Finite Group Databasebeta

← #29665441

Subgroups — #29665441

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#21455629C211yesmaximal, characteristic
3#21455610C211yesmaximal, characteristic
4#21454929C211yesmaximal, characteristic
5#21445279C211yesmaximal, characteristic, complemented
6#21445273C211yesmaximal, characteristic, complemented
7#21445267C211yesmaximal, characteristic, complemented
8#21445189C211yesmaximal, characteristic, complemented
9#21445187C211yesmaximal, characteristic, complemented
10#21445185C211yesmaximal, characteristic, complemented
11#21445183C211yesmaximal, characteristic, complemented
12#21444592C211yesmaximal, characteristic, complemented
13#21444506C211yesmaximal, characteristic, complemented
14#332×D61C211yesmaximal, characteristic, complemented
15C61×#1684C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17C4⋊Q8×D61C2211yescharacteristic
18C4×#11758226C2211yescharacteristic
19#16840142C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840114C2211yescharacteristic
22#16840071—21no—
23#16840065—21no—
24#16840055—21no—
25#16840051—21no—
26#16840049C2211yescharacteristic
27#16840047C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840043C2211yescharacteristic
30#16840043C2211yescharacteristic
31#16840039C2211yescharacteristic
32#16839971—21no—
33#16839971—21no—
34#16839919—21no—
35#16839919—21no—
36Q8×#92418C2211yescharacteristic
37C8×D244—21no—
38C8×D244C2211yescharacteristic
39#16839827—21no—
40#16839827C2211yescharacteristic
41#16839813—21no—
42#16839813C2211yescharacteristic
43#16839801—21no—
44#16839799—21no—
45#16839784C2211yescharacteristic
46C4⋊C8×D61—21no—
47C4⋊C8×D61C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839762—21no—
50#16839757—21no—
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839509C2211yescharacteristic
56#16839509C2211yescharacteristic
57#16839508C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#477C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×#11758226D411yescharacteristic
68C2×Q8×D61—21no—
69C2×Dic244—21no—
70C2×Dic244—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758189—21no—
74C2×C8×D61—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519776—21no—
81#14519776—21no—
82#14519775—21no—
83#14519768—21no—
84#14519757—21no—
85#14519753—21no—
86Q8×Dic61—21no—
87Q8×Dic61D411yescharacteristic
88#14519742—21no—
89C4×D244—21no—
90C4×D244C2311yescharacteristic
91#14519739—21no—
92#14519739D411yescharacteristic
93#14519735—21no—
94#14519735—21no—
95#14519735D411yescharacteristic
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C42×D61—21no—
99C42×D61C2311yescharacteristic
100#14519711—21no—
101#14519711—21no—
102#14519710—21no—
103#14519708—21no—
104#14519708—21no—
105#14519708—21no—
106C4⋊#92418—21no—
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109C4⋊#92418C2311yescharacteristic
110#14519706—21no—
111#14519706—21no—
112#14519706—21no—
113#14519706D411yescharacteristic
114#14519706D411yescharacteristic
115#14519701—21no—
116#14519701—21no—
117#14519700—21no—
118#14519700—21no—
119#14519699—21no—
120#14519699—21no—
121#14519697—21no—
122#14519697—21no—
123#14519689—21no—
124#14519689—21no—
125#14519689D411yescharacteristic
126#14519689D411yescharacteristic
127C4×#92418—21no—
128C4×#92418C2311yescharacteristic
129C4⋊Q8×C61C2311yescharacteristic
130C4⋊Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61—21no—
133#106×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#103×C61—21no—
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147#11758226—22no—
148#11758226Q1614yescomplemented
149Q8×D61—81no—
150Q8×D61—81no—
151C2×D244—21no—
152C2×D244C2×D411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—41no—
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61#4111yescharacteristic
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—41no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×D411yescharacteristic
174C42×C61—21no—
175C42×C61C2411yescharacteristic
176#11758202—81no—
177#11758202—81no—
178#11758198—21no—
179#11758198—21no—
180#11758198—21no—
181#11758198—41no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—41no—
188C4⋊Dic61C2×D411yescharacteristic
189#11758196—41no—
190#11758196—41no—
191#11758196—41no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—41no—
195C4×Dic61C2×D411yescharacteristic
196C4×Dic61C2×D411yescharacteristic
197C4×Dic61#4111yescharacteristic
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203Dic244—81no—
204Dic244—81no—
205#11758189—42no—
206C8×D61—42no—
207C8×D61—81no—
208C8×D61—81no—
209C8×D61—81no—
210C22×D61—21no—
211C22×D61#12111yescharacteristic
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—81no—
216Q8×C61C2×Q1612yes—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61#12111yescharacteristic
228C2×Dic61#12111yescharacteristic
229C2×Dic61#12111yescharacteristic
230D244—24no—
231D244—21no—
232D244C2×Q1612yes—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—24no—
236C4×D61—41no—
237C4×D61—41no—
238C4×D61—41no—
239C4×D61—81no—
240C4×D61—82no—
241C4×D61C2×Q1612yes—
242C4×D61#13512yes—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—82no—
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×C61C2×#12111yescharacteristic
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—22no—
263C2×D61—41no—
264C2×D61—41no—
265C2×D61#46712yes—
266C4×C61—22no—
267C4×C61—21no—
268C4×C61—41no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61C22×Q1611yescharacteristic
272C4×C61C2×#13511yescharacteristic
273C4×C61#57611yescharacteristic
274C4×C61#57711yescharacteristic
275Dic61—21no—
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#1684D6111yescharacteristic, complemented
287#1683—611no—
288#1683—611no—
289#1670—611no—
290#1670—611no—
291#1669—611no—
292#1669—611no—
293#1571—611no—
294#1550—611no—
295C2×#332—611no—
296C2×C61#304511yescharacteristic
297C2×C61C2×#46711yescharacteristic
298C2×C61#306911yescharacteristic
299D61—22no—
300D61—42no—
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—1221no—
311#477C2×D6111yescharacteristic
312#476—611no—
313#476—1221no—
314#475—611no—
315#475—611no—
316#475C2×D6111yescharacteristic
317#474—611no—
318#474—611no—
319#474—611no—
320#462—1221no—
321#462—1221no—
322#451—1221no—
323#451—1221no—
324C8×Q8—611no—
325C8×Q8C2×D6111yescharacteristic
326C8×D4—611no—
327C8×D4—611no—
328C8×D4—1221no—
329C8×D4—1221no—
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—1221no—
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—1221no—
376C4×Q8—1221no—
377C4×Q8C22×D6111yescharacteristic
378C4×D4—611no—
379C4×D4—611no—
380C4×D4—1221no—
381C4×D4—1221no—
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—21no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
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—21no—
413#103—21no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—611no—
423C22⋊C8—611no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C4×C8—21no—
427C4×C8—611no—
428C4×C8—1221no—
429C4×C8C22×D6111yescharacteristic
430#41—614no—
431#41—1222no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—4881no—
437C2×Q8—4881no—
438C2×Q8—2441no—
439C2×Q8D4×D6111yescharacteristic
440C2×D4—611no—
441C2×D4—1221no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—1221no—
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—611no—
458C2×C8—611no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—4881no—
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—21no—
473C4⋊C4—611no—
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—1221no—
485C22⋊C4—1221no—
486C22⋊C4—1221no—
487C22⋊C4—2441no—
488C42—21no—
489C42—611no—
490C42—611no—
491C42—611no—
492C42—1221no—
493C42—1221no—
494C42—2441no—
495C42C23×D6111yescharacteristic
496C23—611no—
497C23—1221no—
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—81no—
502Q8—4882no—
503Q8—2441no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507Q8Q16×D6112yes—
508D4—612no—
509D4—1224no—
510D4—1221no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—611no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—1224no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—4881no—
525C2×C4—4882no—
526C2×C4—2441no—
527C2×C4—2441no—
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—612no—
545C22—1221no—
546C22—1221no—
547C22—1222no—
548C22—2441no—
549C22—2441no—
550C22#121×D6111yescharacteristic
551C4—22no—
552C4—21no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—612no—
557C4—1221no—
558C4—1221no—
559C4—4881no—
560C4—4881no—
561C4—2441no—
562C4C2×Q16×D6111yescharacteristic
563C4#135×D6111yescharacteristic
564C4#1684056411yescharacteristic
565C4#1684057111yescharacteristic
566C2—1222no—
567C2—2442no—
568C2#2145536111yescharacteristic
569C2#467×D6111yescharacteristic
570C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1684C2
socleC22×C61C2×#121
prime radicalC2×#11758226D4
hypercenter#1684D61
hypocenterC61#19411
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1684

Complements

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

NH ≅ G/Ntype
1. #21456768C2semidirect
5. #21445279C2semidirect
6. #21445273C2semidirect
7. #21445267C2semidirect
8. #21445189C2semidirect
9. #21445187C2semidirect
10. #21445185C2semidirect
11. #21445183C2semidirect
12. #21444592C2semidirect
13. #21444506C2semidirect
14. #332×D61C2semidirect
15. C61×#1684C2semidirect
148. #11758226Q16semidirect
286. #1684D61semidirect
352. C61#19411semidirect

Maximal subgroups

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

subgroupindex
#214567682
#214556292
#214556102
#214549292
#214452792
#214452732
#214452672
#214451892
#214451872
#214451852
#214451832
#214445922
#214445062
#332×D612
C61×#16842
#1941161