Finite Group Databasebeta

← #29648193

Subgroups — #29648193

Order 15616.

876 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#21459799C211yesmaximal, characteristic
2#21454837C211yesmaximal, characteristic, complemented
3#21454468C211yesmaximal, characteristic, complemented
4#21444447C212yesmaximal, complemented
5#21444444C212yesmaximal, complemented
6#21444433C211yesmaximal, characteristic, complemented
7#21444426C211yesmaximal, characteristic, complemented
8#21444423C212yesmaximal, complemented
9#21444419C212yesmaximal, complemented
10#21444413C211yesmaximal, characteristic, complemented
11C61×#1524C211yesmaximal, characteristic, complemented
12#16840491C411yescharacteristic
13#16840472C411yescharacteristic
14#16840469C411yescharacteristic
15#16840459C412yes—
16#16840452—21no—
17#16840448—22no—
18#16840446C412yes—
19#16840417—21no—
20#16840417C411yescharacteristic
21#16840374—21no—
22#16840355—22no—
23#16840319C2211yescharacteristic
24#16840291C2212yes—
25#125×D61C2212yes—
26#16840242—21no—
27C4×#11758222C2211yescharacteristic
28C8×Dic122—21no—
29#16839841—21no—
30#16839838—22no—
31#16839824—22no—
32#16839819C2211yescharacteristic
33#16839817C2212yes—
34#16839817C2212yes—
35#16839817C2212yes—
36#16839788—21no—
37#16839728—21no—
38C4×#11758194C2211yescharacteristic
39#16839714C2212yes—
40#16839504C2212yes—
41#16839503C2212yes—
42#16839503C2212yes—
43#16839503C2212yes—
44#16839503C2212yes—
45#16839502C2212yes—
46C61×#529C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C2×#97×C61C2211yescharacteristic
49C61×#330C2211yescharacteristic
50C61×#330C2211yescharacteristic
51C61×#329C2211yescharacteristic
52C61×#329C2211yescharacteristic
53C2×#11758222—21no—
54C2×#11758222D411yescharacteristic
55C2×#11758194—21no—
56C2×#11758189—22no—
57#14519777—21no—
58#14519777C2×C411yescharacteristic
59#14519777C2×C411yescharacteristic
60#14519770—22no—
61#14519770—22no—
62#14519770—22no—
63#14519769—21no—
64#14519769C2312yes—
65#14519768—22no—
66#14519768C2×C412yes—
67#14519768C2×C412yes—
68#14519767—22no—
69#14519767—22no—
70#14519767—22no—
71#14519759—22no—
72#14519759—22no—
73#14519759—22no—
74#14519758—22no—
75#14519756—21no—
76#14519756—21no—
77#14519756—21no—
78#14519756—21no—
79#14519755—22no—
80#14519754—22no—
81#14519753C2×C412yes—
82#14519748—22no—
83#14519748C2×C412yes—
84Q8×Dic61C2×C412yes—
85C4×Dic122—21no—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122D411yescharacteristic
90C4×D244—21no—
91C4×D244C2×C411yescharacteristic
92C4×D244D411yescharacteristic
93D4×Dic61C2×C412yes—
94C4×C22⋊D61C2×C412yes—
95C4×C22⋊D61D412yes—
96#14519737—22no—
97#14519737—22no—
98#14519736C2×C412yes—
99#14519736D411yescharacteristic
100#14519735—22no—
101#14519734D412yes—
102#14519733—22no—
103C2×C4⋊Dic61—21no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61—22no—
106C4⋊C4×D61C2×C412yes—
107C4⋊C4×D61C2×C412yes—
108C42×D61C2312yes—
109C4⋊#92418—21no—
110#14519706—22no—
111#14519706—22no—
112#14519706—22no—
113C22⋊#92418—21no—
114#14519689—22no—
115#14519689—22no—
116#14519689—22no—
117#14519688C2312yes—
118#14519688C2312yes—
119C4×#92418—21no—
120C8×Dic61—22no—
121C2×M16×C61—21no—
122C22×C8×C61—21no—
123#126×C61—21no—
124#125×C61C2311yescharacteristic
125#124×C61C2311yescharacteristic
126#123×C61—21no—
127C22⋊Q8×C61C2×C411yescharacteristic
128#121×C61C2×C411yescharacteristic
129C4×Q8×C61C2×C411yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2×C411yescharacteristic
132#117×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#97×C61—21no—
135#97×C61C2311yescharacteristic
136#97×C61C2311yescharacteristic
137#97×C61C2311yescharacteristic
138#97×C61C2311yescharacteristic
139C4×C8×C61—21no—
140C2×Q8×C61C22×C411yescharacteristic
141C2×D4×C61—21no—
142C2×D4×C61C22×C411yescharacteristic
143C22×C4×C61—21no—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—22no—
148C2×C22⋊D61C22⋊C412yes—
149C22×Dic61—22no—
150#11758222—42no—
151#11758222—44no—
152C2×D244C22⋊C411yescharacteristic
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61C22×C412yes—
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D412yes—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122C22⋊C411yescharacteristic
161M16×C61—42no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—42no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C411yescharacteristic
172C4⋊C4×C61C22×C411yescharacteristic
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C42×C61—21no—
179C42×C61C22×C411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2411yescharacteristic
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61C22⋊C412yes—
187#11758198—22no—
188#11758198C22⋊C412yes—
189#11758198C22⋊C412yes—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22⋊C411yescharacteristic
196C4⋊Dic61C22⋊C411yescharacteristic
197C4⋊Dic61C22×C412yes—
198C4⋊Dic61C22×C412yes—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196C22⋊C412yes—
207#11758196C22⋊C412yes—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61C22×C412yes—
211C4×Dic61C2×D412yes—
212C4×Dic61C2×D412yes—
213#11758194—42no—
214C2×#92418—22no—
215C2×#92418—22no—
216#11758189—44no—
217C23×C61—21no—
218C23×C61C2×C22⋊C411yescharacteristic
219C22×D61C2×C22⋊C412yes—
220Q8×C61—41no—
221D4×C61—41no—
222D4×C61—81no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C22⋊D61—42no—
237C22⋊D61—44no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—82no—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C412yes—
243C2×Dic61C2×C22⋊C412yes—
244D244—22no—
245C4×D61—42no—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—44no—
249C4×D61—82no—
250Dic122—22no—
251Dic122—42no—
252Dic122—81no—
253C8×C61—22no—
254C8×C61—41no—
255C8×C61—41no—
256C8×C61—41no—
257#92418—42no—
258#92418—42no—
259#17682—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61C22×C22⋊C411yescharacteristic
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61#42112yes—
279Dic61—22no—
280Dic61—22no—
281Dic61—22no—
282Dic61—42no—
283Dic61—42no—
284#3515—611no—
285#2971—611no—
286#2923—611no—
287#1535—612no—
288#1534—612no—
289#1531—611no—
290#1529—612no—
291#1529—611no—
292#1528—612no—
293#1526—611no—
294#1524D6111yescharacteristic, complemented
295C2×C61—41no—
296C2×C61—81no—
297C2×C61#288911yescharacteristic
298C2×C61C2×#42111yescharacteristic
299C2×C61#290111yescharacteristic
300D61—42no—
301#557—611no—
302#554—611no—
303#554—611no—
304#553—612no—
305#552—612no—
306#552—1221no—
307#552—1222no—
308#547—611no—
309#547—1221no—
310#541—1221no—
311#539—1222no—
312#533—611no—
313#529—612no—
314#529C2×D6111yescharacteristic
315C2×#125—612no—
316#520—1221no—
317C4×#41—611no—
318C8×Q8—1221no—
319#436—1221no—
320#436—1222no—
321#433—1222no—
322#432—611no—
323#432—612no—
324#432—612no—
325#432—612no—
326#424—1221no—
327#408—1221no—
328C4×M16—611no—
329C4×M16—612no—
330C4×M16C2×D6111yescharacteristic
331C2×#97C2×D6111yescharacteristic
332#330—612no—
333#330—612no—
334#330—612no—
335#330—612no—
336#330—612no—
337#330C2×D6111yescharacteristic
338#330C2×D6111yescharacteristic
339#329—612no—
340#329C2×D6111yescharacteristic
341#329C2×D6111yescharacteristic
342C61#1768211yescharacteristic, complemented, Sylow
343C2×#41—611no—
344C2×#41—1221no—
345C2×M16—21no—
346C2×M16—1222no—
347C2×M16—1221no—
348C22×C8—21no—
349C4⋊Q8—611no—
350C4⋊Q8—611no—
351C4⋊Q8—1221no—
352#126—21no—
353#126—1222no—
354#126—1222no—
355#126—1222no—
356#125—612no—
357#125—612no—
358#125—612no—
359#125—1221no—
360#125—1222no—
361#125—1222no—
362#125—1222no—
363#125—1222no—
364#125C22×D6111yescharacteristic
365#124C22×D6111yescharacteristic
366#123—21no—
367#123—1222no—
368#123—1222no—
369#123—1222no—
370#123—1222no—
371C22⋊Q8—612no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377C22⋊Q8—1221no—
378C22⋊Q8C4×D6111yescharacteristic
379#121—612no—
380#121—1222no—
381#121C4×D6111yescharacteristic
382C4×Q8—611no—
383C4×Q8—611no—
384C4×Q8—612no—
385C4×Q8—1221no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8C4×D6111yescharacteristic
389C4×D4—21no—
390C4×D4—612no—
391C4×D4—612no—
392C4×D4—612no—
393C4×D4—611no—
394C4×D4—611no—
395C4×D4—1221no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4C4×D6111yescharacteristic
399#117—21no—
400#117—611no—
401#117—612no—
402#117—612no—
403#117—1222no—
404#117—1222no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—1222no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1221no—
410C2×C42—612no—
411C2×C42C22×D6111yescharacteristic
412C4⋊C8—1222no—
413C4⋊C8—1222no—
414C4⋊C8—1222no—
415C4⋊C8—1221no—
416C22⋊C8—1222no—
417C22⋊C8—1222no—
418C22⋊C8—1222no—
419C22⋊C8—1221no—
420#97—21no—
421#97—612no—
422#97—612no—
423#97C22×D6111yescharacteristic
424#97C22×D6111yescharacteristic
425#97C22×D6111yescharacteristic
426#97C22×D6111yescharacteristic
427C4×C8—21no—
428C4×C8—1222no—
429C4×C8—1221no—
430#41—2442no—
431#41—2444no—
432C2×Q8—611no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8C2×C4×D6111yescharacteristic
436C2×D4—21no—
437C2×D4—612no—
438C2×D4—611no—
439C2×D4—1222no—
440C2×D4C2×C4×D6111yescharacteristic
441C22×C4—21no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4C2×C4×D6111yescharacteristic
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451M16—42no—
452M16—2444no—
453M16—2442no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—42no—
459C2×C8—1222no—
460C2×C8—1222no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—611no—
466C4⋊C4—612no—
467C4⋊C4—612no—
468C4⋊C4—611no—
469C4⋊C4—612no—
470C4⋊C4—612no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4C2×C4×D6111yescharacteristic
484C4⋊C4C2×C4×D6111yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—612no—
491C22⋊C4—612no—
492C22⋊C4—612no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C42—21no—
498C42—612no—
499C42—612no—
500C42—612no—
501C42—1222no—
502C42—1222no—
503C42C2×C4×D6111yescharacteristic
504C42D4×D6111yescharacteristic
505C42D4×D6111yescharacteristic
506C42C23×D6111yescharacteristic
507C23—21no—
508C23—612no—
509C23C22⋊C4×D6111yescharacteristic
510Q8—41no—
511Q8—1222no—
512Q8—4881no—
513Q8—2442no—
514D4—41no—
515D4—81no—
516D4—1222no—
517D4—2442no—
518D4—2444no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—4882no—
532C2×C4—4882no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2444no—
537C2×C4C22⋊C4×D6111yescharacteristic
538C2×C4C22⋊C4×D6111yescharacteristic
539C2×C4C22⋊C4×D6111yescharacteristic
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C8—22no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—2442no—
548C8—2442no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—1222no—
556C22—1222no—
557C22—1222no—
558C22C2×C22⋊C4×D6111yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—1222no—
568C4—1222no—
569C4—1222no—
570C4—2442no—
571C4—2442no—
572C4#1683977712yes—
573C2—41no—
574C2—81no—
575C2—2442no—
576C2#2145430011yescharacteristic
577C2C2×#1683977711yescharacteristic
578C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1524C2
socleC22×C61C22×C22⋊C4
prime radical#21459799C2
cosocleC42×C61C24
hypercenter#1524D61
hypocenterC61#17682
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
2. #21454837C2semidirect
3. #21454468C2semidirect
4. #21444447C2semidirect
5. #21444444C2semidirect
6. #21444433C2semidirect
7. #21444426C2semidirect
8. #21444423C2semidirect
9. #21444419C2semidirect
10. #21444413C2semidirect
11. C61×#1524C2semidirect
294. #1524D61semidirect
342. C61#17682semidirect

Maximal subgroups

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

subgroupindex
#214597992
#214548372
#214544682
#214444472
#214444442
#214444332
#214444262
#214444232
#214444192
#214444132
C61×#15242
#1768261