Finite Group Databasebeta

← #29647591

Subgroups — #29647591

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459915C211yesmaximal, characteristic
2#21448470C211yesmaximal, characteristic, complemented
3#21444397C212yesmaximal
4#21444384C212yesmaximal, complemented
5#21444371C212yesmaximal, complemented
6#21444358C212yesmaximal
7#21444345C211yesmaximal, characteristic
8#21444332C211yesmaximal, characteristic, complemented
9#21444319C211yesmaximal, characteristic, complemented
10#21444306C211yesmaximal, characteristic
11C61×#2984C211yesmaximal, characteristic, complemented
12#16840491C411yescharacteristic, complemented
13#16840490C411yescharacteristic, complemented
14#16840490C412yescomplemented
15#16840485C411yescharacteristic, complemented
16#16840485C412yescomplemented
17#16840447—22no—
18#16840447—21no—
19Q8×D244C411yescharacteristic, complemented
20Q8×C22⋊D61—21no—
21#16840370—21no—
22#16840370—22no—
23#16840319C2212yes—
24#16840319C2211yescharacteristic
25#16840296C2212yes—
26#16840296C2211yescharacteristic
27#16840238—21no—
28#16839669—21no—
29#16839661—21no—
30#16839648—22no—
31#16839630C2212yes—
32#16839628C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839608C2211yescharacteristic
35#16839500C2212yes—
36#16839500C2212yes—
37#16839500C2211yescharacteristic
38#16839500C2212yes—
39#16839500C2212yes—
40#16839500C2211yescharacteristic
41#16839500C2212yes—
42#16839497C2211yescharacteristic
43#16839497C2212yes—
44#16839497C2211yescharacteristic
45C2×C4×Q8×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#446—22no—
48C8×Q8×C61—21no—
49C61×#432C2212yes—
50C61×#432C2211yescharacteristic
51C61×#431C2212yes—
52C61×#431C2211yescharacteristic
53C2×Q8×D61—21no—
54#14519777C2×C412yes—
55#14519777C2×C411yescharacteristic
56#14519777C2×C412yes—
57#14519777C2×C411yescharacteristic
58#14519775—21no—
59#14519775—22no—
60#14519772—22no—
61#14519772—22no—
62#14519772—22no—
63#14519771—22no—
64#14519771—21no—
65#14519769—22no—
66#14519769—21no—
67#14519769C2×C412yes—
68#14519769C2×C411yescharacteristic
69#14519768—22no—
70#14519768—21no—
71#14519767—22no—
72#14519767—22no—
73#14519767—22no—
74#14519765C2×C412yes—
75#14519765C2×C411yescharacteristic
76#14519760—21no—
77#14519760—22no—
78#14519757—22no—
79#14519757—21no—
80#14519756—21no—
81#14519756—22no—
82#14519756C2312yes—
83#14519756C2311yescharacteristic
84#14519753—21no—
85#14519753—22no—
86#14519751C2311yescharacteristic
87Q8×Dic61—21no—
88C4×Dic122—21no—
89C4×Dic122—22no—
90C4×Dic122C2×C412yes—
91C4×Dic122C2×C412yes—
92C4×Dic122C2×C412yes—
93C4×Dic122C2×C412yes—
94C4×Dic122C2×C411yescharacteristic
95#14519742—22no—
96#14519742—21no—
97C4×D244C2×C412yes—
98C4×D244C2×C411yescharacteristic
99C4×C22⋊D61—21no—
100C4×C22⋊D61—22no—
101#14519736—21no—
102#14519736—22no—
103#14519735—22no—
104#14519735—21no—
105#14519734—21no—
106#14519734—22no—
107C2×C4⋊Dic61—21no—
108C2×C4⋊Dic61—22no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—21no—
111#14519684—21no—
112#14519684—22no—
113#14519684—22no—
114#14519684—22no—
115#14519684C2311yescharacteristic
116#14519684C2311yescharacteristic
117#14519684C2312yes—
118C22×Q8×C61D411yescharacteristic
119C4×Q8×C61C2×C411yescharacteristic
120C4×Q8×C61C2×C412yes—
121C4×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61D411yescharacteristic
124C4×Q8×C61D412yes—
125C2×C4⋊C4×C61D412yes—
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2312yes—
129C4⋊C8×C61—22no—
130C4⋊C8×C61—21no—
131C4⋊C8×C61—21no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61C2311yescharacteristic
134C22⋊C8×C61C2311yescharacteristic
135C22⋊C8×C61C2312yes—
136#97×C61—22no—
137#97×C61—21no—
138C4×C8×C61—21no—
139C4×C8×C61—22no—
140C2×Q8×C61—42no—
141C2×Q8×C61C22⋊C411yescharacteristic
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×Q8×C61C22⋊C412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—22no—
153Q8×D61—44no—
154C2×D244C22×C411yescharacteristic
155C2×C4×D61—22no—
156C2×C4×D61—21no—
157C2×Dic122—22no—
158C2×Dic122—21no—
159C2×Dic122C22×C411yescharacteristic
160C2×Dic122C22×C412yes—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—22no—
164C4⋊C4×C61C22⋊C412yes—
165C4⋊C4×C61C22⋊C411yescharacteristic
166C4⋊C4×C61C22⋊C412yes—
167C4⋊C4×C61C22⋊C411yescharacteristic
168C4⋊C4×C61C22⋊C412yes—
169C4⋊C4×C61C22⋊C412yes—
170C4⋊C4×C61C22⋊C412yes—
171C42×C61C22×C412yes—
172C42×C61C22×C411yescharacteristic
173C42×C61C22×C411yescharacteristic
174C42×C61C22×C412yes—
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D412yes—
177C42×C61C2×D412yes—
178C42×C61C2×D411yescharacteristic
179C22⋊Dic61—22no—
180C22⋊Dic61—21no—
181#11758198—21no—
182#11758198—22no—
183#11758198—21no—
184#11758198—22no—
185#11758198—22no—
186#11758198—21no—
187#11758198—21no—
188C4⋊Dic61—22no—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61—42no—
195C4⋊Dic61—82no—
196C4⋊Dic61C22×C412yes—
197C4⋊Dic61C22×C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199#11758196—21no—
200#11758196—22no—
201#11758196—21no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—21no—
208#11758196—21no—
209#11758196—22no—
210C4×Dic61—22no—
211C4×Dic61—21no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—42no—
216C4×Dic61—81no—
217C23×C61C22×D411yescharacteristic
218C22×D61—21no—
219Q8×C61—22no—
220Q8×C61—22no—
221Q8×C61—42no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—42no—
228C2×C4×C61—41no—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C412yes—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C22⋊C412yes—
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—41no—
240C22⋊D61—41no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—42no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255D244—41no—
256C4×D61—44no—
257C4×D61—42no—
258Dic122—44no—
259Dic122—42no—
260Dic122—41no—
261Dic122—42no—
262C8×C61—41no—
263C8×C61—41no—
264C8×C61—42no—
265#17601—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61C22×C22⋊C411yescharacteristic
269C22×C61#57711yescharacteristic
270C22×C61#57811yescharacteristic
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—21no—
282C4×C61—22no—
283C4×C61#42112yes—
284Dic61—42no—
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288Dic61—42no—
289Dic61—81no—
290Dic61—82no—
291Dic61—81no—
292#3523—611no—
293#2984D6111yescharacteristic, complemented
294#2320—611no—
295#1521—612no—
296#1518—612no—
297#1515—612no—
298#1512—612no—
299#1509—611no—
300#1506—611no—
301#1503—611no—
302#1500—611no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61#288411yescharacteristic
306C2×C61#289811yescharacteristic
307C2×C61C2×#42111yescharacteristic
308D61—81no—
309#557—611no—
310#557—612no—
311#557—611no—
312#556—611no—
313#556—612no—
314#552—1222no—
315#552—1221no—
316D4×Q8—611no—
317D4×Q8—1221no—
318#541—1221no—
319#541—1222no—
320#533—611no—
321#533—612no—
322#529—611no—
323#529—612no—
324#519—1221no—
325C2×C4×Q8C2×D6111yescharacteristic
326#446—21no—
327#446—22no—
328C8×Q8—21no—
329#432C2×D6111yescharacteristic
330#432C2×D6112yes—
331#431C2×D6111yescharacteristic
332#431C2×D6112yes—
333#391—1221no—
334#389—1221no—
335#387—1222no—
336#383—612no—
337#382—612no—
338C4×C4⋊C4—611no—
339#376—611no—
340#328—611no—
341#328—611no—
342#328—612no—
343#328—612no—
344#328—612no—
345#328—612no—
346#328—612no—
347#327—611no—
348#327—611no—
349#327—612no—
350C61#1760111yescharacteristic, complemented, Sylow
351C22×Q8—1221no—
352C22×Q8D24411yescharacteristic
353C4⋊Q8—611no—
354C4⋊Q8—612no—
355C4⋊Q8—611no—
356C4⋊Q8—612no—
357C4⋊Q8—1221no—
358C4⋊Q8—1222no—
359#126—1222no—
360#126—1222no—
361#126—1222no—
362#126—1222no—
363#126—1221no—
364#125—611no—
365#125—612no—
366#125—1222no—
367#125—1222no—
368#125—1222no—
369#125—1222no—
370#125—1221no—
371#125—1222no—
372#125—1221no—
373#124—611no—
374#124—612no—
375#123—1221no—
376#123—1222no—
377C22⋊Q8—611no—
378C22⋊Q8—612no—
379C22⋊Q8—1221no—
380C22⋊Q8—1222no—
381C22⋊Q8—1221no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1221no—
385#121—611no—
386C4×Q8—611no—
387C4×Q8—612no—
388C4×Q8—612no—
389C4×Q8—612no—
390C4×Q8—612no—
391C4×Q8—1221no—
392C4×Q8—1222no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×Q8—1222no—
396C4×Q8C4×D6111yescharacteristic
397C4×Q8C4×D6111yescharacteristic
398C4×Q8C4×D6112yes—
399C4×Q8C22⋊D6111yescharacteristic
400C4×Q8C22⋊D6111yescharacteristic
401C4×Q8C22⋊D6112yes—
402C4×D4—611no—
403C4×D4—612no—
404C4×D4—1221no—
405C4×D4—1222no—
406#117—1221no—
407#117—1222no—
408#117—1222no—
409#117—1221no—
410#117—1221no—
411#117—1222no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4D24412yes—
415C2×C4⋊C4D24411yescharacteristic
416C2×C42—1221no—
417C2×C42—1221no—
418C2×C42C22×D6111yescharacteristic
419C2×C42C22×D6112yes—
420C4⋊C8—22no—
421C4⋊C8—21no—
422C4⋊C8—21no—
423C4⋊C8—22no—
424C22⋊C8C22×D6111yescharacteristic
425C22⋊C8C22×D6111yescharacteristic
426C22⋊C8C22×D6112yes—
427#97—22no—
428#97—21no—
429C4×C8—21no—
430C4×C8—22no—
431#95—611no—
432#95—611no—
433#95—612no—
434#95—1221no—
435#95—1222no—
436#95—1222no—
437#95—1222no—
438C2×Q8—42no—
439C2×Q8—611no—
440C2×Q8—612no—
441C2×Q8—1222no—
442C2×Q8—1221no—
443C2×Q8—2444no—
444C2×Q8#1175819811yescharacteristic
445C2×Q8#1175819811yescharacteristic
446C2×Q8#1175819812yes—
447C2×D4—611no—
448C2×D4—1221no—
449C22×C4—1222no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1222no—
454C22×C4C2×D24411yescharacteristic
455C22×C4C2×D24412yes—
456C22×C4C2×D24411yescharacteristic
457C22×C4C2×D24412yes—
458C22×C4C23×D6111yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C4⋊C4—611no—
463C4⋊C4—612no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
466C4⋊C4—1222no—
467C4⋊C4—1221no—
468C4⋊C4—1222no—
469C4⋊C4—1221no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1221no—
476C4⋊C4—1222no—
477C4⋊C4—1221no—
478C4⋊C4—1222no—
479C4⋊C4—1221no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—4882no—
483C4⋊C4—2442no—
484C4⋊C4#1175819812yes—
485C4⋊C4#1175819811yescharacteristic
486C4⋊C4#1175819812yes—
487C4⋊C4#1175819811yescharacteristic
488C4⋊C4#1175819812yes—
489C4⋊C4#1175819812yes—
490C4⋊C4#1175819812yes—
491C22⋊C4—1222no—
492C22⋊C4—1221no—
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1221no—
498C22⋊C4—1221no—
499C22⋊C4—1222no—
500C42—1222no—
501C42—1222no—
502C42—1222no—
503C42—1222no—
504C42—1221no—
505C42—4881no—
506C42—2442no—
507C42C2×C4×D6112yes—
508C42C2×C4×D6111yescharacteristic
509C42C2×C4×D6111yescharacteristic
510C42C2×C4×D6112yes—
511C42C2×C22⋊D6111yescharacteristic
512C42C2×C22⋊D6112yes—
513C42C2×C22⋊D6112yes—
514C42C2×C22⋊D6111yescharacteristic
515C23—1221no—
516C23C22×D24411yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—42no—
520Q8—2441no—
521Q8—2442no—
522Q8—2444no—
523Q8—2442no—
524D4—2441no—
525D4—2441no—
526D4—2441no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—1222no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—2444no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2442no—
550C2×C4C2×#1175819811yescharacteristic
551C2×C4C2×#1175819812yes—
552C2×C4C2×#1175819812yes—
553C2×C4C2×#1175819812yes—
554C2×C4C2×#1175819811yescharacteristic
555C2×C4C2×#1175819811yescharacteristic
556C2×C4C2×#1175819812yes—
557C2×C4C2×#1175819811yescharacteristic
558C2×C4C22×C4×D6111yescharacteristic
559C2×C4C22×C22⋊D6111yescharacteristic
560C8—41no—
561C8—41no—
562C8—42no—
563C22—21no—
564C22—21no—
565C22—2441no—
566C22—2441no—
567C22—2441no—
568C22C22×#1175819811yescharacteristic
569C22#1684057411yescharacteristic
570C22#1684057911yescharacteristic
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—21no—
576C4—22no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—4881no—
581C4—4882no—
582C4—4881no—
583C4—2441no—
584C4—2442no—
585C4—2441no—
586C4—2442no—
587C4—2441no—
588C4#1683977812yes—
589C2—21no—
590C2—21no—
591C2—4881no—
592C2#2145428211yescharacteristic
593C2#2145435311yescharacteristic
594C2C2×#1683977811yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2984C2
socleC22×C61C22×C22⋊C4
prime radical#14519751C23
cosocleC22×C4×C61C24
hypercenter#2984D61
hypocenterC61#17601
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #21448470C2semidirect
4. #21444384C2semidirect
5. #21444371C2semidirect
8. #21444332C2semidirect
9. #21444319C2semidirect
11. C61×#2984C2semidirect
12. #16840491C4semidirect
13. #16840490C4semidirect
14. #16840490C4semidirect
15. #16840485C4semidirect
16. #16840485C4semidirect
19. Q8×D244C4semidirect
293. #2984D61semidirect
350. C61#17601semidirect

Maximal subgroups

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

subgroupindex
#214599152
#214484702
#214443972
#214443842
#214443712
#214443582
#214443452
#214443322
#214443192
#214443062
C61×#29842
#1760161