Finite Group Databasebeta

← #29656701

Subgroups — #29656701

Order 15616.

882 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#21457202C211yesmaximal, characteristic, complemented
2#21456215C211yesmaximal, characteristic, complemented
3#21455879C211yesmaximal, characteristic
4#21444991C212yesmaximal, complemented
5#21444977C212yesmaximal, complemented
6#21444929C212yesmaximal, complemented
7#21444915C212yesmaximal, complemented
8#21444504C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3509C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840124—22no—
14#16840114C2212yes—
15#16840110C2212yes—
16#16840086—21no—
17#16840057C2211yescharacteristic
18#16840057C2212yes—
19#16840052C2211yescharacteristic
20#16840039—21no—
21#16840010—22no—
22#16839964—21no—
23#16839962C2211yescharacteristic
24#16839962C2212yes—
25#16839957C2211yescharacteristic
26#16839907—21no—
27C4×#11758201—21no—
28C4×#11758199—21no—
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839761—22no—
32C4×#11758194C2211yescharacteristic
33#16839508C2212yes—
34#16839508C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37Q8⋊#92418C2212yes—
38D4⋊#92418C2212yes—
39C61×#557—21no—
40C61×#557C2212yes—
41C61×#556—21no—
42C61×#556C2212yes—
43C61×#552—22no—
44C61×#552—22no—
45C61×#541—22no—
46C61×#536—21no—
47C61×#533—21no—
48C61×#533—21no—
49C61×#533C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#519—21no—
52C4×#41×C61C2211yescharacteristic
53C2×#11758222—21no—
54C2×#11758201—21no—
55C2×#11758201—22no—
56C2×#11758199—21no—
57C2×#11758194—21no—
58C2×#11758194—21no—
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519710—22no—
67#14519710—21no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2311yescharacteristic
71C4⋊#92418C2312yes—
72C4⋊#92418C2311yescharacteristic
73#14519701—22no—
74#14519701—21no—
75#14519700—24no—
76#14519698—21no—
77#14519697—21no—
78#14519697—22no—
79#14519697—21no—
80#14519697—24no—
81C22⋊#92418—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
85C2×#41×C61D411yescharacteristic
86C22×Q8×C61—21no—
87C4⋊Q8×C61—22no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61D411yescharacteristic
91C4⋊Q8×C61D412yes—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61C2312yes—
94#126×C61—24no—
95#126×C61—24no—
96#125×C61—22no—
97#125×C61—24no—
98#125×C61—22no—
99#125×C61—24no—
100#125×C61—21no—
101#124×C61—22no—
102#123×C61—24no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61D412yes—
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C4×Q8×C61—22no—
114C4×Q8×C61—22no—
115C4×Q8×C61—21no—
116C4×Q8×C61D411yescharacteristic
117C4×Q8×C61C2312yes—
118C4×Q8×C61C2311yescharacteristic
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61D411yescharacteristic
122C4×D4×C61C2311yescharacteristic
123#117×C61—22no—
124#117×C61—24no—
125#117×C61—21no—
126#117×C61—22no—
127#117×C61—21no—
128#117×C61D411yescharacteristic
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61D412yes—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#41×C61—41no—
134#41×C61—41no—
135C2×Q8×C61—21no—
136C2×Q8×C61—22no—
137C2×Q8×C61—22no—
138C2×Q8×C61—42no—
139C2×Q8×C61—42no—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×D411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61C2×D411yescharacteristic
144C22×C4×C61—22no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—41no—
152#11758222—44no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×C4×D61—41no—
157C2×Dic122—21no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61C2×D414yes—
173C4⋊C4×C61C2×D412yes—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—24no—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C42×C61—24no—
183C42×C61—24no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—22no—
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2411yescharacteristic
192#11758201—42no—
193#11758201—82no—
194#11758199—42no—
195#11758198—41no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199#11758194—42no—
200#11758194—81no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—22no—
204C2×#92418—22no—
205C23×C61—21no—
206C23×C61#12011yescharacteristic
207C22×D61—41no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61—44no—
213Q8×C61—42no—
214Q8×C61—42no—
215Q8×C61—42no—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—42no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—21no—
230C2×C4×C61—42no—
231C2×C4×C61—44no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—42no—
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C22⋊D61—82no—
242C2×Dic61—41no—
243D244—42no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248C4×D61—82no—
249Dic122—42no—
250#92418—42no—
251#92418—41no—
252#92418—41no—
253#92418—42no—
254#18584—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61C2×#12011yescharacteristic
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—44no—
270C4×C61—41no—
271C4×C61—42no—
272C4×C61—41no—
273C4×C61—42no—
274C4×C61#45412yes—
275C4×C61#57511yescharacteristic
276C4×C61#57511yescharacteristic
277C4×C61#57711yescharacteristic
278C4×C61#57811yescharacteristic
279Dic61—81no—
280#3509D6111yescharacteristic, complemented
281#3254—611no—
282#3141—611no—
283#3106—611no—
284#1638—612no—
285#1636—612no—
286#1628—612no—
287#1626—612no—
288#1549—611no—
289#1489—611no—
290#1485—611no—
291C2×C61—41no—
292C2×C61—81no—
293C2×C61#300511yescharacteristic
294C2×C61#301311yescharacteristic
295C2×C61C2×#45411yescharacteristic
296D61—161no—
297#557—21no—
298#557C2×D6112yes—
299#556—21no—
300#556C2×D6112yes—
301#552—22no—
302#552—22no—
303#541—22no—
304#536—21no—
305#533—21no—
306#533—21no—
307#533C2×D6111yescharacteristic
308#532—611no—
309C2×C4⋊Q8C2×D6111yescharacteristic
310#519—21no—
311C4×#41C2×D6111yescharacteristic
312#489—1222no—
313#487—612no—
314#486—612no—
315#482—1221no—
316#477—611no—
317#477—612no—
318#476—611no—
319#474—1221no—
320#468—1222no—
321#461—1221no—
322#460—611no—
323#460—612no—
324#459—611no—
325#449—1221no—
326C4×SD16—1221no—
327C4×D8—1221no—
328#431—611no—
329#423—611no—
330#417—1222no—
331C4×M16—611no—
332#332—612no—
333#332—612no—
334Q8⋊C8—612no—
335Q8⋊C8—612no—
336Q8⋊C8—612no—
337D4⋊C8—612no—
338C61#1858411yescharacteristic, complemented, Sylow
339C2×#41—1221no—
340C2×#41C22⋊D6111yescharacteristic
341C22×Q8—21no—
342C2×SD16—1222no—
343C2×SD16—1221no—
344C2×D8—1221no—
345C2×M16—1221no—
346C2×M16—1221no—
347C4⋊Q8—22no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—611no—
351C4⋊Q8C22⋊D6111yescharacteristic
352C4⋊Q8C22⋊D6112yes—
353C4⋊Q8C22⋊D6111yescharacteristic
354C4⋊Q8C22×D6112yes—
355#127—611no—
356#126—24no—
357#126—24no—
358#125—22no—
359#125—24no—
360#125—22no—
361#125—24no—
362#125—21no—
363#124—22no—
364#124—1221no—
365#123—24no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8C22⋊D6112yes—
372#121—1221no—
373#121—1221no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×Q8—22no—
380C4×Q8—21no—
381C4×Q8C22⋊D6111yescharacteristic
382C4×Q8C22×D6111yescharacteristic
383C4×Q8C22×D6112yes—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—1221no—
387C4×D4—1221no—
388C4×D4C22⋊D6111yescharacteristic
389C4×D4C22×D6111yescharacteristic
390#117—24no—
391#117—22no—
392#117—21no—
393#117—22no—
394#117—21no—
395#117C22⋊D6111yescharacteristic
396C2×C4⋊C4—21no—
397C2×C4⋊C4C22⋊D6112yes—
398C2×C42—21no—
399C2×C42C22×D6111yescharacteristic
400#107—1221no—
401#107—1222no—
402#106—1221no—
403C4⋊C8—612no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—1221no—
407#103—1221no—
408#103—1222no—
409#103—1224no—
410#102—1221no—
411#102—1224no—
412#102—1221no—
413#102—1221no—
414#102—1222no—
415C22⋊C8—1221no—
416#97—1221no—
417C4×C8—612no—
418C4×C8—1221no—
419#41—41no—
420#41—41no—
421#41—2444no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—1221no—
428C2×Q8C2×C22⋊D6112yes—
429C2×Q8C2×C22⋊D6111yescharacteristic
430C2×D4—21no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—2441no—
435C2×D4C2×C22⋊D6111yescharacteristic
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—2441no—
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444SD16—4882no—
445SD16—2442no—
446D8—2442no—
447M16—4881no—
448M16—2442no—
449C2×C8—1222no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1222no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4C2×C22⋊D6114yes—
470C4⋊C4C2×C22⋊D6111yescharacteristic
471C4⋊C4C2×C22⋊D6112yes—
472C4⋊C4C2×C22⋊D6111yescharacteristic
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—21no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—2441no—
480C22⋊C4—2441no—
481C42—24no—
482C42—24no—
483C42—21no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42C2×C22⋊D6111yescharacteristic
488C42C2×C22⋊D6111yescharacteristic
489C42C2×C22⋊D6111yescharacteristic
490C42C23×D6111yescharacteristic
491C23—21no—
492C23—2441no—
493C23#1451974611yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—22no—
497Q8—22no—
498Q8—44no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—2442no—
503D4—21no—
504D4—21no—
505D4—42no—
506D4—4882no—
507D4—2442no—
508D4—2441no—
509D4—2441no—
510D4—2441no—
511D4—2441no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
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—2442no—
537C8—2441no—
538C8—2441no—
539C8—2442no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—4881no—
547C22—4881no—
548C22—4881no—
549C22C2×#1451974611yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—44no—
559C4—4881no—
560C4#1683993512yes—
561C4#1684056011yescharacteristic
562C4#1684056011yescharacteristic
563C4#1684057011yescharacteristic
564C4#1684057711yescharacteristic
565C2—41no—
566C2—81no—
567C2—9761no—
568C2#2145506111yescharacteristic
569C2#2145513411yescharacteristic
570C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3509C2
socleC22×C61C2×#120
prime radical#21455879C2
hypercenter#3509D61
hypocenterC61#18584
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. #21457202C2semidirect
2. #21456215C2semidirect
4. #21444991C2semidirect
5. #21444977C2semidirect
6. #21444929C2semidirect
7. #21444915C2semidirect
8. #21444504C2semidirect
9. #21444258C2semidirect
10. #21444240C2semidirect
11. C61×#3509C2semidirect
280. #3509D61semidirect
338. C61#18584semidirect

Maximal subgroups

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

subgroupindex
#214572022
#214562152
#214558792
#214449912
#214449772
#214449292
#214449152
#214445042
#214442582
#214442402
C61×#35092
#1858461