Finite Group Databasebeta

← #29677219

Subgroups — #29677219

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#21458714C211yesmaximal, characteristic, complemented
2#21458447C211yesmaximal, characteristic, complemented
3#21458414C211yesmaximal, characteristic
4#21445413C211yesmaximal, characteristic, complemented
5#21445342C211yesmaximal, characteristic, complemented
6#21445323C211yesmaximal, characteristic, complemented
7#21445316C211yesmaximal, characteristic, complemented
8#21445305C211yesmaximal, characteristic, complemented
9#21445045C211yesmaximal, characteristic, complemented
10#21445031C211yesmaximal, characteristic, complemented
11#21444984C211yesmaximal, characteristic, complemented
12#21444970C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3276C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840472C2211yescharacteristic
18#16840168—21no—
19#16840167C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840114C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840102—21no—
25#16840083—21no—
26#16840080—21no—
27#16840074—21no—
28#16840062—21no—
29#16840057C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840052C2211yescharacteristic
33#16840049—21no—
34#16840039C2211yescharacteristic
35#16840039C2211yescharacteristic
36#16840013—21no—
37#16840005—21no—
38#16839893C2211yescharacteristic
39#16839872—21no—
40C4×#11758199C2211yescharacteristic
41#16839841C2211yescharacteristic
42#16839840C2211yescharacteristic
43#16839802—21no—
44#16839801—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C2×#137×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58C61×#547C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#470—21no—
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#452—21no—
64C61×#451—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758199D411yescharacteristic
68C2×#11758194—21no—
69C2×#11758189—21no—
70#14519777C2311yescharacteristic
71#14519777C2311yescharacteristic
72#14519769—21no—
73#14519763—41no—
74#14519755—41no—
75#14519753—21no—
76#14519751—21no—
77#14519748—21no—
78C4×D244—21no—
79C4×D244C2311yescharacteristic
80#14519736—21no—
81#14519735—21no—
82C4⋊C4×D61—21no—
83#14519711—21no—
84#14519711—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710D411yescharacteristic
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701—21no—
101#14519700—41no—
102#14519700—41no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698D411yescharacteristic
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×Q8×C61—21no—
117#137×C61—44no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61D411yescharacteristic
123C2×M16×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126C4⋊Q8×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131#121×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C2×C22⋊D61—41no—
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—41no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166Q16×C61—42no—
167Q16×C61—42no—
168Q16×C61—81no—
169SD16×C61—22no—
170SD16×C61—42no—
171M16×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—41no—
192#11758199—22no—
193#11758198—21no—
194#11758198—41no—
195#11758198—41no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—41no—
202#11758196—41no—
203C4×Dic61—41no—
204#11758194—42no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211#11758189—42no—
212C23×C61—21no—
213C23×C61—21no—
214C22×D61—21no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—82no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—82no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61—82no—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C22⋊D61—161no—
243C2×Dic61—21no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246D244—21no—
247D244—21no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—161no—
251Dic122—81no—
252Dic122—81no—
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#3357—611no—
289#3354—611no—
290#3276D6111yescharacteristic, complemented
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—21no—
309#555—611no—
310#554—611no—
311D4×Q8C2×D6111yescharacteristic
312#547C2×D6111yescharacteristic
313#501—1221no—
314#500—611no—
315#498—611no—
316#498—611no—
317#494C2×D6111yescharacteristic
318#487—611no—
319#487—611no—
320#485—1221no—
321#482—1221no—
322#481—1221no—
323#480—1221no—
324#478—1221no—
325#477—611no—
326#476—611no—
327#476—611no—
328#476—611no—
329#476—1221no—
330#474—611no—
331#474—611no—
332#470—21no—
333#469—1221no—
334#468—1221no—
335#462C2×D6111yescharacteristic
336#461C2×D6111yescharacteristic
337#452—21no—
338#451—21no—
339#446—611no—
340#442—1221no—
341C4×SD16C2×D6111yescharacteristic
342C4×D8—611no—
343#436—611no—
344#436—611no—
345#436C2×D6111yescharacteristic
346#428—1221no—
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—21no—
361C22×Q8—21no—
362#137—44no—
363C2×Q16—21no—
364C2×Q16—21no—
365C2×Q16—21no—
366C2×SD16—21no—
367C2×SD16C22⋊D6111yescharacteristic
368C2×D8—611no—
369C2×M16—21no—
370C2×M16—1221no—
371C2×M16—1221no—
372C4⋊Q8—21no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377#125—1221no—
378#124—2441no—
379#123—41no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—41no—
383C22⋊Q8—1221no—
384C22⋊Q8—2441no—
385#121—21no—
386#121—1221no—
387#121—1221no—
388C4×Q8C22×D6111yescharacteristic
389C4×D4—21no—
390C4×D4—611no—
391C4×D4—1221no—
392C4×D4C22×D6111yescharacteristic
393C4×D4C22×D6111yescharacteristic
394#117—1221no—
395#117—1221no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—1221no—
398#107—611no—
399#107—1221no—
400#107—1221no—
401#107—1221no—
402#107—1221no—
403#107—1221no—
404#106—1221no—
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—21no—
419#103—1221no—
420#103—2441no—
421#103—2441no—
422#103C22⋊D6111yescharacteristic
423#102—611no—
424#102—611no—
425#102—1221no—
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—42no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—81no—
445C2×Q8—1221no—
446C2×Q8—1221no—
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—41no—
457C22×C4—1221no—
458C22×C4—2441no—
459Q16—42no—
460Q16—42no—
461Q16—81no—
462SD16—22no—
463SD16—42no—
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—1221no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C42—21no—
503C42—2441no—
504C42C23×D6111yescharacteristic
505C23—21no—
506C23—21no—
507C23—1221no—
508Q8—21no—
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—82no—
515Q8—4881no—
516Q8—4881no—
517D4—21no—
518D4—21no—
519D4—21no—
520D4—21no—
521D4—82no—
522D4—1221no—
523D4—1221no—
524D4—9761no—
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—82no—
536C2×C4—82no—
537C2×C4—1221no—
538C2×C4—9761no—
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#1451980312yes—
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×#1451980311yescharacteristic
576C4#137×D6111yescharacteristic
577C4#1684057311yescharacteristic
578C4#1684057811yescharacteristic
579C2—81no—
580C2—81no—
581C2—4881no—
582C2#2145741511yescharacteristic
583C2#2145745911yescharacteristic
584C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3276C2
socleC22×C61D42
prime radical#21458414C2
hypercenter#3276D61
hypocenterC61#20315
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3276

Complements

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

NH ≅ G/Ntype
1. #21458714C2semidirect
2. #21458447C2semidirect
4. #21445413C2semidirect
5. #21445342C2semidirect
6. #21445323C2semidirect
7. #21445316C2semidirect
8. #21445305C2semidirect
9. #21445045C2semidirect
10. #21445031C2semidirect
11. #21444984C2semidirect
12. #21444970C2semidirect
13. #21444710C2semidirect
14. #21444702C2semidirect
15. C61×#3276C2semidirect
261. #92418#137semidirect
290. #3276D61semidirect
359. C61#20315semidirect

Maximal subgroups

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

subgroupindex
#214587142
#214584472
#214584142
#214454132
#214453422
#214453232
#214453162
#214453052
#214450452
#214450312
#214449842
#214449702
#214447102
#214447022
C61×#32762
#2031561