Finite Group Databasebeta

← #29656514

Subgroups — #29656514

Order 15616.

980 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457202C211yesmaximal, characteristic, complemented
2#21456276C211yesmaximal, characteristic, complemented
3#21455879C211yesmaximal, characteristic
4#21444959C212yesmaximal, complemented
5#21444953C212yesmaximal, complemented
6#21444897C212yesmaximal, complemented
7#21444891C212yesmaximal, complemented
8#21444500C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3503C211yesmaximal, characteristic, complemented
12C2×#14519808—22no—
13#16840312C2211yescharacteristic
14#16840154—22no—
15#16840141C2212yes—
16#16840136C2212yes—
17#16840086—21no—
18#16840057C2211yescharacteristic
19#16840057C2212yes—
20#16840052C2211yescharacteristic
21#16840039—21no—
22#16840010—22no—
23#16839964—21no—
24#16839962C2211yescharacteristic
25#16839962C2212yes—
26#16839957C2211yescharacteristic
27#16839907—21no—
28C4×#11758201—21no—
29C4×#11758199—21no—
30#16839815C2211yescharacteristic
31#16839786C2211yescharacteristic
32C4×#11758194C2211yescharacteristic
33#16839511C2212yes—
34#16839511C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37Q8⋊#92418C2212yes—
38D4⋊#92418C2212yes—
39Q82×C61C2212yes—
40C61×#557—21no—
41C61×#554—22no—
42C61×#554C2212yes—
43D4×Q8×C61—22no—
44D4×Q8×C61—21no—
45C61×#544—22no—
46C61×#544—21no—
47C61×#533—21no—
48C61×#533C2211yescharacteristic
49C2×C4⋊Q8×C61—21no—
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#520—21no—
52C4×#41×C61C2211yescharacteristic
53C2×#11758222—21no—
54#14519808—48no—
55C2×#11758202—24no—
56C2×#11758201—21no—
57C2×#11758201—24no—
58C2×#11758201—22no—
59C2×#11758199—21no—
60C2×#11758194—21no—
61C2×#11758194—21no—
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519710—22no—
70#14519710—21no—
71#14519708—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75C4⋊#92418C2311yescharacteristic
76#14519701—22no—
77#14519701—21no—
78#14519698—21no—
79#14519697—21no—
80#14519697—22no—
81#14519697—21no—
82C22⋊#92418—21no—
83#14519688—21no—
84C4×#92418—21no—
85C4×#92418C2312yes—
86C2×#41×C61D411yescharacteristic
87C22×Q8×C61—22no—
88C22×Q8×C61D412yes—
89C4⋊Q8×C61—22no—
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61—24no—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61—21no—
95C4⋊Q8×C61—21no—
96C4⋊Q8×C61D412yes—
97C4⋊Q8×C61D411yescharacteristic
98C4⋊Q8×C61D411yescharacteristic
99C4⋊Q8×C61C2312yes—
100#125×C61—22no—
101#125×C61—24no—
102#125×C61—22no—
103#125×C61—21no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—24no—
106C22⋊Q8×C61—24no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—24no—
109C22⋊Q8×C61—21no—
110C22⋊Q8×C61D412yes—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C4×Q8×C61—22no—
114C4×Q8×C61—21no—
115C4×Q8×C61D411yescharacteristic
116C4×Q8×C61C2312yes—
117C4×Q8×C61C2311yescharacteristic
118C4×D4×C61—24no—
119C4×D4×C61—22no—
120C4×D4×C61—21no—
121C4×D4×C61—21no—
122C4×D4×C61D411yescharacteristic
123C4×D4×C61C2311yescharacteristic
124#117×C61—22no—
125#117×C61—21no—
126#117×C61—21no—
127#117×C61D411yescharacteristic
128C2×C4⋊C4×C61—22no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—22no—
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#41×C61—22no—
135#41×C61—22no—
136C2×Q8×C61—24no—
137C2×Q8×C61—22no—
138C2×Q8×C61—21no—
139C2×Q8×C61—44no—
140C2×Q8×C61—84no—
141C2×Q8×C61C2×D412yes—
142C2×Q8×C61C2×D411yescharacteristic
143C2×Q8×C61C2×D414yes—
144C2×D4×C61—21no—
145C2×D4×C61C2×D411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—24no—
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C2×C22⋊D61—41no—
155#11758222—44no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×C4×D61—41no—
160C2×Dic122—21no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C2×D412yes—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—24no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C42×C61—24no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—22no—
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2411yescharacteristic
193#11758202—48no—
194#11758201—48no—
195#11758201—42no—
196#11758201—82no—
197#11758199—42no—
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202#11758194—24no—
203#11758194—81no—
204C2×#92418—22no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—22no—
208C23×C61—21no—
209C23×C61#12011yescharacteristic
210C22×D61—41no—
211Q8×C61—21no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—44no—
215Q8×C61—44no—
216Q8×C61—42no—
217Q8×C61—84no—
218Q8×C61#13712yescomplemented
219D4×C61—24no—
220D4×C61—21no—
221D4×C61#13712yescomplemented
222C2×C4×C61—24no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—82no—
238C2×C4×C61—84no—
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C22⋊D61—82no—
246C2×Dic61—41no—
247D244—42no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252C4×D61—82no—
253Dic122—42no—
254#92418—24no—
255#92418—41no—
256#92418—41no—
257#92418—42no—
258#18567—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—22no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61C2×#12011yescharacteristic
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—42no—
275C4×C61—41no—
276C4×C61—44no—
277C4×C61—42no—
278C4×C61#45312yes—
279C4×C61C2×#13711yescharacteristic
280C4×C61C2×#13711yescharacteristic
281C4×C61#57711yescharacteristic
282C4×C61#57811yescharacteristic
283Dic61—81no—
284#3503D6111yescharacteristic, complemented
285#3254—611no—
286#3152—611no—
287#3106—611no—
288#1633—612no—
289#1632—612no—
290#1623—612no—
291#1622—612no—
292#1548—611no—
293#1489—611no—
294#1485—611no—
295C2×C61—41no—
296C2×C61—42no—
297C2×C61#300411yescharacteristic
298C2×C61#301311yescharacteristic
299C2×C61C2×#45311yescharacteristic
300D61—161no—
301C2×#137—1222no—
302Q82C2×D6112yes—
303#557—21no—
304#554—22no—
305#554C2×D6112yes—
306D4×Q8—22no—
307D4×Q8—21no—
308#544—22no—
309#544—21no—
310#533—21no—
311#533C2×D6111yescharacteristic
312#532—611no—
313C2×C4⋊Q8—21no—
314C2×C4⋊Q8C2×D6111yescharacteristic
315#520—21no—
316C4×#41C2×D6111yescharacteristic
317#497—1222no—
318#494—612no—
319#492—612no—
320#482—1221no—
321#477—611no—
322#477—612no—
323#476—611no—
324#474—1221no—
325#468—1222no—
326#461—1221no—
327#460—611no—
328#460—612no—
329#459—611no—
330#449—1221no—
331C4×SD16—1221no—
332C4×D8—1221no—
333#431—611no—
334#423—611no—
335C4×M16—611no—
336#333—612no—
337#333—612no—
338Q8⋊C8—612no—
339Q8⋊C8—612no—
340Q8⋊C8—612no—
341D4⋊C8—612no—
342C61#1856711yescharacteristic, complemented, Sylow
343C2×#41—1221no—
344C2×#41C22⋊D6111yescharacteristic
345C22×Q8—22no—
346C22×Q8C22⋊D6112yes—
347#137—2448no—
348C2×Q16—1224no—
349C2×SD16—1222no—
350C2×SD16—1224no—
351C2×SD16—1221no—
352C2×D8—1221no—
353C2×M16—1221no—
354C2×M16—1221no—
355C4⋊Q8—22no—
356C4⋊Q8—22no—
357C4⋊Q8—24no—
358C4⋊Q8—22no—
359C4⋊Q8—22no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—611no—
363C4⋊Q8C22⋊D6112yes—
364C4⋊Q8C22⋊D6111yescharacteristic
365C4⋊Q8C22⋊D6111yescharacteristic
366C4⋊Q8C22×D6112yes—
367#127—611no—
368#125—22no—
369#125—24no—
370#125—22no—
371#125—21no—
372#124—1221no—
373C22⋊Q8—22no—
374C22⋊Q8—24no—
375C22⋊Q8—24no—
376C22⋊Q8—22no—
377C22⋊Q8—24no—
378C22⋊Q8—21no—
379C22⋊Q8C22⋊D6112yes—
380#121—1221no—
381#121—1221no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—21no—
386C4×Q8C22⋊D6111yescharacteristic
387C4×Q8C22×D6111yescharacteristic
388C4×Q8C22×D6112yes—
389C4×D4—24no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—1221no—
394C4×D4—1221no—
395C4×D4C22⋊D6111yescharacteristic
396C4×D4C22×D6111yescharacteristic
397#117—22no—
398#117—21no—
399#117—21no—
400#117C22⋊D6111yescharacteristic
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C42—21no—
406C2×C42C22×D6111yescharacteristic
407#107—1221no—
408#107—1222no—
409#106—1221no—
410C4⋊C8—612no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414#103—1221no—
415#103—1222no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1222no—
420C22⋊C8—1221no—
421#97—1221no—
422C4×C8—612no—
423C4×C8—1221no—
424#41—22no—
425#41—22no—
426#41—2444no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—44no—
431C2×Q8—84no—
432C2×Q8—1221no—
433C2×Q8C2×C22⋊D6112yes—
434C2×Q8C2×C22⋊D6111yescharacteristic
435C2×Q8C2×C22⋊D6114yes—
436C2×D4—21no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—2441no—
441C2×D4C2×C22⋊D6111yescharacteristic
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—2441no—
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451Q16—2448no—
452SD16—4882no—
453SD16—2448no—
454SD16—2442no—
455D8—2442no—
456M16—1224no—
457M16—4881no—
458C2×C8—1222no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1222no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4C2×C22⋊D6111yescharacteristic
480C4⋊C4C2×C22⋊D6112yes—
481C4⋊C4C2×C22⋊D6111yescharacteristic
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C42—24no—
490C42—21no—
491C42—22no—
492C42—21no—
493C42—21no—
494C42C2×C22⋊D6111yescharacteristic
495C42C2×C22⋊D6111yescharacteristic
496C42C2×C22⋊D6111yescharacteristic
497C42C23×D6111yescharacteristic
498C23—21no—
499C23—2441no—
500C23#1451974611yescharacteristic
501Q8—21no—
502Q8—22no—
503Q8—22no—
504Q8—44no—
505Q8—44no—
506Q8—42no—
507Q8—84no—
508Q8—2442no—
509Q8#1451980812yescomplemented
510D4—24no—
511D4—21no—
512D4—4882no—
513D4—2442no—
514D4—2441no—
515D4—2441no—
516D4—2441no—
517D4—2441no—
518D4#1451980812yescomplemented
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—4882no—
537C2×C4—2441no—
538C2×C4#1451974611yescharacteristic
539C2×C4#1451974611yescharacteristic
540C2×C4#1451974611yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C8—1224no—
545C8—2441no—
546C8—2441no—
547C8—2442no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—41no—
553C22—41no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22C2×#1451974611yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—22no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—44no—
566C4—42no—
567C4—4881no—
568C4#1683993012yes—
569C4C2×#1451980811yescharacteristic
570C4C2×#1451980811yescharacteristic
571C4#1684057011yescharacteristic
572C4#1684057711yescharacteristic
573C2—41no—
574C2—42no—
575C2—9761no—
576C2#2145505311yescharacteristic
577C2#2145513411yescharacteristic
578C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3503C2
socleC22×C61C2×#120
prime radical#21455879C2
hypercenter#3503D61
hypocenterC61#18567
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. #21457202C2semidirect
2. #21456276C2semidirect
4. #21444959C2semidirect
5. #21444953C2semidirect
6. #21444897C2semidirect
7. #21444891C2semidirect
8. #21444500C2semidirect
9. #21444258C2semidirect
10. #21444240C2semidirect
11. C61×#3503C2semidirect
218. Q8×C61#137semidirect
221. D4×C61#137semidirect
284. #3503D61semidirect
342. C61#18567semidirect
509. Q8#14519808semidirect
518. D4#14519808semidirect

Maximal subgroups

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

subgroupindex
#214572022
#214562762
#214558792
#214449592
#214449532
#214448972
#214448912
#214445002
#214442582
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C61×#35032
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