11758196—21no—
99C2×#11758196D411yescharacteristic
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61D411yescharacteristic
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×#11758198D411yescharacteristic
105C22⋊C4×D61—22no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108#14519706—22no—
109C22⋊#92418—21no—
110C22⋊#92418C2312yes—
111#14519689—22no—
112C8×Dic61—22no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684C2311yescharacteristic
119#14519684C2311yescharacteristic
120C22×Q8×C61C2×C411yescharacteristic, complemented
121C23×C4×C61C2311yescharacteristic
122C2×M16×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61C2×C412yescomplemented
126C22⋊Q8×C61C2312yes—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61—21no—
131C2×C42×C61—21no—
132C22⋊C8×C61—21no—
133C22⋊C8×C61C2312yes—
134#95×C61C2311yescharacteristic
135#95×C61C2311yescharacteristic
136C24×C61C2×D411yescharacteristic
137C23×D61—21no—
138C23×D61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—81no—
141C2×Q8×C61C22×C412yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—81no—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—42no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61C22⋊C412yes—
160C2×C22⋊D61C22⋊C412yes—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167#11758226—82no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—21no—
171C2×D244—81no—
172C2×D244C22×C412yes—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—81no—
178C2×C4×D61—81no—
179C2×C4×D61C22×C412yes—
180C2×C4×D61C2×D412yes—
181M16×C61—81no—
182C2×C8×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C22×C412yes—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—42no—
190C42×C61—81no—
191C22⋊Dic61—81no—
192C22⋊Dic61C22⋊C412yes—
193C22⋊Dic61C22⋊C412yes—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—42no—
203#11758198C22⋊C412yes—
204#11758198C22⋊C412yes—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61C22×C412yes—
209#11758196—22no—
210#11758196—22no—
211#11758196—81no—
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—42no—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219#11758194—81no—
220C2×#92418—22no—
221#11758189—82no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61C2×C22⋊C411yescharacteristic
227C23×C61C2×C22⋊C411yescharacteristic
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—22no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—42no—
235C22×D61—81no—
236C22×D61—81no—
237C22×D61C2×C22⋊C412yes—
238Q8×C61—42no—
239Q8×C61—81no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—81no—
255C2×C4×C61—81no—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C22⋊D61—42no—
261C22⋊D61—42no—
262C22⋊D61—44no—
263C22⋊D61—42no—
264C22⋊D61—42no—
265C2×Dic61—21no—
266C2×Dic61—21no—
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×Dic61C2×C22⋊C412yes—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280D244—42no—
281D244—81no—
282D244—82no—
283D244—82no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61—82no—
287C4×D61—82no—
288C4×D61—84no—
289C8×C61—42no—
290#92418—42no—
291#17623—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61C22×C22⋊C411yescharacteristic
303C22×C61#57511yescharacteristic
304C22×C61#57511yescharacteristic
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—162no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61—81no—
322Dic61—22no—
323Dic61—22no—
324Dic61—22no—
325Dic61—42no—
326Dic61—81no—
327Dic61—81no—
328#3505—611no—
329#2975—611no—
330#2353—611no—
331#1521—612no—
332#1513—612no—
333#1511—611no—
334#1510—611no—
335#1508D6111yescharacteristic, complemented
336#1507—611no—
337#1504—612no—
338#1500—612no—
339C2×C61—21no—
340C2×C61—21no—
341C2×C61—41no—
342C2×C61—41no—
343C2×C61#288311yescharacteristic
344C2×C61#289611yescharacteristic
345C2×C61#290111yescharacteristic
346D61—42no—
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⋊Q8C2×D6111yescharacteristic
360C2×#121—611no—
361#518—1221no—
362C2×C4×D4—611no—
363#436—1222no—
364#436—1222no—
365#431—612no—
366#407—611no—
367#407—612no—
368#407C2×D6111yescharacteristic
369#388—1221no—
370#387—1221no—
371#386—1221no—
372#383—611no—
373#381—611no—
374#380—611no—
375C4×C22⋊C4—1221no—
376C4×C22⋊C4C2×D6111yescharacteristic
377#376—611no—
378C2×#95—611no—
379#328—612no—
380#328—612no—
381#328C2×D6112yescomplemented
382#328C2×D6112yescomplemented
383#327—612no—
384#327—612no—
385#327—612no—
386#327—612no—
387C61#1762311yescharacteristic, complemented, Sylow
388C2×#41—612no—
389C22×Q8C4×D6111yescharacteristic, complemented
390C22×D4—611no—
391C22×D4—611no—
392C22×D4—1221no—
393C23×C4—611no—
394C23×C4—1221no—
395C23×C4C22×D6111yescharacteristic
396C2×M16—21no—
397C2×M16—1222no—
398C2×M16—1221no—
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—21no—
412C22⋊Q8—21no—
413C22⋊Q8—612no—
414C22⋊Q8—1222no—
415C22⋊Q8C4×D6112yescomplemented
416C22⋊Q8C22×D6112yes—
417#121—612no—
418#121—612no—
419#121—612no—
420#121—1222no—
421#121—1222no—
422#121—1222no—
423#121—1221no—
424#121—1221no—
425#120—1222no—
426#120—1222no—
427C4×Q8—612no—
428C4×D4—612no—
429C4×D4—612no—
430C4×D4—612no—
431C4×D4—612no—
432C4×D4—612no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—1222no—
438#117—612no—
439#117—1222no—
440#117—1222no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4C4×D6111yescharacteristic, complemented
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—611no—
449C2×C22⋊C4—611no—
450C2×C22⋊C4—1222no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C42—21no—
455C2×C42—611no—
456C2×C42—1221no—
457C4⋊C8—1222no—
458C22⋊C8—21no—
459C22⋊C8—612no—
460C22⋊C8—1222no—
461C22⋊C8—1221no—
462C22⋊C8C22×D6112yes—
463C4×C8—1222no—
464#95—611no—
465#95—611no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95C22×D6111yescharacteristic
472#95C22×D6111yescharacteristic
473C24—611no—
474C24—1221no—
475C24D4×D6111yescharacteristic
476#41—4882no—
477C2×Q8—21no—
478C2×Q8—81no—
479C2×Q8C2×C4×D6112yes—
480C2×D4—612no—
481C2×D4—612no—
482C2×D4—612no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—1221no—
488C2×D4—4881no—
489C2×D4—2441no—
490C2×D4—2441no—
491C2×D4—2442no—
492C2×D4—2442no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—81no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—612no—
502C22×C4—612no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—4881no—
509C22×C4—4881no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4C2×C4×D6111yescharacteristic
513C22×C4C2×C4×D6111yescharacteristic
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518M16—81no—
519M16—4882no—
520M16—4881no—
521C2×C8—22no—
522C2×C8—1222no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—612no—
527C4⋊C4—612no—
528C4⋊C4—612no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1221no—
534C4⋊C4—4881no—
535C4⋊C4C2×C4×D6112yes—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—42no—
539C22⋊C4—612no—
540C22⋊C4—612no—
541C22⋊C4—612no—
542C22⋊C4—612no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—4881no—
552C22⋊C4—2442no—
553C42—81no—
554C42—612no—
555C42—612no—
556C42—1222no—
557C42—1222no—
558C42—2442no—
559C23—21no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23—612no—
564C23—1221no—
565C23—1221no—
566C23—1222no—
567C23—4881no—
568C23—4881no—
569C23—2441no—
570C23—2441no—
571C23—2442no—
572C23C22⋊C4×D6111yescharacteristic
573C23C22⋊C4×D6111yescharacteristic
574C23C2×D4×D6111yescharacteristic
575Q8—42no—
576Q8—81no—
577D4—4881no—
578D4—4882no—
579D4—4882no—
580D4—2442no—
581D4—2442no—
582D4—2442no—
583D4—2444no—
584D4—2442no—
585D4—2442no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1222no—
608C2×C4—4882no—
609C2×C4—4882no—
610C2×C4—4884no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2442no—
622C2×C4C22⋊C4×D6111yescharacteristic
623C2×C4C22⋊C4×D6111yescharacteristic
624C2×C4C22×C4×D6111yescharacteristic
625C2×C4C2×D4×D6111yescharacteristic
626C8—42no—
627C8—2442no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—81no—
637C22—81no—
638C22—1222no—
639C22—1222no—
640C22—1222no—
641C22—9762no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22C2×C22⋊C4×D6111yescharacteristic
648C22#1684056111yescharacteristic
649C22#1684056111yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—81no—
658C4—1222no—
659C4—1222no—
660C4—1222no—
661C4—4881no—
662C4—4881no—
663C4—2442no—
664C2—21no—
665C2—21no—
666C2—41no—
667C2—41no—
668C2—4882no—
669C2—2442no—
670C2#2145427711yescharacteristic
671C2#2145433711yescharacteristic
672C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1508C2
socleC22×C61C22×C22⋊C4
prime radical#21459660C2
cosocleC22×C4×C61C24
hypercenter#1508D61
hypocenterC61#17623
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
2. #21454858C2semidirect
3. #21448762C2semidirect
4. #21444395C2semidirect
5. #21444361C2semidirect
6. #21444352C2semidirect
7. #21444349C2semidirect
8. #21444334C2semidirect
9. #21444322C2semidirect
10. #21444304C2semidirect
11. C61×#1508C2semidirect
13. #16840460C4semidirect
14. #16840454C4semidirect
16. #16840423C4semidirect
17. #16840403C4semidirect
28. #16839812C22semidirect
30. #16839724C22semidirect
34. #16839629C22semidirect
36. #16839621C22semidirect
38. #16839607C22semidirect
39. #95×D61C22semidirect
40. #16839499C22semidirect
41. #16839499C22semidirect
49. C61×#328C22semidirect
50. C61×#328C22semidirect
52. C22×D244C2×C4semidirect
63. #14519768C2×C4semidirect
65. #14519766C2×C4semidirect
79. #14519751C2×C4semidirect
83. Q8×Dic61C2×C4semidirect
85. #14519739C2×C4semidirect
97. C2×C4⋊Dic61C2×C4semidirect
120. C22×Q8×C61C2×C4semidirect
125. C22⋊Q8×C61C2×C4semidirect
128. C2×C4⋊C4×C61C2×C4semidirect
335. #1508D61semidirect
381. #328C2×D61semidirect
382. #328C2×D61semidirect
387. C61#17623semidirect
389. C22×Q8C4×D61semidirect
415. C22⋊Q8C4×D61semidirect
445. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596602
#214548582
#214487622
#214443952
#214443612
#214443522
#214443492
#214443342
#214443222
#214443042
C61×#15082
#1762361