Finite Group Databasebeta

← #29647584

Subgroups — #29647584

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#21459913C211yesmaximal, characteristic
2#21454906C211yesmaximal, characteristic, complemented
3#21448471C211yesmaximal, characteristic, complemented
4#21444393C212yesmaximal
5#21444381C212yesmaximal, complemented
6#21444368C212yesmaximal, complemented
7#21444354C212yesmaximal
8#21444344C211yesmaximal, characteristic
9#21444330C211yesmaximal, characteristic, complemented
10#21444319C211yesmaximal, characteristic, complemented
11C61×#1500C211yesmaximal, characteristic
12#16840491C411yescharacteristic, complemented
13#16840490C411yescharacteristic, complemented
14#16840489C412yescomplemented
15#16840482C411yescharacteristic, complemented
16#16840481C412yescomplemented
17#16840452—21no—
18#16840449—22no—
19D4×Dic122—21no—
20D4×Dic122C411yescharacteristic, complemented
21#16840374—21no—
22#16840369—22no—
23#16840319C2211yescharacteristic
24#16840314C2212yes—
25#16840293C2212yes—
26#16840239—21no—
27C2×C4×Dic122C2211yescharacteristic
28#16839892—21no—
29#16839891—22no—
30C8×Dic122—21no—
31#16839819C2211yescharacteristic
32#16839816C2212yes—
33#16839812C2212yes—
34#16839668—21no—
35#16839660—21no—
36#16839644—22no—
37#16839629C2212yes—
38#16839626C2212yes—
39C4×C4⋊Dic61C2211yescharacteristic
40#16839500C2212yes—
41#16839500C2212yes—
42#16839499C2212yes—
43#16839498C2212yes—
44#16839498C2212yes—
45#16839496C2212yes—
46C61×#529C2211yescharacteristic
47C61×#431C2211yescharacteristic
48C61×#376C2211yescharacteristic
49C61×#328C2211yescharacteristic
50C61×#328C2211yescharacteristic
51C61×#327C2211yescharacteristic
52C61×#327C2211yescharacteristic
53C22×Dic122—21no—
54C22×Dic122D411yescharacteristic
55#14519777—21no—
56#14519777C2×C411yescharacteristic
57#14519777C2×C411yescharacteristic
58#14519776—22no—
59#14519776C2×C412yes—
60#14519775C2×C412yes—
61#14519770—22no—
62#14519770—22no—
63#14519770—22no—
64#14519770—22no—
65#14519769—21no—
66#14519769—21no—
67#14519768—22no—
68#14519768C2×C412yes—
69#14519767—22no—
70#14519767—22no—
71#14519767—22no—
72#14519767—22no—
73#14519764C2×C412yes—
74#14519758—22no—
75#14519756—21no—
76#14519756—21no—
77#14519756—21no—
78#14519756C2312yes—
79#14519754—22no—
80#14519754—22no—
81#14519754—22no—
82Q8×Dic61C2×C412yes—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122C2×C412yes—
87C4×Dic122C2×C412yes—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122C2×C411yescharacteristic
90C4×Dic122D411yescharacteristic
91C4×Dic122D412yes—
92C4×Dic122D411yescharacteristic
93#14519742—22no—
94#14519742—22no—
95#14519742C2×C412yes—
96#14519742C2×C412yes—
97D4×Dic61—22no—
98D4×Dic61C2×C412yes—
99#14519736—21no—
100#14519736—21no—
101#14519733—22no—
102#14519733—22no—
103#14519733—22no—
104C2×C4⋊Dic61—21no—
105C2×C4⋊Dic61D411yescharacteristic
106C2×#11758196—22no—
107C2×#11758196D412yes—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2312yes—
110C4⋊#92418—21no—
111#14519706—22no—
112#14519706—22no—
113C22⋊#92418C2312yes—
114#14519688—21no—
115#14519687—22no—
116C8×Dic61—22no—
117#14519684—22no—
118#14519684—22no—
119#14519684—22no—
120#14519684C2312yes—
121#126×C61—21no—
122#125×C61C2×C411yescharacteristic
123#124×C61C2×C411yescharacteristic
124#123×C61—21no—
125C22⋊Q8×C61C2311yescharacteristic
126#121×C61C2311yescharacteristic
127C4×Q8×C61C2×C411yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2×C411yescharacteristic
130#117×C61—21no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61—21no—
134C22⋊C8×C61C2311yescharacteristic
135C22⋊C8×C61C2311yescharacteristic
136C4×C8×C61—21no—
137#95×C61—21no—
138#95×C61C2311yescharacteristic
139#95×C61C2311yescharacteristic
140C2×Q8×C61C22×C411yescharacteristic
141C2×D4×C61—21no—
142C2×D4×C61C22×C411yescharacteristic
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61C2×D412yes—
152C22×Dic61C2×D412yes—
153C2×Dic122—21no—
154C2×Dic122—22no—
155C2×Dic122—42no—
156C2×Dic122—44no—
157C2×Dic122C22⋊C411yescharacteristic
158C2×Dic122C22⋊C411yescharacteristic
159C2×Dic122C22⋊C412yes—
160C2×Dic122C22×C412yes—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61C22×C411yescharacteristic
168C4⋊C4×C61C22×C411yescharacteristic
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C42×C61—21no—
175C42×C61—42no—
176C42×C61C22×C411yescharacteristic
177C42×C61C22×C411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—42no—
190C4⋊Dic61C22⋊C411yescharacteristic
191C4⋊Dic61C22⋊C411yescharacteristic
192C4⋊Dic61C22⋊C412yes—
193C4⋊Dic61C22×C412yes—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—82no—
203#11758196C22⋊C412yes—
204#11758196C22⋊C412yes—
205#11758196C22⋊C412yes—
206#11758196C22⋊C412yes—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—81no—
212C4×Dic61C22×C412yes—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C2×#92418—22no—
217C23×C61—21no—
218C23×C61C22×D411yescharacteristic
219Q8×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—42no—
248C2×Dic61—44no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256Dic122—22no—
257Dic122—22no—
258Dic122—42no—
259Dic122—44no—
260Dic122—42no—
261Dic122—42no—
262C8×C61—41no—
263C8×C61—41no—
264#92418—42no—
265#17601—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57811yescharacteristic
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61#42112yes—
285Dic61—22no—
286Dic61—22no—
287Dic61—22no—
288Dic61—22no—
289Dic61—42no—
290Dic61—42no—
291Dic61—82no—
292#3523—611no—
293#2984—611no—
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#1500D6111yescharacteristic
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—81no—
306C2×C61#288411yescharacteristic
307C2×C61#289811yescharacteristic
308C2×C61C2×#42111yescharacteristic
309#557—612no—
310#557—611no—
311#557—611no—
312#556—612no—
313#556—611no—
314#552—1221no—
315#552—1222no—
316D4×Q8—611no—
317D4×Q8—1221no—
318#541—1221no—
319#541—1222no—
320#533—612no—
321#533—611no—
322#529—612no—
323#529C2×D6111yescharacteristic
324#519—1221no—
325C2×C4×Q8—611no—
326#446—1221no—
327#446—1222no—
328C8×Q8—1221no—
329#432—611no—
330#432—612no—
331#431—612no—
332#431C2×D6111yescharacteristic
333#391—1221no—
334#389—1221no—
335#387—1222no—
336#383—612no—
337#382—612no—
338C4×C4⋊C4—611no—
339#376C2×D6111yescharacteristic
340#328—612no—
341#328—612no—
342#328—612no—
343#328—612no—
344#328—612no—
345#328C2×D6111yescharacteristic
346#328C2×D6111yescharacteristic
347#327—612no—
348#327C2×D6111yescharacteristic
349#327C2×D6111yescharacteristic
350C61#1760111yescharacteristic, complemented, Sylow
351C22×Q8—611no—
352C22×Q8—1221no—
353C4⋊Q8—612no—
354C4⋊Q8—612no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358C4⋊Q8—1222no—
359#126—21no—
360#126—1222no—
361#126—1222no—
362#126—1222no—
363#126—1222no—
364#125—612no—
365#125—1221no—
366#125—1222no—
367#125—1222no—
368#125—1222no—
369#125—1222no—
370#125—1222no—
371#125—1221no—
372#125C4×D6111yescharacteristic
373#124—612no—
374#124C4×D6111yescharacteristic
375#123—21no—
376#123—1222no—
377C22⋊Q8—612no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—1221no—
384C22⋊Q8C22×D6111yescharacteristic
385#121C22×D6111yescharacteristic
386C4×Q8—611no—
387C4×Q8—612no—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—611no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—612no—
394C4×Q8—612no—
395C4×Q8—612no—
396C4×Q8—1221no—
397C4×Q8—1221no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1221no—
401C4×Q8C4×D6111yescharacteristic
402C4×D4—21no—
403C4×D4—612no—
404C4×D4—1222no—
405C4×D4C4×D6111yescharacteristic
406#117—21no—
407#117—1222no—
408#117—1222no—
409#117—1222no—
410#117—1221no—
411#117—1221no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1221no—
416C2×C42—21no—
417C2×C42—612no—
418C2×C42—1221no—
419C2×C42C22×D6111yescharacteristic
420C4⋊C8—21no—
421C4⋊C8—1222no—
422C4⋊C8—1222no—
423C4⋊C8—1221no—
424C22⋊C8—612no—
425C22⋊C8C22×D6111yescharacteristic
426C22⋊C8C22×D6111yescharacteristic
427#97—1222no—
428#97—1221no—
429C4×C8—21no—
430C4×C8—1222no—
431#95—21no—
432#95—612no—
433#95—1222no—
434#95—1222no—
435#95—1222no—
436#95C22×D6111yescharacteristic
437#95C22×D6111yescharacteristic
438C2×Q8—611no—
439C2×Q8—611no—
440C2×Q8—612no—
441C2×Q8—612no—
442C2×Q8—1222no—
443C2×Q8—1221no—
444C2×Q8—2442no—
445C2×Q8—2444no—
446C2×Q8C2×C4×D6111yescharacteristic
447C2×D4—21no—
448C2×D4C2×C4×D6111yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4D4×D6111yescharacteristic
457C22×C4D4×D6111yescharacteristic
458C22×C4C23×D6111yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—1222no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—612no—
467C4⋊C4—612no—
468C4⋊C4—612no—
469C4⋊C4—612no—
470C4⋊C4—611no—
471C4⋊C4—611no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4882no—
488C4⋊C4—2442no—
489C4⋊C4C2×C4×D6111yescharacteristic
490C4⋊C4C2×C4×D6111yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C42—21no—
501C42—42no—
502C42—612no—
503C42—612no—
504C42—612no—
505C42—612no—
506C42—1222no—
507C42—1222no—
508C42—1222no—
509C42—1222no—
510C42—4881no—
511C42C2×C4×D6111yescharacteristic
512C42C2×C4×D6111yescharacteristic
513C42D4×D6111yescharacteristic
514C42D4×D6111yescharacteristic
515C23—21no—
516C23C2×D4×D6111yescharacteristic
517Q8—41no—
518Q8—1222no—
519Q8—1222no—
520Q8—2442no—
521Q8—2442no—
522Q8—2444no—
523Q8—2442no—
524D4—41no—
525D4—41no—
526D4—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—2442no—
550C2×C4—2444no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4C22⋊C4×D6111yescharacteristic
555C2×C4C22⋊C4×D6111yescharacteristic
556C2×C4C22⋊C4×D6111yescharacteristic
557C2×C4C22⋊C4×D6111yescharacteristic
558C2×C4C22×C4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C8—41no—
561C8—41no—
562C8—2442no—
563C22—21no—
564C22—21no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22C2×C22⋊C4×D6111yescharacteristic
569C22#1684056911yescharacteristic
570C22#1684057611yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—1222no—
582C4—1222no—
583C4—1222no—
584C4—1222no—
585C4—4882no—
586C4—2442no—
587C4—2442no—
588C4#1683977712yes—
589C2—21no—
590C2—21no—
591C2—81no—
592C2#2145428111yescharacteristic
593C2#2145434811yescharacteristic
594C2C2×#1683977711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1500C2
socleC22×C61C22×C22⋊C4
prime radical#121×C61C23
cosocleC22×C4×C61C24
hypercenter#1500D61
hypocenterC61#17601
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1500

Complements

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

NH ≅ G/Ntype
2. #21454906C2semidirect
3. #21448471C2semidirect
5. #21444381C2semidirect
6. #21444368C2semidirect
9. #21444330C2semidirect
10. #21444319C2semidirect
12. #16840491C4semidirect
13. #16840490C4semidirect
14. #16840489C4semidirect
15. #16840482C4semidirect
16. #16840481C4semidirect
20. D4×Dic122C4semidirect
350. C61#17601semidirect

Maximal subgroups

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

subgroupindex
#214599132
#214549062
#214484712
#214443932
#214443812
#214443682
#214443542
#214443442
#214443302
#214443192
C61×#15002
#1760161