Finite Group Databasebeta

← #29648183

Subgroups — #29648183

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459780C211yesmaximal, characteristic
2#21454839C211yesmaximal, characteristic, complemented
3#21444451C212yesmaximal, complemented
4#21444442C212yesmaximal, complemented
5#21444429C211yesmaximal, characteristic, complemented
6#21444427C212yesmaximal, complemented
7#21444424C211yesmaximal, characteristic, complemented
8#21444418C212yesmaximal, complemented
9#21444410C211yesmaximal, characteristic, complemented
10#21444405C211yesmaximal, characteristic, complemented
11C61×#2923C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic
13#16840485C411yescharacteristic
14#16840475—21no—
15#16840436—22no—
16D4×Dic122C411yescharacteristic
17#16840421C412yes—
18#16840408C412yes—
19#16840400—21no—
20D4×D244C411yescharacteristic
21#16840384—21no—
22#16840345—22no—
23#16840312C2211yescharacteristic
24#16840296C2211yescharacteristic
25#16840296C2212yes—
26C2×#14519765C2212yes—
27#16840234—21no—
28Q8×#92418—21no—
29#16839840—21no—
30#16839840—22no—
31#16839819C2211yescharacteristic, complemented
32#16839819C2212yescomplemented
33C4×#11758194C2211yescharacteristic, complemented
34C4×#11758194C2212yescomplemented
35#16839503C2211yescharacteristic, complemented
36#16839503C2211yescharacteristic, complemented
37#16839503C2212yescomplemented
38#16839501C2211yescharacteristic
39#16839501C2211yescharacteristic
40#16839501C2212yes—
41#16839501C2212yescomplemented
42#16839501C2212yescomplemented
43#16839501C2212yescomplemented
44#16839501C2212yescomplemented
45C4×#41×C61C2211yescharacteristic
46C61×#433—22no—
47C61×#432C2212yescomplemented
48C61×#432C2212yescomplemented
49C61×#424—21no—
50C61×#408—21no—
51C4×M16×C61C2211yescharacteristic, complemented
52C2×#97×C61C2211yescharacteristic, complemented
53C2×#11758222C2×C412yes—
54C22×Dic122C2×C412yes—
55C22×D244C2×C412yes—
56C2×D4×D61—21no—
57C2×#11758194—22no—
58C2×#11758194—21no—
59#14519777C2×C411yescharacteristic
60#14519776—21no—
61#14519774C2×C411yescharacteristic
62#14519772—22no—
63#14519771—22no—
64#14519771—21no—
65#14519769C2311yescharacteristic
66#14519766—21no—
67#14519765—22no—
68#14519765—22no—
69#14519765C2×C412yes—
70#14519765C2×C412yes—
71#14519765C2311yescharacteristic
72#14519765C2312yes—
73#14519763—22no—
74#14519762—22no—
75#14519762—22no—
76#14519760—21no—
77#14519760—22no—
78#14519756C2×C412yes—
79#14519756C2×C411yescharacteristic
80#14519755—22no—
81#14519754—22no—
82#14519753—21no—
83#14519752—21no—
84#14519751—21no—
85#14519751—21no—
86#14519751C2×C412yes—
87#14519751C2×C411yescharacteristic
88#14519750—22no—
89#14519749—22no—
90#14519747—22no—
91#14519745—22no—
92C4×Dic122C2×C412yes—
93C4×Dic122C2×C411yescharacteristic
94C4×D244—21no—
95C4×D244—21no—
96C4×D244C2×C411yescharacteristic
97C4×D244C2×C412yes—
98D4×Dic61—21no—
99#14519739—21no—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—22no—
102#14519737—22no—
103#14519735—21no—
104#14519734—22no—
105#14519734—21no—
106C2×#11758198—22no—
107C2×#11758198—22no—
108C22⋊C4×D61—22no—
109C4⋊#92418—21no—
110C4⋊#92418—22no—
111C22⋊#92418—22no—
112C22⋊#92418—21no—
113#14519688C2311yescharacteristic
114#14519688C2311yescharacteristic
115#14519688C2312yes—
116C4×#92418—21no—
117C4×#92418—22no—
118C2×#41×C61D411yescharacteristic
119C2×M16×C61—21no—
120C22×C8×C61—21no—
121C4×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2×C412yes—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127#117×C61D411yescharacteristic
128#117×C61D412yes—
129C2×C42×C61C2312yes—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—22no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135#97×C61—21no—
136#97×C61C2311yescharacteristic
137#97×C61C2311yescharacteristic
138#97×C61C2312yes—
139C4×C8×C61—21no—
140C23×D61—22no—
141#41×C61—42no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C412yes—
145C22×C4×C61C22×C412yes—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—21no—
154C22×Dic61—22no—
155D4×D61—44no—
156#11758222—82no—
157C2×D244—21no—
158C2×D244—22no—
159C2×D244—21no—
160C2×D244—82no—
161C2×D244C22×C411yescharacteristic
162C2×D244C22×C412yes—
163C2×C4×D61—21no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×Dic122—22no—
167C2×Dic122—82no—
168C2×Dic122C22×C411yescharacteristic
169C2×Dic122C22×C412yes—
170M16×C61—42no—
171C2×C8×C61—22no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—42no—
175C4⋊C4×C61C22⋊C411yescharacteristic
176C4⋊C4×C61C22⋊C412yes—
177C4⋊C4×C61C22⋊C411yescharacteristic
178C4⋊C4×C61C22⋊C412yes—
179C22⋊C4×C61C22⋊C412yes—
180C22⋊C4×C61C22⋊C412yes—
181C22⋊C4×C61C22⋊C412yes—
182C42×C61C22×C412yes—
183C42×C61C22×C411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D412yes—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D412yes—
188C42×C61C2411yescharacteristic
189C22⋊Dic61—22no—
190C22⋊Dic61—21no—
191#11758198—21no—
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—21no—
199#11758198—21no—
200#11758198—21no—
201#11758198—22no—
202#11758198—22no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C22×C411yescharacteristic
206C4⋊Dic61C22×C411yescharacteristic
207#11758196—21no—
208#11758196—22no—
209#11758196—22no—
210#11758196—21no—
211#11758196—21no—
212#11758196—21no—
213C4×Dic61—22no—
214C4×Dic61—21no—
215#11758194—44no—
216#11758194—42no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—22no—
220C23×C61C2×C22⋊C412yes—
221C23×C61C2×C22⋊C411yescharacteristic
222C22×D61—21no—
223C22×D61—22no—
224C22×D61—44no—
225C22×D61—82no—
226C22×D61—82no—
227Q8×C61—22no—
228D4×C61—22no—
229D4×C61—42no—
230C2×C4×C61—22no—
231C2×C4×C61—42no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C22⋊D61—44no—
246C22⋊D61—82no—
247C22⋊D61—81no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—82no—
253D244—41no—
254D244—42no—
255D244—42no—
256D244—81no—
257D244—82no—
258C4×D61—42no—
259C4×D61—82no—
260C4×D61—82no—
261Dic122—41no—
262Dic122—42no—
263Dic122—82no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—42no—
267#92418—22no—
268#92418—41no—
269#92418—42no—
270#17681—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—21no—
273C22×C61—22no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—22no—
277C22×C61C22×C22⋊C411yescharacteristic
278C2×D61—42no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—41no—
284C2×D61—164no—
285C4×C61—21no—
286C4×C61—22no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—22no—
292C4×C61—21no—
293C4×C61#42112yes—
294Dic61—41no—
295Dic61—42no—
296Dic61—41no—
297Dic61—41no—
298#3514—611no—
299#2971—611no—
300#2923D6111yescharacteristic, complemented
301#1535—612no—
302#1533—612no—
303#1530—611no—
304#1529—612no—
305#1529—611no—
306#1527—612no—
307#1525—611no—
308#1524—611no—
309C2×C61—42no—
310C2×C61—41no—
311C2×C61#288811yescharacteristic
312C2×C61C2×#42111yescharacteristic
313C2×C61#290111yescharacteristic
314D61—82no—
315D61—81no—
316#559—611no—
317#556—611no—
318#555—1221no—
319#551—1222no—
320D4×Q8—611no—
321#548—612no—
322#546—612no—
323#546—1221no—
324D42—611no—
325#542—1221no—
326#538—1222no—
327#532—611no—
328#529—611no—
329#529—612no—
330C2×#124—612no—
331#518—1221no—
332C4×#41C2×D6111yescharacteristic
333C8×Q8—1221no—
334#436—1221no—
335#436—1222no—
336#433—22no—
337#432—611no—
338#432—612no—
339#432C2×D6112yescomplemented
340#432C2×D6112yescomplemented
341#424—21no—
342#408—21no—
343C4×M16—611no—
344C4×M16—612no—
345C4×M16C2×D6111yescharacteristic, complemented
346C2×#97C2×D6111yescharacteristic, complemented
347#330—611no—
348#330—611no—
349#330—612no—
350#329—611no—
351#329—611no—
352#329—612no—
353#329—612no—
354#329—612no—
355#329—612no—
356#329—612no—
357C61#1768111yescharacteristic, complemented, Sylow
358C2×#41—612no—
359C2×#41C22⋊D6111yescharacteristic
360C22×Q8—612no—
361C22×D4—612no—
362C22×D4—1221no—
363C2×M16—21no—
364C2×M16—1222no—
365C2×M16—1221no—
366C22×C8—21no—
367C4⋊Q8—611no—
368C4⋊Q8—1221no—
369#127—611no—
370#126—1222no—
371#126—1222no—
372#126—1221no—
373#125—611no—
374#124—612no—
375#124—612no—
376#124—611no—
377#124—612no—
378#124—1221no—
379#124—1222no—
380#124—1222no—
381#124—1222no—
382#123—1221no—
383#123—1222no—
384#123—1222no—
385#123—1222no—
386C22⋊Q8—612no—
387C22⋊Q8—611no—
388C22⋊Q8—1221no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391#121—612no—
392#121—611no—
393#121—1222no—
394#121—1221no—
395#121—1221no—
396#121—1221no—
397#121—1222no—
398#120—1222no—
399#120—1222no—
400C4×Q8—612no—
401C4×Q8—611no—
402C4×Q8C4×D6111yescharacteristic
403C4×Q8D24411yescharacteristic
404C4×D4—611no—
405C4×D4—612no—
406C4×D4—1221no—
407C4×D4—1222no—
408C4×D4—1221no—
409C4×D4—1221no—
410C4×D4—1222no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4C4×D6111yescharacteristic
414C4×D4C4×D6112yes—
415C4×D4D24411yescharacteristic
416C4×D4D24412yes—
417#117—1221no—
418#117—1221no—
419#117—1222no—
420#117C22⋊D6112yes—
421#117C22⋊D6111yescharacteristic
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1222no—
425C2×C42C22×D6112yes—
426C2×C42C22×D6111yescharacteristic
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8—1221no—
430C4⋊C8—1222no—
431C22⋊C8—22no—
432C22⋊C8—22no—
433C22⋊C8—1222no—
434C22⋊C8—1221no—
435#97—21no—
436#97—611no—
437#97—611no—
438#97—612no—
439#97C22×D6111yescharacteristic
440#97C22×D6111yescharacteristic
441#97C22×D6112yes—
442C4×C8—21no—
443C4×C8—1221no—
444C4×C8—1222no—
445C24—1222no—
446#41—42no—
447#41—4882no—
448C2×Q8—611no—
449C2×Q8—612no—
450C2×Q8—1222no—
451C2×Q8—4882no—
452C2×Q8#1175819811yescharacteristic
453C2×D4—611no—
454C2×D4—612no—
455C2×D4—1221no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1221no—
459C2×D4—1222no—
460C2×D4—1221no—
461C2×D4—4882no—
462C2×D4—2444no—
463C2×D4#1175819811yescharacteristic
464C2×D4#1175819812yes—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4C2×C4×D6112yes—
470C22×C4C2×C4×D6111yescharacteristic
471C22×C4C2×D24412yes—
472C22×C4C2×D24411yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6112yes—
475M16—42no—
476M16—2444no—
477M16—2442no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1222no—
485C4⋊C4—611no—
486C4⋊C4—611no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4#1175819811yescharacteristic
496C4⋊C4#1175819811yescharacteristic
497C4⋊C4#1175819812yes—
498C4⋊C4#1175819812yes—
499C22⋊C4—1222no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—1222no—
503C22⋊C4—1221no—
504C22⋊C4—1222no—
505C22⋊C4—1221no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1221no—
512C22⋊C4—1222no—
513C22⋊C4#1175819812yes—
514C22⋊C4#1175819812yes—
515C22⋊C4#1175819812yes—
516C42—1222no—
517C42—1221no—
518C42C2×C4×D6112yes—
519C42C2×C4×D6111yescharacteristic
520C42C2×D24411yescharacteristic
521C42C2×D24412yes—
522C42C2×C22⋊D6111yescharacteristic
523C42C2×C22⋊D6112yes—
524C42C23×D6111yescharacteristic
525C23—1221no—
526C23—1222no—
527C23—4882no—
528C23—4882no—
529C23—2444no—
530C23C2×#1175819812yes—
531C23C2×#1175819811yescharacteristic
532Q8—22no—
533Q8—4882no—
534Q8—2441no—
535Q8—2442no—
536D4—22no—
537D4—42no—
538D4—4881no—
539D4—4882no—
540D4—4881no—
541D4—4882no—
542D4—2441no—
543D4—2442no—
544D4—2444no—
545D4—2442no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—4882no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—2442no—
560C2×C4C2×#1175819812yes—
561C2×C4C2×#1175819812yes—
562C2×C4C2×#1175819811yescharacteristic
563C2×C4C2×#1175819811yescharacteristic
564C2×C4C2×#1175819812yes—
565C2×C4C2×#1175819811yescharacteristic
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4C22×C22⋊D6111yescharacteristic
568C2×C4C22×D24411yescharacteristic
569C8—41no—
570C8—41no—
571C8—42no—
572C8—1222no—
573C8—2441no—
574C8—2442no—
575C22—21no—
576C22—22no—
577C22—21no—
578C22—22no—
579C22—22no—
580C22—21no—
581C22—9764no—
582C22—2441no—
583C22—2442no—
584C22—2442no—
585C22—2441no—
586C22—2442no—
587C22—2441no—
588C22C22×#1175819811yescharacteristic
589C4—21no—
590C4—22no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—22no—
597C4—2441no—
598C4—2441no—
599C4—2442no—
600C4—2441no—
601C4#1683978012yes—
602C2—42no—
603C2—41no—
604C2—4881no—
605C2—4882no—
606C2#2145429711yescharacteristic
607C2C2×#1683978011yescharacteristic
608C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2923C2
socleC22×C61C22×C22⋊C4
prime radical#21459780C2
cosocleC42×C61C24
hypercenter#2923D61
hypocenterC61#17681
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2923

Complements

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

NH ≅ G/Ntype
2. #21454839C2semidirect
3. #21444451C2semidirect
4. #21444442C2semidirect
5. #21444429C2semidirect
6. #21444427C2semidirect
7. #21444424C2semidirect
8. #21444418C2semidirect
9. #21444410C2semidirect
10. #21444405C2semidirect
11. C61×#2923C2semidirect
31. #16839819C22semidirect
32. #16839819C22semidirect
33. C4×#11758194C22semidirect
34. C4×#11758194C22semidirect
35. #16839503C22semidirect
36. #16839503C22semidirect
37. #16839503C22semidirect
41. #16839501C22semidirect
42. #16839501C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
47. C61×#432C22semidirect
48. C61×#432C22semidirect
51. C4×M16×C61C22semidirect
52. C2×#97×C61C22semidirect
300. #2923D61semidirect
339. #432C2×D61semidirect
340. #432C2×D61semidirect
345. C4×M16C2×D61semidirect
346. C2×#97C2×D61semidirect
357. C61#17681semidirect

Maximal subgroups

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

subgroupindex
#214597802
#214548392
#214444512
#214444422
#214444292
#214444272
#214444242
#214444182
#214444102
#214444052
C61×#29232
#1768161