Finite Group Databasebeta

← #29656613

Subgroups — #29656613

Order 15616.

980 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457210C211yesmaximal, characteristic, complemented
2#21456193C211yesmaximal, characteristic, complemented
3#21455890C211yesmaximal, characteristic, complemented
4#21445019C212yesmaximal, complemented
5#21445012C212yesmaximal, complemented
6#21445005C212yesmaximal, complemented
7#21445005C212yesmaximal, complemented
8#21444487C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3508C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840126—22no—
14#16840108C2212yes—
15#16840108C2212yes—
16#16840103—21no—
17#16840047C2211yescharacteristic, complemented
18#16840047C2212yescomplemented
19#16840044—21no—
20#16840042C2211yescharacteristic, complemented
21#16840018—22no—
22#16839972C2211yescharacteristic, complemented
23#16839972C2212yescomplemented
24#16839969—21no—
25#16839967C2211yescharacteristic, complemented
26#16839924—21no—
27C4×#11758202—21no—
28C4×#11758200—21no—
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839767—22no—
32C4×#11758194C2211yescharacteristic
33#16839506C2212yes—
34#16839506C2212yes—
35Q8⋊#92418C2212yescomplemented
36Q8⋊#92418C2212yescomplemented
37Q8⋊#92418C2212yescomplemented
38D4⋊#92418C2212yescomplemented
39C61×#559—21no—
40C61×#559C2212yes—
41C61×#550—21no—
42C61×#550C2212yes—
43C61×#546—22no—
44C61×#546—22no—
45C61×#540—22no—
46C61×#535—21no—
47C61×#533C2211yescharacteristic
48C61×#532—21no—
49C61×#532—21no—
50C2×#127×C61C2211yescharacteristic
51C61×#518—21no—
52C4×#41×C61C2211yescharacteristic
53C2×#11758202—21no—
54C2×#11758202—22no—
55C2×#11758200—21no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66#14519710—21no—
67#14519708—22no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2311yescharacteristic
71C4⋊#92418C2312yes—
72C4⋊#92418C2311yescharacteristic
73#14519701—22no—
74#14519701—21no—
75#14519700—21no—
76#14519700—22no—
77#14519700—21no—
78#14519698—24no—
79#14519698—24no—
80#14519698—21no—
81C22⋊#92418—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61D411yescharacteristic
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C22×D4×C61D412yes—
91C4⋊Q8×C61D411yescharacteristic
92C4⋊Q8×C61D411yescharacteristic
93#127×C61—22no—
94#127×C61—21no—
95#127×C61—21no—
96#127×C61D412yes—
97#127×C61C2312yes—
98#125×C61—21no—
99#124×C61—22no—
100#124×C61—22no—
101#124×C61—22no—
102#123×C61—24no—
103#123×C61—24no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61D412yes—
106#121×C61—22no—
107#121×C61—22no—
108#121×C61—22no—
109#121×C61—24no—
110#121×C61—24no—
111#121×C61—22no—
112#120×C61—24no—
113C4×Q8×C61—21no—
114C4×Q8×C61D411yescharacteristic
115C4×Q8×C61C2312yes—
116C4×Q8×C61C2311yescharacteristic
117C4×D4×C61—22no—
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—21no—
121C4×D4×C61—21no—
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124C4×D4×C61—22no—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61C2311yescharacteristic
127#117×C61—21no—
128#117×C61—21no—
129#117×C61D411yescharacteristic
130C2×C22⋊C4×C61—24no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133C24×C61—22no—
134#41×C61—41no—
135#41×C61—41no—
136#41×C61—82no—
137#41×C61—82no—
138#41×C61—84no—
139C2×Q8×C61—21no—
140C2×Q8×C61C2×D411yescharacteristic
141C2×Q8×C61C2×D412yes—
142C2×D4×C61—21no—
143C2×D4×C61—24no—
144C2×D4×C61—24no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—42no—
151C2×D4×C61—42no—
152C2×D4×C61—44no—
153C2×D4×C61—81no—
154C2×D4×C61—81no—
155C2×D4×C61C2×D411yescharacteristic
156C2×D4×C61C2×D414yes—
157C22×C4×C61—24no—
158C22×C4×C61—24no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D412yes—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—22no—
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194#11758202—42no—
195#11758202—82no—
196#11758200—42no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201#11758196—41no—
202#11758196—41no—
203C4×Dic61—41no—
204#11758194—42no—
205#11758194—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—22no—
209C2×#92418—22no—
210C23×C61—24no—
211C23×C61—21no—
212C23×C61—44no—
213C23×C61—82no—
214C23×C61—82no—
215C23×C61#12011yescharacteristic
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61—42no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—44no—
224D4×C61—44no—
225D4×C61—42no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—42no—
229D4×C61—42no—
230D4×C61—44no—
231D4×C61—82no—
232D4×C61—84no—
233D4×C61—84no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—42no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—84no—
249C2×C4×C61—84no—
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×Dic61—41no—
257C2×Dic61—41no—
258Dic122—41no—
259Dic122—41no—
260Dic122—41no—
261Dic122—41no—
262#92418—42no—
263#92418—41no—
264#92418—41no—
265#92418—42no—
266#18576—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—41no—
271C22×C61—44no—
272C22×C61—41no—
273C22×C61—44no—
274C22×C61—41no—
275C22×C61—44no—
276C22×C61—164no—
277C22×C61C2×#12011yescharacteristic
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—42no—
284C4×C61—41no—
285C4×C61—42no—
286C4×C61#45512yes—
287C4×C61#57511yescharacteristic
288C4×C61#57511yescharacteristic
289C4×C61#57611yescharacteristic
290C4×C61#57711yescharacteristic
291Dic61—81no—
292Dic61—81no—
293#3508D6111yescharacteristic
294#3255—611no—
295#3138—611no—
296#3107—611no—
297#1642—612no—
298#1641—612no—
299#1640—612no—
300#1640—612no—
301#1545—611no—
302#1489—611no—
303#1485—611no—
304C2×C61—41no—
305C2×C61—81no—
306C2×C61—84no—
307C2×C61#300511yescharacteristic
308C2×C61#301211yescharacteristic
309C2×C61C2×#45511yescharacteristic
310#559—21no—
311#559C2×D6112yes—
312#550—21no—
313#550C2×D6112yes—
314#546—22no—
315#546—22no—
316#540—22no—
317#535—21no—
318#533—611no—
319#533C2×D6111yescharacteristic
320#532—21no—
321#532—21no—
322C2×#127C2×D6111yescharacteristic
323#518—21no—
324C4×#41C2×D6111yescharacteristic
325#490—1222no—
326#486—612no—
327#486—612no—
328#485—1221no—
329#475—611no—
330#475—612no—
331#475—1221no—
332#474—611no—
333#470—1222no—
334#462—611no—
335#462—612no—
336#462—1221no—
337#461—611no—
338#452—1221no—
339C4×Q16—1221no—
340C4×SD16—1221no—
341#431—611no—
342#423—611no—
343#418—1222no—
344C4×M16—611no—
345#331—612no—
346#331—612no—
347Q8⋊C8—612no—
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350D4⋊C8—612no—
351C61#1857611yescharacteristic, complemented, Sylow
352C2×#41—22no—
353C2×#41—22no—
354C2×#41C22⋊D6111yescharacteristic
355C22×D4—21no—
356C22×D4—21no—
357C22×D4C22⋊D6112yes—
358C2×Q16—1222no—
359C2×Q16—1221no—
360C2×SD16—1221no—
361C2×M16—1221no—
362C2×M16—1221no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8C22⋊D6111yescharacteristic
366C4⋊Q8C22⋊D6111yescharacteristic
367#127—22no—
368#127—21no—
369#127—21no—
370#127C22⋊D6112yes—
371#127C22×D6112yes—
372#125—21no—
373#125—1221no—
374#124—22no—
375#124—22no—
376#124—22no—
377#123—24no—
378#123—24no—
379C22⋊Q8—21no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8C22⋊D6112yes—
383#121—22no—
384#121—22no—
385#121—22no—
386#121—24no—
387#121—24no—
388#121—22no—
389#120—24no—
390C4×Q8—21no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8C22⋊D6111yescharacteristic
394C4×Q8C22×D6111yescharacteristic
395C4×Q8C22×D6112yes—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4C22⋊D6111yescharacteristic
405C4×D4C22×D6111yescharacteristic
406#117—21no—
407#117—21no—
408#117—1221no—
409#117C22⋊D6111yescharacteristic
410C2×C22⋊C4—24no—
411C2×C42—21no—
412C2×C42C22×D6111yescharacteristic
413#107—1221no—
414#106—1221no—
415#106—1222no—
416C4⋊C8—612no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420#103—1221no—
421#103—1222no—
422#103—1221no—
423#103—1221no—
424#103—1222no—
425#102—1224no—
426#102—1224no—
427#102—1221no—
428C22⋊C8—1221no—
429#97—1221no—
430C4×C8—612no—
431C4×C8—1221no—
432C24—22no—
433#41—41no—
434#41—41no—
435#41—82no—
436#41—82no—
437#41—84no—
438C2×Q8—21no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8C2×C22⋊D6111yescharacteristic
442C2×Q8C2×C22⋊D6112yes—
443C2×D4—21no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—81no—
455C2×D4—81no—
456C2×D4C2×C22⋊D6111yescharacteristic
457C2×D4C2×C22⋊D6114yes—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466Q16—4882no—
467Q16—2442no—
468SD16—2442no—
469M16—4881no—
470M16—2442no—
471C2×C8—1222no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1222no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2441no—
487C4⋊C4—2441no—
488C4⋊C4C2×C22⋊D6111yescharacteristic
489C4⋊C4C2×C22⋊D6112yes—
490C4⋊C4C2×C22⋊D6111yescharacteristic
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—2441no—
500C42—21no—
501C42—22no—
502C42—21no—
503C42—21no—
504C42—2441no—
505C42C2×C22⋊D6111yescharacteristic
506C42C2×C22⋊D6111yescharacteristic
507C42C2×C22⋊D6111yescharacteristic
508C42C23×D6111yescharacteristic
509C23—24no—
510C23—21no—
511C23—44no—
512C23—82no—
513C23—82no—
514C23#1451974611yescharacteristic
515Q8—21no—
516Q8—21no—
517Q8—22no—
518Q8—22no—
519Q8—42no—
520Q8—2441no—
521Q8—2441no—
522Q8—2441no—
523Q8—2441no—
524D4—21no—
525D4—21no—
526D4—44no—
527D4—44no—
528D4—42no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—82no—
535D4—84no—
536D4—84no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4#1451974611yescharacteristic
556C2×C4#1451974611yescharacteristic
557C2×C4#1451974611yescharacteristic
558C2×C4C22×C22⋊D6111yescharacteristic
559C2×C4C22×C22⋊D6111yescharacteristic
560C2×C4C22×C22⋊D6111yescharacteristic
561C8—2442no—
562C8—2441no—
563C8—2441no—
564C8—2442no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—41no—
569C22—44no—
570C22—41no—
571C22—44no—
572C22—41no—
573C22—44no—
574C22—164no—
575C22C2×#1451974611yescharacteristic
576C4—21no—
577C4—21no—
578C4—21no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—4881no—
585C4—4881no—
586C4#1683994012yes—
587C4#1684056011yescharacteristic
588C4#1684056011yescharacteristic
589C4#1684056511yescharacteristic
590C4#1684057311yescharacteristic
591C2—41no—
592C2—84no—
593C2—81no—
594C2#2145506111yescharacteristic
595C2#2145512311yescharacteristic
596C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#18576
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21457210C2semidirect
2. #21456193C2semidirect
3. #21455890C2semidirect
4. #21445019C2semidirect
5. #21445012C2semidirect
6. #21445005C2semidirect
7. #21445005C2semidirect
8. #21444487C2semidirect
9. #21444258C2semidirect
10. #21444240C2semidirect
17. #16840047C22semidirect
18. #16840047C22semidirect
20. #16840042C22semidirect
22. #16839972C22semidirect
23. #16839972C22semidirect
25. #16839967C22semidirect
35. Q8⋊#92418C22semidirect
36. Q8⋊#92418C22semidirect
37. Q8⋊#92418C22semidirect
38. D4⋊#92418C22semidirect
351. C61#18576semidirect

Maximal subgroups

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

subgroupindex
#214572102
#214561932
#214558902
#214450192
#214450122
#214450052
#214444872
#214442582
#214442402
C61×#35082
#1857661