Finite Group Databasebeta

← #29656800

Subgroups — #29656800

Order 15616.

916 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457206C211yesmaximal, characteristic, complemented
2#21456361C211yesmaximal, characteristic, complemented
3#21455890C211yesmaximal, characteristic, complemented
4#21445103C212yesmaximal, complemented
5#21445089C212yesmaximal, complemented
6#21445075C212yesmaximal, complemented
7#21445054C212yesmaximal, complemented
8#21444491C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3511C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840168—22no—
14#16840165C2212yes—
15#16840158C2212yes—
16#16840103—21no—
17#16840054—21no—
18#16840047C2211yescharacteristic
19#16840042C2211yescharacteristic
20#16840042C2212yes—
21#16840006—22no—
22#16839972C2211yescharacteristic
23#16839967C2211yescharacteristic
24#16839967C2212yes—
25#16839959—21no—
26#16839924—21no—
27C4×#11758202—21no—
28C4×#11758200—21no—
29#16839815C2211yescharacteristic
30#16839802—22no—
31#16839786C2211yescharacteristic
32C4×#11758194C2211yescharacteristic
33#16839508C2212yes—
34#16839508C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38D4⋊#92418C2212yes—
39C61×#556C2212yes—
40C61×#555—21no—
41C61×#554—21no—
42C61×#553—22no—
43C61×#548—22no—
44C61×#547C2212yes—
45C61×#542—22no—
46C61×#537—21no—
47C61×#533—21no—
48C61×#532—21no—
49C61×#532C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#520—21no—
52C4×#41×C61C2211yescharacteristic
53C2×#11758202—21no—
54C2×#11758200—21no—
55C2×#11758200—22no—
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—22no—
67#14519710—21no—
68#14519710—24no—
69#14519708—21no—
70#14519708—24no—
71C4⋊#92418—21no—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75#14519701—21no—
76#14519700—22no—
77#14519700—21no—
78#14519700—21no—
79#14519698—22no—
80#14519698—21no—
81C22⋊#92418—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
85C2×#41×C61—22no—
86C2×#41×C61—21no—
87C2×#41×C61D411yescharacteristic
88C22×Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61D412yes—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61C2312yes—
94#127×C61D411yescharacteristic
95#126×C61—24no—
96#125×C61—22no—
97#124×C61—24no—
98#124×C61—22no—
99#124×C61—21no—
100#123×C61—24no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105#121×C61—22no—
106#121×C61—22no—
107#121×C61—24no—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61D412yes—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C4×Q8×C61D411yescharacteristic
114C4×Q8×C61C2311yescharacteristic
115C4×D4×C61—22no—
116C4×D4×C61—22no—
117C4×D4×C61—24no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61D411yescharacteristic
122C4×D4×C61C2312yes—
123C4×D4×C61C2311yescharacteristic
124#117×C61—22no—
125#117×C61—22no—
126#117×C61—21no—
127#117×C61D411yescharacteristic
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—22no—
130C2×C4⋊C4×C61D412yes—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#41×C61—41no—
134#41×C61—41no—
135#41×C61—42no—
136#41×C61—42no—
137#41×C61—82no—
138#41×C61—82no—
139C2×Q8×C61—21no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61—81no—
143C2×Q8×C61—81no—
144C2×Q8×C61C2×D411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—22no—
147C2×D4×C61—24no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C2×D4×C61C2×D412yes—
152C22×C4×C61—24no—
153C22×C4×C61—22no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—22no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—21no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D414yes—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D412yes—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—22no—
183C42×C61—24no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2411yescharacteristic
191C22⋊Dic61—41no—
192#11758202—42no—
193#11758200—42no—
194#11758200—82no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—41no—
202#11758194—24no—
203#11758194—81no—
204C2×#92418—22no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—22no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61#12011yescharacteristic
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—42no—
214Q8×C61—42no—
215Q8×C61—42no—
216Q8×C61—82no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—22no—
220D4×C61—22no—
221D4×C61—42no—
222D4×C61—42no—
223D4×C61—44no—
224D4×C61—42no—
225D4×C61—84no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—24no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—82no—
243C2×C4×C61—84no—
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×Dic61—41no—
251C2×Dic61—41no—
252Dic122—41no—
253Dic122—41no—
254Dic122—41no—
255Dic122—41no—
256#92418—24no—
257#92418—41no—
258#92418—41no—
259#92418—42no—
260#18593—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—41no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61—42no—
268C22×C61—41no—
269C22×C61—42no—
270C22×C61C2×#12011yescharacteristic
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61—41no—
277C4×C61—44no—
278C4×C61—41no—
279C4×C61#45612yes—
280C4×C61#57511yescharacteristic
281C4×C61#57511yescharacteristic
282C4×C61#57611yescharacteristic
283C4×C61#57711yescharacteristic
284Dic61—81no—
285Dic61—81no—
286#3511D6111yescharacteristic
287#3254—611no—
288#3165—611no—
289#3107—611no—
290#1654—612no—
291#1652—612no—
292#1650—612no—
293#1647—612no—
294#1546—611no—
295#1489—611no—
296#1484—611no—
297C2×C61—41no—
298C2×C61—81no—
299C2×C61—82no—
300C2×C61#300511yescharacteristic
301C2×C61#301211yescharacteristic
302C2×C61C2×#45611yescharacteristic
303#556C2×D6112yes—
304#555—21no—
305#554—21no—
306#553—22no—
307#548—22no—
308#547C2×D6112yes—
309#542—22no—
310#537—21no—
311#533—21no—
312#533—611no—
313#532—21no—
314#532C2×D6111yescharacteristic
315C2×C4⋊Q8C2×D6111yescharacteristic
316#520—21no—
317C4×#41C2×D6111yescharacteristic
318#501—1222no—
319#500—612no—
320#498—612no—
321#485—1221no—
322#477—1221no—
323#475—611no—
324#474—611no—
325#474—612no—
326#468—1222no—
327#462—611no—
328#461—611no—
329#461—612no—
330#460—1221no—
331#452—1221no—
332C4×Q16—1221no—
333C4×SD16—1221no—
334#431—611no—
335#428—1222no—
336#423—611no—
337C4×M16—611no—
338#332—612no—
339#332—612no—
340Q8⋊C8—612no—
341D4⋊C8—612no—
342D4⋊C8—612no—
343D4⋊C8—612no—
344C61#1859311yescharacteristic, complemented, Sylow
345C2×#41—22no—
346C2×#41—21no—
347C2×#41C22⋊D6111yescharacteristic
348C22×Q8—21no—
349C2×Q16—1221no—
350C2×SD16—1222no—
351C2×SD16—1221no—
352C2×M16—1221no—
353C2×M16—1221no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8C22⋊D6112yes—
359C4⋊Q8C22⋊D6111yescharacteristic
360C4⋊Q8C22×D6112yes—
361#127C22⋊D6111yescharacteristic
362#126—24no—
363#125—22no—
364#125—1221no—
365#124—24no—
366#124—22no—
367#124—21no—
368#123—24no—
369C22⋊Q8—22no—
370C22⋊Q8—24no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375#121—22no—
376#121—22no—
377#121—24no—
378#121—22no—
379#121—21no—
380#121C22⋊D6112yes—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×Q8C22⋊D6111yescharacteristic
386C4×Q8C22×D6111yescharacteristic
387C4×D4—22no—
388C4×D4—24no—
389C4×D4—22no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4C22⋊D6111yescharacteristic
394C4×D4C22×D6111yescharacteristic
395C4×D4C22×D6112yes—
396#117—22no—
397#117—22no—
398#117—21no—
399#117—1221no—
400#117C22⋊D6111yescharacteristic
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4C22⋊D6112yes—
404C2×C42—21no—
405C2×C42C22×D6111yescharacteristic
406#107—1224no—
407#107—1221no—
408#107—1222no—
409#106—1224no—
410#106—1221no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—612no—
414C4⋊C8—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1222no—
419#102—1221no—
420#102—1222no—
421C22⋊C8—1221no—
422#97—1221no—
423C4×C8—612no—
424C4×C8—1221no—
425#41—41no—
426#41—41no—
427#41—42no—
428#41—42no—
429#41—82no—
430#41—82no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—81no—
435C2×Q8—81no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8C2×C22⋊D6111yescharacteristic
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C22⋊D6111yescharacteristic
445C2×D4C2×C22⋊D6112yes—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455Q16—2442no—
456SD16—4882no—
457SD16—2442no—
458M16—1224no—
459M16—4881no—
460C2×C8—1222no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1222no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C4⋊C4C2×C22⋊D6111yescharacteristic
480C4⋊C4C2×C22⋊D6112yes—
481C4⋊C4C2×C22⋊D6114yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—2441no—
491C42—24no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—2441no—
496C42C2×C22⋊D6111yescharacteristic
497C42C2×C22⋊D6111yescharacteristic
498C42C2×C22⋊D6111yescharacteristic
499C42C23×D6111yescharacteristic
500C23—21no—
501C23—22no—
502C23#1451974611yescharacteristic
503Q8—21no—
504Q8—21no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—82no—
509Q8—2441no—
510Q8—2441no—
511Q8—2441no—
512Q8—2441no—
513D4—21no—
514D4—21no—
515D4—22no—
516D4—22no—
517D4—42no—
518D4—42no—
519D4—44no—
520D4—42no—
521D4—84no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—82no—
539C2×C4—84no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4#1451974611yescharacteristic
543C2×C4#1451974611yescharacteristic
544C2×C4#1451974611yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4C22×C22⋊D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C8—1224no—
549C8—2441no—
550C8—2441no—
551C8—2442no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—42no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22C2×#1451974611yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—44no—
567C4—41no—
568C4—41no—
569C4—42no—
570C4—4881no—
571C4—4881no—
572C4#1683994512yes—
573C4#1684056011yescharacteristic
574C4#1684056011yescharacteristic
575C4#1684056511yescharacteristic
576C4#1684057311yescharacteristic
577C2—41no—
578C2—81no—
579C2—82no—
580C2#2145506111yescharacteristic
581C2#2145512311yescharacteristic
582C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#18593
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21457206C2semidirect
2. #21456361C2semidirect
3. #21455890C2semidirect
4. #21445103C2semidirect
5. #21445089C2semidirect
6. #21445075C2semidirect
7. #21445054C2semidirect
8. #21444491C2semidirect
9. #21444258C2semidirect
10. #21444236C2semidirect
344. C61#18593semidirect

Maximal subgroups

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

subgroupindex
#214572062
#214563612
#214558902
#214451032
#214450892
#214450752
#214450542
#214444912
#214442582
#214442362
C61×#35112
#1859361