Finite Group Databasebeta

← #29663423

Subgroups — #29663423

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456994C211yesmaximal, characteristic, complemented
2#21456791C211yesmaximal, characteristic
3#21456378C211yesmaximal, characteristic, complemented
4#21445325C211yesmaximal, characteristic, complemented
5#21445318C211yesmaximal, characteristic, complemented
6#21445291C211yesmaximal, characteristic, complemented
7#21445284C211yesmaximal, characteristic, complemented
8#21445249C211yesmaximal, characteristic, complemented
9#21445242C211yesmaximal, characteristic, complemented
10#21445235C211yesmaximal, characteristic, complemented
11#21445228C211yesmaximal, characteristic, complemented
12#21444567C211yesmaximal, characteristic, complemented
13#21444530C211yesmaximal, characteristic, complemented
14#21444525C211yesmaximal, characteristic, complemented
15C61×#2966C211yesmaximal, characteristic, complemented
16#16840319C2211yescharacteristic
17#16840312C2211yescharacteristic
18#16840161—21no—
19#16840160C2211yescharacteristic
20#16840144—21no—
21#16840135C2211yescharacteristic
22#16840119—21no—
23#16840116—21no—
24#16840113C2211yescharacteristic
25#16840109C2211yescharacteristic
26#16840100—21no—
27#16840083—21no—
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840047C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16839991—21no—
33#16839991—21no—
34#16839972C2211yescharacteristic
35#16839967C2211yescharacteristic
36#16839962C2211yescharacteristic
37#16839957C2211yescharacteristic
38#16839922—21no—
39#16839905—21no—
40C4×#11758190—21no—
41C4×#11758190—21no—
42#16839815C2211yescharacteristic
43#16839815C2211yescharacteristic
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#529C2211yescharacteristic
57C61×#446C2211yescharacteristic
58C8×Q8×C61C2211yescharacteristic
59C61×#436C2211yescharacteristic
60C61×#435—21no—
61C8×D4×C61—21no—
62C8×D4×C61C2211yescharacteristic
63C61×#433—21no—
64C61×#432C2211yescharacteristic
65C61×#405—21no—
66C2×C4×C8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758202—21no—
69C2×#11758190—21no—
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758199—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519774C2311yescharacteristic
78#14519769—21no—
79#14519765—21no—
80#14519756—21no—
81#14519756—21no—
82#14519751—21no—
83#14519751—21no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×D244—21no—
87C4×D244—21no—
88#14519736—21no—
89#14519710—21no—
90#14519710—21no—
91#14519709—21no—
92#14519709—21no—
93#14519708—21no—
94#14519708—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519701—21no—
103#14519700—21no—
104#14519700—21no—
105#14519699—21no—
106#14519699—21no—
107#14519698—21no—
108#14519698—21no—
109#14519697—21no—
110#14519697—21no—
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113C4×#92418—21no—
114C4×#92418—21no—
115C2×M16×C61—21no—
116C22×C8×C61—21no—
117C22×C8×C61D411yescharacteristic
118C22×C8×C61D411yescharacteristic
119#126×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61D411yescharacteristic
122#123×C61—21no—
123C22⋊Q8×C61D411yescharacteristic
124#121×C61D411yescharacteristic
125C4×Q8×C61C2311yescharacteristic
126C4×D4×C61—21no—
127C4×D4×C61C2311yescharacteristic
128#117×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C4⋊C8×C61—21no—
131C4⋊C8×C61—21no—
132C4⋊C8×C61C2311yescharacteristic
133C4⋊C8×C61C2311yescharacteristic
134C22⋊C8×C61—21no—
135C22⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137#97×C61—21no—
138#97×C61—21no—
139C4×C8×C61—21no—
140C4×C8×C61D411yescharacteristic
141C4×C8×C61D411yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C2×D411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61C2×D411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61#4111yescharacteristic
151C2×C22⋊D61—41no—
152#11758222—81no—
153#11758222—81no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×C4×D61—41no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161M16×C61—42no—
162C2×C8×C61—22no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—41no—
166C2×C8×C61—42no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
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×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C42×C61—21no—
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61#4111yescharacteristic
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188#11758202—81no—
189#11758201—81no—
190#11758200—81no—
191#11758199—81no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200#11758196—41no—
201#11758196—41no—
202C4×Dic61—41no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207#11758190—42no—
208#11758190—42no—
209C23×C61—21no—
210C23×C61#12111yescharacteristic
211C22×D61—41no—
212Q8×C61—21no—
213Q8×C61—21no—
214D4×C61—21no—
215D4×C61—21no—
216D4×C61—42no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C22⋊D61—82no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237D244—41no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—81no—
242C4×D61—82no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—81no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61#13512yes—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#19224—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61C2×#12111yescharacteristic
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61C2×#13511yescharacteristic
277C4×C61#57511yescharacteristic
278C4×C61#57711yescharacteristic
279C4×C61#57711yescharacteristic
280Dic61—81no—
281Dic61—81no—
282Dic61—81no—
283#3233—611no—
284#3212—611no—
285#3167—611no—
286#2966D6111yescharacteristic, complemented
287#1692—611no—
288#1691—611no—
289#1686—611no—
290#1685—611no—
291#1680—611no—
292#1679—611no—
293#1678—611no—
294#1677—611no—
295#1564—611no—
296#1554—611no—
297#1553—611no—
298C2×C61—41no—
299C2×C61—81no—
300C2×C61#303511yescharacteristic
301C2×C61#305711yescharacteristic
302C2×C61#306711yescharacteristic
303D61—161no—
304#533—611no—
305#532—611no—
306#529C2×D6111yescharacteristic
307#499—1221no—
308#498—611no—
309#495—1221no—
310#492—611no—
311#488—1221no—
312#488—1221no—
313#487—611no—
314#486—611no—
315#485—1221no—
316#482—1221no—
317#477—611no—
318#476—611no—
319#475—611no—
320#474—611no—
321#465—1221no—
322#465—1221no—
323#462—611no—
324#461—611no—
325#460—611no—
326#459—611no—
327#452—1221no—
328#449—1221no—
329#446C2×D6111yescharacteristic
330C8×Q8C2×D6111yescharacteristic
331C4×SD16—1221no—
332C4×SD16—1221no—
333#436C2×D6111yescharacteristic
334#435—21no—
335C8×D4—21no—
336C8×D4C2×D6111yescharacteristic
337#433—21no—
338#432C2×D6111yescharacteristic
339#431—611no—
340#431—611no—
341#405—21no—
342C2×C4×C8C2×D6111yescharacteristic
343#335—611no—
344#335—611no—
345#334—611no—
346#334—611no—
347#333—611no—
348#333—611no—
349#332—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353#331—611no—
354#331—611no—
355C61#1922411yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×Q16—1221no—
358C2×SD16—1221no—
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×D8—1221no—
363C2×M16—21no—
364C22×C8—21no—
365C22×C8C22⋊D6111yescharacteristic
366C22×C8C22⋊D6111yescharacteristic
367C4⋊Q8—611no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370#127—611no—
371#126—21no—
372#125—1221no—
373#125D24411yescharacteristic
374#124—1221no—
375#124D24411yescharacteristic
376#123—21no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379C22⋊Q8D24411yescharacteristic
380#121—1221no—
381#121—1221no—
382#121D24411yescharacteristic
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×Q8C22×D6111yescharacteristic
386C4×D4—21no—
387C4×D4—1221no—
388C4×D4—1221no—
389C4×D4C22×D6111yescharacteristic
390#117—21no—
391#117—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—1221no—
394#107—1221no—
395#106—1221no—
396#106—1221no—
397#106—1221no—
398#106—1221no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8C22×D6111yescharacteristic
406C4⋊C8C22×D6111yescharacteristic
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#102—1221no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419C22⋊C8—21no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—21no—
425#97—21no—
426C4×C8—21no—
427C4×C8—1221no—
428C4×C8—1221no—
429C4×C8C22⋊D6111yescharacteristic
430C4×C8C22⋊D6111yescharacteristic
431C4×C8C22×D6111yescharacteristic
432C4×C8C22×D6111yescharacteristic
433#41—4881no—
434#41—4881no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8C2×D24411yescharacteristic
439C2×D4—21no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—2441no—
444C2×D4C2×D24411yescharacteristic
445C22×C4—21no—
446C22×C4—2441no—
447C22×C4C2×D24411yescharacteristic
448C22×C4#1175822211yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450Q16—4881no—
451SD16—4881no—
452SD16—4881no—
453SD16—2442no—
454SD16—2442no—
455D8—4881no—
456M16—42no—
457C2×C8—22no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—41no—
461C2×C8—42no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8C2×C22⋊D6111yescharacteristic
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C2×C8C2×C22⋊D6111yescharacteristic
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4C2×D24411yescharacteristic
483C4⋊C4C2×D24411yescharacteristic
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C42—21no—
493C42—2441no—
494C42C2×D24411yescharacteristic
495C42#1175822211yescharacteristic
496C42C2×C22⋊D6111yescharacteristic
497C42C23×D6111yescharacteristic
498C23—21no—
499C23—2441no—
500C23#1451975111yescharacteristic
501Q8—21no—
502Q8—21no—
503Q8—4881no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508D4—21no—
509D4—21no—
510D4—42no—
511D4—4881no—
512D4—4882no—
513D4—2441no—
514D4—2441no—
515D4—2441no—
516D4—2441no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—4882no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4#1451975111yescharacteristic
532C2×C4#1451975111yescharacteristic
533C2×C4#1451975111yescharacteristic
534C2×C4C22×C22⋊D6111yescharacteristic
535C2×C4C22×D24411yescharacteristic
536C2×C4C2×#1175822211yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—41no—
541C8—41no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8#1451979712yes—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22C2×#1451975111yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—4881no—
565C4—4881no—
566C4—4881no—
567C4C2×#1451979711yescharacteristic
568C4#1684056011yescharacteristic
569C4#1684057411yescharacteristic
570C4#1684057411yescharacteristic
571C2—41no—
572C2—81no—
573C2—9761no—
574C2#2145527411yescharacteristic
575C2#2145545811yescharacteristic
576C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2966C2
socleC22×C61C2×#121
prime radical#21456791C2
hypercenter#2966D61
hypocenterC61#19224
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
1. #21456994C2semidirect
3. #21456378C2semidirect
4. #21445325C2semidirect
5. #21445318C2semidirect
6. #21445291C2semidirect
7. #21445284C2semidirect
8. #21445249C2semidirect
9. #21445242C2semidirect
10. #21445235C2semidirect
11. #21445228C2semidirect
12. #21444567C2semidirect
13. #21444530C2semidirect
14. #21444525C2semidirect
15. C61×#2966C2semidirect
286. #2966D61semidirect
355. C61#19224semidirect

Maximal subgroups

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

subgroupindex
#214569942
#214567912
#214563782
#214453252
#214453182
#214452912
#214452842
#214452492
#214452422
#214452352
#214452282
#214445672
#214445302
#214445252
C61×#29662
#1922461