Finite Group Databasebeta

← #29670951

Subgroups — #29670951

Order 15616.

596 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#21458445C211yesmaximal, characteristic, complemented
2#21458170C211yesmaximal, characteristic, complemented
3#21457749C211yesmaximal, characteristic
4#21445495C211yesmaximal, characteristic, complemented
5#21445320C211yesmaximal, characteristic, complemented
6#21445277C211yesmaximal, characteristic, complemented
7#21445216C211yesmaximal, characteristic, complemented
8#21445215C211yesmaximal, characteristic, complemented
9#21445026C211yesmaximal, characteristic, complemented
10#21444968C211yesmaximal, characteristic, complemented
11#21444963C211yesmaximal, characteristic, complemented
12#21444953C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15C61×#3380C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840416C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840105—21no—
24#16840100—21no—
25#16840090—21no—
26#16840080—21no—
27#16840075—21no—
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840049C2211yescharacteristic
31#16840049C2211yescharacteristic
32#16840047C2211yescharacteristic
33#16840010—21no—
34#16839993—21no—
35#16839990—21no—
36#16839967C2211yescharacteristic
37#16839904—21no—
38C4×#11758201C2211yescharacteristic
39#16839840C2211yescharacteristic
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42#16839802—21no—
43C2×#14519708—21no—
44#16839755—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839509C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#554C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#499—21no—
60C61×#478—21no—
61C61×#478—21no—
62C61×#476C2211yescharacteristic
63C61×#476C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×D8×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758224—21no—
68C2×#11758202—41no—
69C2×#11758200—21no—
70C2×#11758201D411yescharacteristic
71C2×#11758194—21no—
72C22×#92418—21no—
73C2×C8×D61—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519763—41no—
77#14519761—41no—
78#14519757—21no—
79#14519751—21no—
80#14519751—21no—
81#14519748—21no—
82C4×D244C2311yescharacteristic
83#14519736—21no—
84#14519735—21no—
85C2×C4⋊Dic61—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708D411yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—41no—
104#14519701D411yescharacteristic
105#14519700—21no—
106#14519698—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×D8×C61D411yescharacteristic
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#125×C61—21no—
119#125×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61—41no—
123C22⋊Q8×C61—41no—
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61C2311yescharacteristic
128C4×D4×C61C2311yescharacteristic
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131#107×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—41no—
152C22×C4×C61—41no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155C22×Dic61—41no—
156#11758224—42no—
157#11758224—81no—
158C2×D244C2×D411yescharacteristic
159C2×C4×D61—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162D8×C61—22no—
163C2×C8×C61—21no—
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185#11758202—82no—
186#11758201—22no—
187#11758200—81no—
188#11758198—21no—
189#11758198—41no—
190#11758198—41no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197C4×Dic61—41no—
198#11758194—42no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—42no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C8×D61—81no—
207C23×C61—21no—
208C23×C61—21no—
209C22×D61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—81no—
213Q8×C61—81no—
214D4×C61—21no—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61#13712yes—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—161no—
229C2×C4×C61—161no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C22⋊D61—82no—
234C22⋊D61—161no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—82no—
239D244—21no—
240D244—21no—
241C4×D61—41no—
242C4×D61—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—81no—
246C8×C61—21no—
247C8×C61—21no—
248C8×C61—41no—
249#92418—21no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418#13512yes—
255#19861—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61D4211yescharacteristic
263C2×D61—41no—
264C2×D61—41no—
265C2×D61—41no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61C2×#13511yescharacteristic
274C4×C61C2×#13711yescharacteristic
275C4×C61#57711yescharacteristic
276C4×C61#57811yescharacteristic
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280#3380D6111yescharacteristic, complemented
281#3357—611no—
282#3335—611no—
283#3303—611no—
284#1723—611no—
285#1691—611no—
286#1684—611no—
287#1675—611no—
288#1675—611no—
289#1643—611no—
290#1635—611no—
291#1634—611no—
292#1632—611no—
293#1581—611no—
294SD16⋊C8—611no—
295C2×C61—42no—
296C2×C61—81no—
297C2×C61#326611yescharacteristic
298C2×C61#328311yescharacteristic
299C2×C61#328811yescharacteristic
300D61—81no—
301#555—611no—
302#554C2×D6111yescharacteristic
303#554C2×D6111yescharacteristic
304#547—611no—
305#500—611no—
306#500C2×D6111yescharacteristic
307#499—21no—
308#498—611no—
309#494—611no—
310#487—611no—
311#487—611no—
312#485—1221no—
313#485—1221no—
314#483—1221no—
315#481—1221no—
316#480—1221no—
317#478—21no—
318#478—21no—
319#477—611no—
320#476—611no—
321#476—611no—
322#476—611no—
323#476C2×D6111yescharacteristic
324#476C2×D6111yescharacteristic
325#475—611no—
326#468—1221no—
327#465—1221no—
328#465—1221no—
329#461—611no—
330#449—1221no—
331C8×Q8C2×D6111yescharacteristic
332C4×SD16—611no—
333C4×D8—21no—
334C4×D8C2×D6111yescharacteristic
335#436—611no—
336C8×D4—611no—
337C8×D4—611no—
338#428—1221no—
339C2×#106—1221no—
340#416—1221no—
341#335—611no—
342#335—611no—
343#333—611no—
344#333—611no—
345#333—611no—
346#332—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350D4⋊C8—611no—
351#322—611no—
352C61#1986111yescharacteristic, complemented, Sylow
353C2×#41—1221no—
354C2×Q16—2441no—
355C2×SD16—611no—
356C2×SD16—1221no—
357C2×D8C22⋊D6111yescharacteristic
358C2×M16—1221no—
359C22×C8—1221no—
360C22×C8—1221no—
361C4⋊Q8—611no—
362C4⋊Q8—611no—
363C4⋊Q8C22×D6111yescharacteristic
364C4⋊Q8C22×D6111yescharacteristic
365#125—21no—
366#125—21no—
367#124—2441no—
368#123—2441no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—41no—
372C22⋊Q8—41no—
373C22⋊Q8—1221no—
374#121—1221no—
375#121—1221no—
376#121—1221no—
377C4×Q8C22×D6111yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—611no—
381C4×D4C22×D6111yescharacteristic
382C4×D4C22×D6111yescharacteristic
383#117—1221no—
384#117—1221no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—1221no—
388#107—21no—
389#107—21no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#107C22⋊D6111yescharacteristic
394#106—21no—
395#106—21no—
396#106—611no—
397#106—1221no—
398#106—1221no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402#106—1221no—
403C4⋊C8—21no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8C22×D6111yescharacteristic
408#103—611no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—2441no—
415#102—21no—
416#102—21no—
417#102—611no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102C22⋊D6111yescharacteristic
422#102C22⋊D6111yescharacteristic
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C4×C8—21no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8C22×D6111yescharacteristic
430#41—4881no—
431#41—2442no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8D4×D6111yescharacteristic
437C2×D4—611no—
438C2×D4—2441no—
439C2×D4—2441no—
440C2×D4C2×C22⋊D6111yescharacteristic
441C2×D4C2×C22⋊D6111yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—1221no—
447C22×C4—2441no—
448Q16—4882no—
449SD16—1222no—
450SD16—4881no—
451D8—22no—
452M16—2442no—
453C2×C8—21no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—4881no—
461C2×C8—2442no—
462C2×C8C2×C22⋊D6111yescharacteristic
463C2×C8C2×C22⋊D6111yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—611no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—2441no—
479C4⋊C4D4×D6111yescharacteristic
480C4⋊C4C2×C22⋊D6111yescharacteristic
481C4⋊C4C2×C22⋊D6111yescharacteristic
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—1221no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C42—21no—
491C42—2441no—
492C42C23×D6111yescharacteristic
493C23—21no—
494C23—21no—
495C23—1221no—
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—81no—
500Q8—4881no—
501Q8—2441no—
502Q8—2441no—
503D4—21no—
504D4—21no—
505D4—21no—
506D4—1221no—
507D4—1221no—
508D4—9761no—
509D4—4882no—
510D4#1451980812yes—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—161no—
522C2×C4—161no—
523C2×C4—1221no—
524C2×C4—4882no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4C2×D4×D6111yescharacteristic
530C2×C4C22×C22⋊D6111yescharacteristic
531C2×C4#1451982211yescharacteristic
532C8—21no—
533C8—21no—
534C8—41no—
535C8—612no—
536C8—1221no—
537C8—2441no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—2441no—
548C22—2441no—
549C22—2441no—
550C22D4×C22⋊D6111yescharacteristic
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—4881no—
559C4—4881no—
560C4—2441no—
561C4#135×D6111yescharacteristic
562C4C2×#1451980811yescharacteristic
563C4#1684057011yescharacteristic
564C4#1684057611yescharacteristic
565C2—42no—
566C2—81no—
567C2—4881no—
568C2#2145733011yescharacteristic
569C2#2145753511yescharacteristic
570C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3380C2
socleC22×C61D42
prime radical#21457749C2
hypercenter#3380D61
hypocenterC61#19861
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #21458445C2semidirect
2. #21458170C2semidirect
4. #21445495C2semidirect
5. #21445320C2semidirect
6. #21445277C2semidirect
7. #21445216C2semidirect
8. #21445215C2semidirect
9. #21445026C2semidirect
10. #21444968C2semidirect
11. #21444963C2semidirect
12. #21444953C2semidirect
13. #21444649C2semidirect
14. #21444629C2semidirect
15. C61×#3380C2semidirect
280. #3380D61semidirect
352. C61#19861semidirect

Maximal subgroups

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

subgroupindex
#214584452
#214581702
#214577492
#214454952
#214453202
#214452772
#214452162
#214452152
#214450262
#214449682
#214449632
#214449532
#214446492
#214446292
C61×#33802
#1986161