Finite Group Databasebeta

← #29647811

Subgroups — #29647811

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic
2#21454904C211yesmaximal, characteristic, complemented
3#21444394C212yesmaximal, complemented
4#21444369C211yesmaximal, characteristic, complemented
5#21444365C211yesmaximal, characteristic
6#21444355C212yesmaximal, complemented
7#21444342C212yesmaximal, complemented
8#21444330C211yesmaximal, characteristic, complemented
9#21444326C211yesmaximal, characteristic
10#21444303C212yesmaximal, complemented
11C61×#2356C211yesmaximal, characteristic, complemented
12#16840497C412yescomplemented
13#16840490C412yescomplemented
14#16840485C412yescomplemented
15#16840484—22no—
16#16840472C412yescomplemented
17#16840453—22no—
18#16840359—21no—
19#16840359—21no—
20#16840336—21no—
21#16840319C2211yescharacteristic
22#16840312C2211yescharacteristic
23#16840296C2212yes—
24#16840296C2212yes—
25#16840243—21no—
26#16839893—22no—
27#16839893—22no—
28#16839819C2212yes—
29#16839819C2211yescharacteristic
30#16839819C2211yescharacteristic
31#16839815C2212yes—
32#16839498C2212yes—
33#16839498C2212yes—
34#16839498C2212yes—
35#16839498C2212yes—
36#16839495C2212yes—
37#16839495C2212yes—
38#16839495C2212yes—
39#16839495C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#391—21no—
42C61×#387—21no—
43C61×#387—21no—
44C61×#383C2211yescharacteristic
45C61×#383C2211yescharacteristic
46C61×#382C2211yescharacteristic
47C4×C4⋊C4×C61—21no—
48C4×C4⋊C4×C61C2211yescharacteristic
49C61×#376C2211yescharacteristic
50C61×#376C2211yescharacteristic
51C2×#11758222—21no—
52#14519777C2×C411yescharacteristic
53#14519777C2×C411yescharacteristic
54#14519777C2×C411yescharacteristic
55#14519775—22no—
56#14519774C2×C411yescharacteristic
57#14519772—22no—
58#14519772—22no—
59#14519771—22no—
60#14519771—22no—
61#14519769—21no—
62#14519769C2×C412yes—
63#14519769C2×C412yes—
64#14519767—22no—
65#14519767—22no—
66#14519766—22no—
67#14519765—21no—
68#14519765C2×C412yes—
69#14519765C2×C412yes—
70#14519762—22no—
71#14519762—22no—
72#14519760—22no—
73#14519760—22no—
74#14519756—21no—
75#14519756C2312yes—
76#14519755—22no—
77#14519755—22no—
78#14519753—22no—
79#14519753—22no—
80#14519751—21no—
81#14519751C2312yes—
82C4×Dic122—21no—
83C4×Dic122C2×C412yes—
84C4×Dic122C2×C412yes—
85C4×Dic122C2×C412yes—
86#14519742—22no—
87C4×D244—21no—
88C4×D244C2×C412yes—
89C4×D244C2×C412yes—
90C4×D244C2×C412yes—
91#14519739—22no—
92C4×C22⋊D61—22no—
93C4×C22⋊D61—22no—
94#14519736—21no—
95#14519735—22no—
96#14519734—22no—
97#14519734—22no—
98C4⋊C4×D61—22no—
99C4⋊#92418—22no—
100C4⋊#92418—22no—
101C4⋊#92418—21no—
102C4⋊#92418—21no—
103C22⋊#92418C2312yes—
104C22⋊#92418C2312yes—
105#14519688—22no—
106#14519688—21no—
107#14519688—21no—
108C4×#92418—22no—
109C22×Q8×C61D411yescharacteristic
110C4×Q8×C61C2×C412yes—
111C4×Q8×C61C2×C412yes—
112C4×Q8×C61D412yes—
113C4×Q8×C61D412yes—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61D411yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61C2311yescharacteristic
133#95×C61C2311yescharacteristic
134#95×C61C2311yescharacteristic
135#95×C61C2311yescharacteristic
136C2×Q8×C61—41no—
137C2×Q8×C61—41no—
138C2×Q8×C61C22⋊C412yes—
139C2×Q8×C61C22⋊C412yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—22no—
152#11758222—42no—
153#11758222—42no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244C22×C412yes—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122C22×C412yes—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61C22⋊C412yes—
166C4⋊C4×C61C22⋊C412yes—
167C4⋊C4×C61C22⋊C412yes—
168C4⋊C4×C61C22⋊C412yes—
169C4⋊C4×C61C22⋊C412yes—
170C4⋊C4×C61C22⋊C412yes—
171C42×C61—42no—
172C42×C61—42no—
173C42×C61—42no—
174C42×C61C22×C411yescharacteristic
175C42×C61C22×C412yes—
176C42×C61C22×C412yes—
177C42×C61C22×C411yescharacteristic
178C42×C61C2×D412yes—
179C42×C61C2×D412yes—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C22⋊Dic61—22no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C412yes—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212Q8×C61—42no—
213Q8×C61—42no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C22⋊D61—42no—
247C22⋊D61—42no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251D244—42no—
252D244—42no—
253D244—81no—
254C4×D61—44no—
255C4×D61—82no—
256C4×D61—82no—
257Dic122—42no—
258Dic122—42no—
259Dic122—81no—
260#92418—42no—
261#92418—42no—
262#17624—611nomaximal, Sylow
263C22×C61—21no—
264C22×C61—21no—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61#57511yescharacteristic
267C22×C61#57511yescharacteristic
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—42no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—22no—
276C4×C61—22no—
277C4×C61—21no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61—81no—
285Dic61—42no—
286Dic61—42no—
287Dic61—42no—
288#3507—611no—
289#2983—611no—
290#2356D6111yescharacteristic, complemented
291#1521—612no—
292#1515—611no—
293#1514—611no—
294#1512—612no—
295#1509—612no—
296#1506—611no—
297#1505—611no—
298#1500—612no—
299C2×C61—21no—
300C2×C61—21no—
301C2×C61#288411yescharacteristic
302C2×C61#289611yescharacteristic
303C2×C61#290111yescharacteristic
304D61—82no—
305#559—612no—
306#557—612no—
307#556—612no—
308#556—1222no—
309#554—612no—
310#552—1222no—
311#539—1221no—
312#539—1221no—
313#536—1221no—
314#533—611no—
315#532—611no—
316#529—612no—
317#529—612no—
318#520—1221no—
319C2×C4×Q8C2×D6111yescharacteristic
320#446—1222no—
321#446—1222no—
322#432—611no—
323#432—611no—
324#432—612no—
325#431—612no—
326#391—21no—
327#387—21no—
328#387—21no—
329#383C2×D6111yescharacteristic
330#383C2×D6111yescharacteristic
331#382C2×D6111yescharacteristic
332C4×C4⋊C4—21no—
333C4×C4⋊C4C2×D6111yescharacteristic
334#376C2×D6111yescharacteristic
335#376C2×D6111yescharacteristic
336#328—612no—
337#328—612no—
338#328—612no—
339#328—612no—
340#327—612no—
341#327—612no—
342#327—612no—
343#327—612no—
344C61#1762411yescharacteristic, complemented, Sylow
345C2×#41—1221no—
346C22×Q8C22⋊D6111yescharacteristic
347C4⋊Q8—611no—
348C4⋊Q8—611no—
349C4⋊Q8—611no—
350C4⋊Q8—1222no—
351#127—611no—
352#126—1222no—
353#126—1222no—
354#126—1222no—
355#126—1222no—
356#125—612no—
357#125—612no—
358#125—1222no—
359#125—1222no—
360#125—1221no—
361#124—612no—
362#124—612no—
363#124—1222no—
364#124—1221no—
365#123—1222no—
366#123—1222no—
367#123—1222no—
368#123—1222no—
369C22⋊Q8—612no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8—1221no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375#121—612no—
376#121—1221no—
377C4×Q8—612no—
378C4×Q8—612no—
379C4×Q8—612no—
380C4×Q8—1222no—
381C4×Q8—1221no—
382C4×Q8C4×D6112yes—
383C4×Q8C4×D6112yes—
384C4×Q8D24412yes—
385C4×Q8D24412yes—
386C4×D4—612no—
387C4×D4—612no—
388C4×D4—612no—
389C4×D4—1222no—
390C4×D4—1222no—
391C4×D4—1222no—
392C4×D4—1221no—
393#117—1221no—
394#117—1222no—
395#117—1222no—
396#117—1222no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4C22⋊D6111yescharacteristic
401C2×C4⋊C4C22⋊D6111yescharacteristic
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408C2×C42C22×D6111yescharacteristic
409C4⋊C8—1222no—
410C4⋊C8—1222no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C22⋊C8—612no—
414C22⋊C8—612no—
415#97—1222no—
416#97—1221no—
417#97—1221no—
418C4×C8—1222no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95C22×D6111yescharacteristic
427#95C22×D6111yescharacteristic
428#95C22×D6111yescharacteristic
429#95C22×D6111yescharacteristic
430#41—2442no—
431#41—2442no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—612no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8#1175819812yes—
438C2×Q8#1175819812yes—
439C2×D4—612no—
440C2×D4—1222no—
441C2×D4—1221no—
442C2×D4—1221no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457C2×C8—1222no—
458C2×C8—1222no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—612no—
462C4⋊C4—612no—
463C4⋊C4—1222no—
464C4⋊C4—1221no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4#1175819812yes—
476C4⋊C4#1175819812yes—
477C4⋊C4#1175819812yes—
478C4⋊C4#1175819812yes—
479C4⋊C4#1175819812yes—
480C4⋊C4#1175819812yes—
481C22⋊C4—1222no—
482C22⋊C4—1222no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—1222no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—1222no—
494C42—1222no—
495C42—1222no—
496C42C2×C4×D6111yescharacteristic
497C42C2×C4×D6112yes—
498C42C2×C4×D6112yes—
499C42C2×C4×D6111yescharacteristic
500C42C2×D24412yes—
501C42C2×D24412yes—
502C42C2×D24411yescharacteristic
503C42C2×D24411yescharacteristic
504C23—1222no—
505C23C22×C22⋊D6111yescharacteristic
506Q8—42no—
507Q8—42no—
508Q8—4881no—
509Q8—2442no—
510Q8—2442no—
511D4—4881no—
512D4—2442no—
513D4—2442no—
514D4—2442no—
515D4—2442no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—4882no—
542C2×C4—4882no—
543C2×C4—2444no—
544C2×C4C2×#1175819811yescharacteristic
545C2×C4C2×#1175819812yes—
546C2×C4C2×#1175819812yes—
547C2×C4C2×#1175819812yes—
548C2×C4C2×#1175819812yes—
549C2×C4C2×#1175819811yescharacteristic
550C2×C4C2×#1175819811yescharacteristic
551C2×C4C2×#1175819811yescharacteristic
552C2×C4C22×C4×D6111yescharacteristic
553C2×C4C22×D24411yescharacteristic
554C8—2442no—
555C8—2442no—
556C22—21no—
557C22—21no—
558C22—2442no—
559C22—2442no—
560C22—2442no—
561C22C22×#1175819811yescharacteristic
562C22#1684056011yescharacteristic
563C22#1684056011yescharacteristic
564C4—21no—
565C4—21no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—2442no—
579C4—2442no—
580C4—2442no—
581C2—21no—
582C2—21no—
583C2—4882no—
584C2#2145428211yescharacteristic
585C2#2145433611yescharacteristic
586C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2356C2
socleC22×C61C22×C22⋊C4
prime radical#16840312C22
cosocleC22×C4×C61C24
hypercenter#2356D61
hypocenterC61#17624
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
2. #21454904C2semidirect
3. #21444394C2semidirect
4. #21444369C2semidirect
6. #21444355C2semidirect
7. #21444342C2semidirect
8. #21444330C2semidirect
10. #21444303C2semidirect
11. C61×#2356C2semidirect
12. #16840497C4semidirect
13. #16840490C4semidirect
14. #16840485C4semidirect
16. #16840472C4semidirect
290. #2356D61semidirect
344. C61#17624semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214549042
#214443942
#214443692
#214443652
#214443552
#214443422
#214443302
#214443262
#214443032
C61×#23562
#1762461