Finite Group Databasebeta

← #29677213

Subgroups — #29677213

Order 15616.

624 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458718C211yesmaximal, characteristic, complemented
2#21458408C211yesmaximal, characteristic
3#21457466C211yesmaximal, characteristic, complemented
4#21445417C211yesmaximal, characteristic, complemented
5#21445339C211yesmaximal, characteristic, complemented
6#21445327C211yesmaximal, characteristic, complemented
7#21445320C211yesmaximal, characteristic, complemented
8#21445308C211yesmaximal, characteristic, complemented
9#21445047C211yesmaximal, characteristic, complemented
10#21445033C211yesmaximal, characteristic, complemented
11#21444982C211yesmaximal, characteristic, complemented
12#21444968C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3357C211yesmaximal, characteristic, complemented
16C2×#14519806—21no—
17Q8×D244C2211yescharacteristic
18#16840416C2211yescharacteristic
19#16840172—21no—
20#16840158C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840142C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840086—21no—
26#16840078—21no—
27#16840059—21no—
28#16840052—21no—
29#16840049C2211yescharacteristic
30#16840049C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16840042C2211yescharacteristic
33#16840021—21no—
34#16840013—21no—
35#16839972C2211yescharacteristic
36#16839967C2211yescharacteristic
37#16839925—21no—
38#16839919—21no—
39#16839893C2211yescharacteristic
40#16839872—21no—
41C4×#11758199C2211yescharacteristic
42#16839841C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839805—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839509C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#555C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#485—21no—
60C61×#480—21no—
61C61×#477C2211yescharacteristic
62C61×#476C2211yescharacteristic
63C61×#468—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#428—21no—
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69#14519806—44no—
70C2×Dic244—21no—
71C2×Dic244—21no—
72C2×#11758202—21no—
73C2×#11758200—21no—
74C2×#11758199D411yescharacteristic
75C2×#11758194—21no—
76C2×#11758189—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519761—41no—
81#14519757—21no—
82#14519757—21no—
83#14519757—41no—
84#14519751—21no—
85C4×D244—21no—
86C4×D244C2311yescharacteristic
87C2×C4⋊Dic61—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701—41no—
102#14519701—41no—
103#14519700—21no—
104#14519700—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×SD16×C61D411yescharacteristic
117C2×M16×C61—21no—
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#125×C61—21no—
121#124×C61—41no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—41no—
124#121×C61—21no—
125#121×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131#117×C61—21no—
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61—21no—
136#106×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—41no—
149C2×D4×C61C2×D411yescharacteristic
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C2×C22⋊D61—41no—
155C22×Dic61—41no—
156Q8×D61—42no—
157Q8×D61—81no—
158#11758224—42no—
159#11758224—81no—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—41no—
166SD16×C61—22no—
167M16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—41no—
188C42×C61C2411yescharacteristic
189#11758202—42no—
190#11758200—42no—
191#11758199—22no—
192#11758198—21no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—41no—
200C4⋊Dic61—41no—
201C4⋊Dic61C2×D411yescharacteristic
202#11758194—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209Dic244—42no—
210Dic244—81no—
211#11758189—42no—
212C23×C61—21no—
213C23×C61—21no—
214C22×D61—21no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—81no—
218Q8×C61—81no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—161no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—161no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C22⋊D61—82no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—82no—
243D244—21no—
244D244—21no—
245C4×D61—41no—
246C4×D61—41no—
247C4×D61—82no—
248Dic122—41no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—82no—
253C8×C61—21no—
254C8×C61—41no—
255C8×C61AGL(1,ℤ/8)12yes—
256#92418—21no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418#13712yescomplemented
262#20315—611nomaximal, Sylow
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61D4211yescharacteristic
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C4×C61C2×#13711yescharacteristic
282C4×C61#57711yescharacteristic
283C4×C61#57811yescharacteristic
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287#3384—611no—
288#3357D6111yescharacteristic, complemented
289#3354—611no—
290#3276—611no—
291#1707—611no—
292#1695—611no—
293#1692—611no—
294#1691—611no—
295#1689—611no—
296#1646—611no—
297#1644—611no—
298#1637—611no—
299#1635—611no—
300#1590—611no—
301#1589—611no—
302C2×C61—81no—
303C2×C61—81no—
304C2×C61#327211yescharacteristic
305C2×C61#327611yescharacteristic
306C2×C61#329011yescharacteristic
307D61—81no—
308C2×#137—1221no—
309#555C2×D6111yescharacteristic
310#554C2×D6111yescharacteristic
311D4×Q8—611no—
312#547—611no—
313#501—1221no—
314#500C2×D6111yescharacteristic
315#498—611no—
316#498—611no—
317#494—611no—
318#487—611no—
319#487—611no—
320#485—21no—
321#482—1221no—
322#481—1221no—
323#480—21no—
324#478—1221no—
325#477C2×D6111yescharacteristic
326#476—611no—
327#476—611no—
328#476—1221no—
329#476C2×D6111yescharacteristic
330#474—611no—
331#474—611no—
332#470—1221no—
333#469—1221no—
334#468—21no—
335#462—611no—
336#461—611no—
337#452—1221no—
338#451—1221no—
339#446—611no—
340#442—1221no—
341C4×SD16C2×D6111yescharacteristic
342C4×D8—611no—
343#436—611no—
344#436—611no—
345#436C2×D6111yescharacteristic
346#428—21no—
347#428—1221no—
348#335—611no—
349#334—611no—
350#333—611no—
351#333—611no—
352#332—611no—
353#332—611no—
354Q8⋊C8—611no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358C82—611no—
359C61#2031511yescharacteristic, complemented, Sylow
360C2×#41—1221no—
361C22×Q8—1221no—
362#137—2444no—
363C2×Q16—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×SD16—1221no—
367C2×SD16C22⋊D6111yescharacteristic
368C2×D8—611no—
369C2×M16—21no—
370C2×M16—1221no—
371C2×M16—1221no—
372C4⋊Q8—611no—
373C4⋊Q8—611no—
374C4⋊Q8—1221no—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377#125—21no—
378#124—41no—
379#123—2441no—
380C22⋊Q8—21no—
381C22⋊Q8—41no—
382C22⋊Q8—1221no—
383C22⋊Q8—1221no—
384C22⋊Q8—2441no—
385#121—21no—
386#121—21no—
387#121—1221no—
388C4×Q8C22×D6111yescharacteristic
389C4×D4—21no—
390C4×D4—611no—
391C4×D4—1221no—
392C4×D4C22×D6111yescharacteristic
393C4×D4C22×D6111yescharacteristic
394#117—21no—
395#117—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—1221no—
398#107—21no—
399#107—21no—
400#107—21no—
401#107—611no—
402#107—1221no—
403#107—1221no—
404#106—21no—
405#106—1221no—
406#106—1221no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410#106—1221no—
411#106C22⋊D6111yescharacteristic
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—1221no—
416C4⋊C8C22×D6111yescharacteristic
417#103—21no—
418#103—1221no—
419#103—1221no—
420#103—2441no—
421#103—2441no—
422#103C22⋊D6111yescharacteristic
423#102—21no—
424#102—611no—
425#102—611no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102C22⋊D6111yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434#97—1221no—
435C4×C8—611no—
436C4×C8—611no—
437C4×C8C22×D6111yescharacteristic
438#41—4881no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—4881no—
445C2×Q8—2441no—
446C2×Q8—2442no—
447C2×Q8C2×C22⋊D6111yescharacteristic
448C2×D4—41no—
449C2×D4—611no—
450C2×D4—2441no—
451C2×D4D4×D6111yescharacteristic
452C2×D4C2×C22⋊D6111yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—1221no—
457C22×C4—2441no—
458C22×C4—2441no—
459Q16—4881no—
460Q16—2442no—
461Q16—2442no—
462SD16—22no—
463SD16—2442no—
464D8—1222no—
465M16—42no—
466M16—2442no—
467M16—2442no—
468C2×C8—21no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8C2×C22⋊D6111yescharacteristic
476C2×C8C2×C22⋊D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4D4×D6111yescharacteristic
492C4⋊C4C2×C22⋊D6111yescharacteristic
493C4⋊C4C2×C22⋊D6111yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—1221no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C42—21no—
503C42—41no—
504C42C23×D6111yescharacteristic
505C23—21no—
506C23—21no—
507C23—1221no—
508Q8—21no—
509Q8—21no—
510Q8—81no—
511Q8—81no—
512Q8—4882no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517D4—21no—
518D4—21no—
519D4—21no—
520D4—21no—
521D4—161no—
522D4—1221no—
523D4—1221no—
524D4—4882no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—161no—
536C2×C4—1221no—
537C2×C4—4882no—
538C2×C4—4882no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4#1451982211yescharacteristic
546C8—21no—
547C8—41no—
548C8—612no—
549C8—1221no—
550C8—2441no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C8#1451980212yes—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22D4×C22⋊D6111yescharacteristic
565C4—21no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—4881no—
573C4—4881no—
574C4—2441no—
575C4C2×#1451980211yescharacteristic
576C4#137×D6111yescharacteristic
577C4#1684057011yescharacteristic
578C4#1684057811yescharacteristic
579C2—81no—
580C2—81no—
581C2—4881no—
582C2#2145741911yescharacteristic
583C2#2145745911yescharacteristic
584C2#2145760111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3357C2
socleC22×C61D42
prime radical#21458408C2
hypercenter#3357D61
hypocenterC61#20315
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3357

Complements

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

NH ≅ G/Ntype
1. #21458718C2semidirect
3. #21457466C2semidirect
4. #21445417C2semidirect
5. #21445339C2semidirect
6. #21445327C2semidirect
7. #21445320C2semidirect
8. #21445308C2semidirect
9. #21445047C2semidirect
10. #21445033C2semidirect
11. #21444982C2semidirect
12. #21444968C2semidirect
13. #21444710C2semidirect
14. #21444702C2semidirect
15. C61×#3357C2semidirect
261. #92418#137semidirect
288. #3357D61semidirect
359. C61#20315semidirect

Maximal subgroups

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

subgroupindex
#214587182
#214584082
#214574662
#214454172
#214453392
#214453272
#214453202
#214453082
#214450472
#214450332
#214449822
#214449682
#214447102
#214447022
C61×#33572
#2031561