Finite Group Databasebeta

← #29657019

Subgroups — #29657019

Order 15616.

1140 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456810C211yesmaximal, characteristic, complemented
2#21456553C211yesmaximal, characteristic, complemented
3#21455850C211yesmaximal, characteristic, complemented
4#21445082C212yesmaximal, complemented
5#21445019C212yesmaximal, complemented
6#21444922C212yesmaximal, complemented
7#21444884C212yesmaximal, complemented
8#21444524C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3497C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840158C2211yescharacteristic
14#16840149—21no—
15#16840138C2211yescharacteristic
16#16840124—21no—
17#16840110C2211yescharacteristic
18#16840108C2211yescharacteristic
19#16840080—21no—
20#16840047C2212yes—
21#16840013—22no—
22#16839967C2212yes—
23#16839962C2212yes—
24#16839957C2212yes—
25#16839905—22no—
26#16839901—21no—
27#16839872—22no—
28C4×#11758201—22no—
29#16839819C2211yescharacteristic
30#16839815C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839508C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C2×2−1+4×C61—21no—
39C61×#550—21no—
40C61×#550C2212yes—
41D4×Q8×C61—21no—
42D4×Q8×C61C2212yes—
43C61×#548—22no—
44C61×#548—22no—
45C61×#537—22no—
46C61×#537—21no—
47C61×#532—22no—
48C61×#532C2211yescharacteristic
49C61×#532C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758222—21no—
52C2×#11758200—22no—
53C2×#11758201—22no—
54C2×#11758199—22no—
55C2×#11758199—22no—
56C2×#11758194—21no—
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×D244—21no—
63C4×D244—21no—
64#14519708—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519701—22no—
69#14519700—22no—
70#14519698—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74C22⋊#92418—21no—
75C22⋊#92418—21no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
812−1+4×C61—48no—
82C2×#41×C61—24no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—21no—
86C2×#41×C61—21no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61D411yescharacteristic
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61D411yescharacteristic
92C4⋊Q8×C61C2311yescharacteristic
93#127×C61—22no—
94#127×C61D411yescharacteristic
95#127×C61C2311yescharacteristic
96#124×C61—24no—
97#124×C61—21no—
98#124×C61—21no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—24no—
103#121×C61—24no—
104#121×C61—22no—
105#121×C61—24no—
106#121×C61—24no—
107#121×C61—21no—
108#121×C61—21no—
109#121×C61—22no—
110#121×C61D412yes—
111#121×C61D412yes—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61D412yes—
115C4×Q8×C61C2312yes—
116C4×D4×C61—24no—
117C4×D4×C61—22no—
118C4×D4×C61—22no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61D412yes—
122C4×D4×C61C2312yes—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#41×C61—416no—
128#41×C61—44no—
129#41×C61—44no—
130#41×C61—48no—
131#41×C61—44no—
132#41×C61—44no—
133C2×Q8×C61—24no—
134C2×Q8×C61—24no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—48no—
139C2×Q8×C61—44no—
140C2×Q8×C61—44no—
141C2×Q8×C61C2×D412yes—
142C2×D4×C61—24no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—24no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D412yes—
151C2×D4×C61C2×D412yes—
152C22×C4×C61—21no—
153C22×C4×C61—24no—
154C22×C4×C61—24no—
155C22×C4×C61—21no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C2×C22⋊D61—41no—
162#11758222—82no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D412yes—
176C4⋊C4×C61C2×D412yes—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C42×C61—22no—
184C42×C61—22no—
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2411yescharacteristic
189#11758201—44no—
190#11758200—82no—
191#11758199—44no—
192#11758199—82no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197#11758194—42no—
198C2×#92418—21no—
199C2×#92418—22no—
200C2×#92418—21no—
201C2×#92418—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61#12011yescharacteristic
205C22×D61—41no—
206Q8×C61—24no—
207Q8×C61—22no—
208Q8×C61—48no—
209Q8×C61—42no—
210Q8×C61—42no—
211Q8×C61AGL(1,ℤ/8)14yes—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—42no—
215D4×C61—48no—
216D4×C61—42no—
217D4×C61—44no—
218D4×C61—44no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—81no—
222D4×C61—84no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61—48no—
232C2×C4×C61—48no—
233C2×C4×C61—44no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—81no—
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C22⋊D61—82no—
244C2×Dic61—41no—
245D244—42no—
246D244—42no—
247D244—42no—
248C4×D61—82no—
249Dic122—42no—
250#92418—22no—
251#92418—41no—
252#92418—42no—
253#92418—42no—
254#18610—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61C2×#12011yescharacteristic
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C4×C61—24no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61C2×AGL(1,ℤ/8)12yes—
277C4×C61#57712yes—
278Dic61—81no—
279#3497D6111yescharacteristic, complemented
280#3214—611no—
281#3184—611no—
282#3103—611no—
283#1651—612no—
284#1642—612no—
285#1627—612no—
286#1621—612no—
287#1553—611no—
288#1491—611no—
289#1484—611no—
290C2×C61—41no—
291C2×C61—82no—
292C2×C61—82no—
293C2×C61#300311yescharacteristic
294C2×C61#301011yescharacteristic
295C2×C61#301811yescharacteristic
296D61—161no—
297C2×2−1+4—21no—
298#550—21no—
299#550C2×D6112yes—
300D4×Q8—21no—
301D4×Q8C2×D6112yes—
302#548—22no—
303#548—22no—
304#537—22no—
305#537—21no—
306#532—22no—
307#532—611no—
308#532C2×D6111yescharacteristic
309#532C2×D6111yescharacteristic
310C2×C4×Q8C2×D6111yescharacteristic
311#498—611no—
312#496—1221no—
313#493—611no—
314#489—1221no—
315#486—611no—
316#486—611no—
317#481—1221no—
318#475—612no—
319#469—1222no—
320#461—612no—
321#460—612no—
322#459—612no—
323#449—1222no—
324#448—1221no—
325#442—1222no—
326C4×SD16—1222no—
327#432—611no—
328#431—611no—
329C4×M16—611no—
330#332—612no—
331#331—612no—
332Q8⋊C8—612no—
333Q8⋊C8—612no—
334D4⋊C8—612no—
335D4⋊C8—612no—
336C61#1861011yescharacteristic, complemented, Sylow
3372−1+4—48no—
338C2×#41—24no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41—1221no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8C22⋊D6111yescharacteristic
347C2×SD16—1222no—
348C2×SD16—1222no—
349C2×D8—1222no—
350C2×D8—1222no—
351C2×M16—1221no—
352C4⋊Q8—22no—
353C4⋊Q8—611no—
354C4⋊Q8C22⋊D6111yescharacteristic
355C4⋊Q8C22×D6111yescharacteristic
356#127—22no—
357#127—611no—
358#127C22⋊D6111yescharacteristic
359#127C22×D6111yescharacteristic
360#124—24no—
361#124—21no—
362#124—21no—
363#124—1221no—
364C22⋊Q8—22no—
365C22⋊Q8—24no—
366C22⋊Q8—22no—
367C22⋊Q8—24no—
368#121—24no—
369#121—22no—
370#121—24no—
371#121—24no—
372#121—21no—
373#121—21no—
374#121—22no—
375#121—1221no—
376#121—1221no—
377#121C22⋊D6112yes—
378#121C22⋊D6112yes—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8C22⋊D6112yes—
382C4×Q8C22×D6112yes—
383C4×D4—24no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—1221no—
389C4×D4—1221no—
390C4×D4C22⋊D6112yes—
391C4×D4C22×D6112yes—
392C2×C4⋊C4—21no—
393C2×C4⋊C4C22⋊D6111yescharacteristic
394C2×C42—21no—
395C2×C42C22×D6111yescharacteristic
396#106—1222no—
397#106—1222no—
398C4⋊C8—612no—
399C4⋊C8—612no—
400#103—1222no—
401#103—1222no—
402#102—1222no—
403#102—1222no—
404#102—1222no—
405#102—1222no—
406C22⋊C8—1221no—
407C22⋊C8—1221no—
408#97—1221no—
409#97—1221no—
410C4×C8—611no—
411C4×C8—611no—
412C4×C8—1221no—
413#41—416no—
414#41—44no—
415#41—44no—
416#41—48no—
417#41—44no—
418#41—44no—
419#41—4882no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—1221no—
429C2×Q8C2×C22⋊D6112yes—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—2441no—
442C2×D4C2×C22⋊D6112yes—
443C2×D4C2×C22⋊D6112yes—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—2441no—
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454SD16—4882no—
455SD16—2444no—
456D8—4882no—
457D8—2444no—
458M16—2442no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1222no—
462C2×C8—1222no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4C2×C22⋊D6112yes—
472C4⋊C4C2×C22⋊D6112yes—
473C4⋊C4C2×C22⋊D6112yes—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—2441no—
481C22⋊C4—2441no—
482C42—22no—
483C42—22no—
484C42C2×C22⋊D6111yescharacteristic
485C42C2×C22⋊D6111yescharacteristic
486C42C2×C22⋊D6111yescharacteristic
487C42C23×D6111yescharacteristic
488C23—22no—
489C23—22no—
490C23—2441no—
491C23#1451974611yescharacteristic
492Q8—24no—
493Q8—22no—
494Q8—48no—
495Q8—42no—
496Q8—42no—
497Q8—2442no—
498Q8#1451980314yes—
499D4—22no—
500D4—22no—
501D4—42no—
502D4—48no—
503D4—42no—
504D4—44no—
505D4—44no—
506D4—42no—
507D4—42no—
508D4—81no—
509D4—84no—
510D4—4882no—
511D4—2442no—
512D4—2442no—
513D4—2442no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—4882no—
529C2×C4—2441no—
530C2×C4#1451974611yescharacteristic
531C2×C4#1451974611yescharacteristic
532C2×C4#1451974611yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C22×C22⋊D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C8—1222no—
537C8—2441no—
538C8—2442no—
539C8—2442no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—4881no—
550C22—4881no—
551C22—4881no—
552C22C2×#1451974611yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—24no—
557C4—21no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—4881no—
562C4C2×#1451980312yes—
563C4#1684057012yes—
564C2—41no—
565C2—82no—
566C2—82no—
567C2—9761no—
568C2#2145504611yescharacteristic
569C2#2145510511yescharacteristic
570C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18610
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21456810C2semidirect
2. #21456553C2semidirect
3. #21455850C2semidirect
4. #21445082C2semidirect
5. #21445019C2semidirect
6. #21444922C2semidirect
7. #21444884C2semidirect
8. #21444524C2semidirect
9. #21444266C2semidirect
10. #21444236C2semidirect
11. C61×#3497C2semidirect
279. #3497D61semidirect
336. C61#18610semidirect

Maximal subgroups

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

subgroupindex
#214568102
#214565532
#214558502
#214450822
#214450192
#214449222
#214448842
#214445242
#214442662
#214442362
C61×#34972
#1861061