Finite Group Databasebeta

← #29665439

Subgroups — #29665439

Order 15616.

620 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#21456768C211yesmaximal, characteristic, complemented
2#21455627C211yesmaximal, characteristic
3#21455608C211yesmaximal, characteristic
4#21454926C211yesmaximal, characteristic
5#21445280C211yesmaximal, characteristic, complemented
6#21445274C211yesmaximal, characteristic, complemented
7#21445272C211yesmaximal, characteristic, complemented
8#21445190C211yesmaximal, characteristic, complemented
9#21445188C211yesmaximal, characteristic, complemented
10#21445184C211yesmaximal, characteristic, complemented
11#21445182C211yesmaximal, characteristic, complemented
12#21444592C211yesmaximal, characteristic, complemented
13#21444506C211yesmaximal, characteristic, complemented
14#332×D61C211yesmaximal, characteristic, complemented
15C61×#1683C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17C4⋊Q8×D61C2211yescharacteristic
18C4×#11758224C2211yescharacteristic
19#16840142C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840114C2211yescharacteristic
22#16840072—21no—
23#16840064—21no—
24#16840056—21no—
25#16840054C2211yescharacteristic
26#16840050—21no—
27#16840048C2211yescharacteristic
28#16840048C2211yescharacteristic
29#16840044C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16839971—21no—
33#16839971—21no—
34#16839919—21no—
35#16839919—21no—
36C8×Dic122C2211yescharacteristic
37#16839886C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39C8×C22⋊D61—21no—
40#16839825—21no—
41#16839813C2211yescharacteristic
42#16839812—21no—
43#16839803—21no—
44#16839797—21no—
45#16839784C2211yescharacteristic
46C2×#14519706—21no—
47C4⋊C8×D61C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839760—21no—
50#16839755—21no—
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839509C2211yescharacteristic
56#16839509C2211yescharacteristic
57#16839508C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#476C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×#11758224D411yescharacteristic
68C2×Q8×D61—21no—
69C2×Dic244—21no—
70C2×Dic244—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758189—21no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519776—21no—
80#14519776—21no—
81#14519775—21no—
82#14519768—21no—
83#14519757—21no—
84#14519753—21no—
85Q8×Dic61—21no—
86C4×Dic122C2311yescharacteristic
87#14519742—21no—
88#14519742D411yescharacteristic
89D4×Dic61D411yescharacteristic
90C4×C22⋊D61—21no—
91#14519737—21no—
92#14519735—21no—
93#14519735D411yescharacteristic
94#14519733—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C2×C4×Dic61—21no—
98C42×D61C2311yescharacteristic
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519709—21no—
103#14519709—21no—
104#14519708—21no—
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519706—21no—
113#14519706D411yescharacteristic
114#14519706D411yescharacteristic
115#14519702—21no—
116#14519702—21no—
117#14519700—21no—
118#14519700—21no—
119#14519700—21no—
120#14519700—21no—
121#14519698—21no—
122#14519698—21no—
123C22⋊#92418—21no—
124#14519689D411yescharacteristic
125#14519689D411yescharacteristic
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C22×C8×C61—21no—
130C4⋊Q8×C61C2311yescharacteristic
131C4⋊Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#106×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×D4×C61C2×D411yescharacteristic
147C22×C4×C61—21no—
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151Q8×D61—81no—
152Q8×D61—81no—
153#11758224—22no—
154#11758224Q1614yescomplemented
155C2×C4×D61—41no—
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61#4111yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162C2×Dic122C2×D411yescharacteristic
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—81no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C42×C61C2411yescharacteristic
176C22⋊Dic61—21no—
177#11758202—81no—
178#11758202—81no—
179#11758198—21no—
180#11758198—41no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—41no—
186C4⋊Dic61C2×D411yescharacteristic
187#11758196—21no—
188#11758196—21no—
189#11758196—41no—
190#11758196—41no—
191#11758196—41no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—41no—
196C4×Dic61C2×D411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61#4111yescharacteristic
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—81no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205Dic244—81no—
206Dic244—81no—
207#11758189—42no—
208C8×D61—42no—
209C8×D61—81no—
210C23×C61—21no—
211C22×D61#12111yescharacteristic
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215D4×C61—21no—
216D4×C61C2×Q1612yes—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C22⋊D61—24no—
226C2×Dic61—21no—
227C2×Dic61—24no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—81no—
232C2×Dic61#12111yescharacteristic
233C2×Dic61#12111yescharacteristic
234C2×Dic61#12111yescharacteristic
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—41no—
238C4×D61—82no—
239C4×D61C2×Q1612yes—
240C4×D61#13512yes—
241Dic122—21no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—82no—
247Dic122C2×Q1612yes—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251C8×C61—41no—
252C8×C61—41no—
253#92418—21no—
254#92418—21no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#19411—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61C2×#12111yescharacteristic
263C2×D61—21no—
264C2×D61—21no—
265C2×D61#46712yes—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C22×Q1611yescharacteristic
271C4×C61C2×#13511yescharacteristic
272C4×C61#57611yescharacteristic
273C4×C61#57711yescharacteristic
274Dic61—21no—
275Dic61—22no—
276Dic61—21no—
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280Dic61#46712yes—
281#3209—611no—
282#3079—611no—
283#3076—611no—
284#2988—611no—
285#1684—611no—
286#1684—611no—
287#1683—611no—
288#1683D6111yescharacteristic, complemented
289#1670—611no—
290#1670—611no—
291#1669—611no—
292#1669—611no—
293#1571—611no—
294#1550—611no—
295C2×#332—611no—
296C2×C61—42no—
297C2×C61#304511yescharacteristic
298C2×C61C2×#46711yescharacteristic
299C2×C61#306911yescharacteristic
300D61—22no—
301#533—611no—
302C2×C4⋊Q8—611no—
303C4×#41—611no—
304#494—611no—
305#494—611no—
306#487—611no—
307#487C2×D6111yescharacteristic
308#479—1221no—
309#478—1221no—
310#477—611no—
311#477—1221no—
312#476—1221no—
313#476C2×D6111yescharacteristic
314#475—611no—
315#475—611no—
316#475—611no—
317#474—611no—
318#474—611no—
319#474C2×D6111yescharacteristic
320#462—1221no—
321#462—1221no—
322#451—1221no—
323#451—1221no—
324C8×Q8—611no—
325C8×Q8—611no—
326C8×D4—611no—
327C8×D4—1221no—
328C8×D4—1221no—
329C8×D4C2×D6111yescharacteristic
330#431—611no—
331#431—1221no—
332#428—1221no—
333#427—1221no—
334#423—611no—
335C2×C4⋊C8—611no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8—1221no—
338#417—1221no—
339#416—1221no—
340#335—611no—
341#335—611no—
342#335—611no—
343#335C2×D6111yescharacteristic
344#333—611no—
345#333C2×D6111yescharacteristic
346#332—611no—
347#332—611no—
348#332—611no—
349#332—611no—
350#332—611no—
351#332C2×D6111yescharacteristic
352C61#1941111yescharacteristic, complemented, Sylow
353C2×#41—611no—
354C22×Q8—1221no—
355C2×Q16—1221no—
356C2×Q16—1221no—
357C2×Q16—1221no—
358C2×Q16—1221no—
359C2×M16—1221no—
360C22×C8—21no—
361C22×C8—1221no—
362C22×C8—1221no—
363C22×C8—1221no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8C22×D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371#125—1221no—
372C22⋊Q8—1221no—
373C22⋊Q8—1221no—
374C4×Q8—611no—
375C4×Q8—611no—
376C4×Q8—1221no—
377C4×Q8—1221no—
378C4×D4—611no—
379C4×D4—1221no—
380C4×D4—1221no—
381C4×D4C22×D6111yescharacteristic
382#117—611no—
383#117—1221no—
384#117—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C42—611no—
388C2×C42—1221no—
389#107—21no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#106—21no—
394#106—1221no—
395#106—1221no—
396#106—1221no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8C22×D6111yescharacteristic
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
410#103—21no—
411#103—21no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#102—21no—
419#102—21no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—21no—
423C22⋊C8—611no—
424C22⋊C8—611no—
425C22⋊C8—1221no—
426C4×C8—611no—
427C4×C8—1221no—
428C4×C8—1221no—
429C4×C8C22×D6111yescharacteristic
430#41—614no—
431#41—1222no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—611no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—4881no—
438C2×Q8—4881no—
439C2×Q8—2441no—
440C2×D4—1221no—
441C2×D4D4×D6111yescharacteristic
442C22×C4—21no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—2441no—
449Q16—4881no—
450Q16—4881no—
451Q16—4881no—
452Q16—4881no—
453M16—2442no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—81no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—4881no—
464C2×C8—4881no—
465C2×C8—2442no—
466C2×C8D4×D6111yescharacteristic
467C2×C8D4×D6111yescharacteristic
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—611no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4D4×D6111yescharacteristic
484C22⋊C4—21no—
485C22⋊C4—1221no—
486C22⋊C4—1221no—
487C22⋊C4—2441no—
488C42—611no—
489C42—611no—
490C42—611no—
491C42—1221no—
492C42—1221no—
493C42—1221no—
494C42—2441no—
495C42C23×D6111yescharacteristic
496C23—21no—
497C23—611no—
498Q8—41no—
499Q8—41no—
500Q8—81no—
501Q8—612no—
502Q8—1221no—
503Q8—4882no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508D4—21no—
509D4—1224no—
510D4Q16×D6112yes—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—1224no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—4882no—
527C2×C4—4881no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4C2×D4×D6111yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4#41×D6111yescharacteristic
534C8—21no—
535C8—21no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—1221no—
540C8—1221no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—612no—
548C22—1221no—
549C22—1221no—
550C22#121×D6111yescharacteristic
551C4—21no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—612no—
556C4—1222no—
557C4—1221no—
558C4—1221no—
559C4—4881no—
560C4—4881no—
561C4—2441no—
562C4C2×Q16×D6111yescharacteristic
563C4#135×D6111yescharacteristic
564C4#1684056611yescharacteristic
565C4#1684056911yescharacteristic
566C2—42no—
567C2—1222no—
568C2#2145535911yescharacteristic
569C2#467×D6111yescharacteristic
570C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1683C2
socleC22×C61C2×#121
prime radicalC2×#11758224D4
hypercenter#1683D61
hypocenterC61#19411
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1683

Complements

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

NH ≅ G/Ntype
1. #21456768C2semidirect
5. #21445280C2semidirect
6. #21445274C2semidirect
7. #21445272C2semidirect
8. #21445190C2semidirect
9. #21445188C2semidirect
10. #21445184C2semidirect
11. #21445182C2semidirect
12. #21444592C2semidirect
13. #21444506C2semidirect
14. #332×D61C2semidirect
15. C61×#1683C2semidirect
154. #11758224Q16semidirect
288. #1683D61semidirect
352. C61#19411semidirect

Maximal subgroups

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

subgroupindex
#214567682
#214556272
#214556082
#214549262
#214452802
#214452742
#214452722
#214451902
#214451882
#214451842
#214451822
#214445922
#214445062
#332×D612
C61×#16832
#1941161