Finite Group Databasebeta

← #29668007

Subgroups — #29668007

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456959C211yesmaximal, characteristic, complemented
2#21456761C211yesmaximal, characteristic
3#21456404C211yesmaximal, characteristic, complemented
4#21456196C211yesmaximal, characteristic, complemented
5#21456190C211yesmaximal, characteristic
6#486×D61C211yesmaximal, characteristic
7#21445481C211yesmaximal, characteristic, complemented
8#21445476C211yesmaximal, characteristic, complemented
9#21445474C211yesmaximal, characteristic, complemented
10#21445469C211yesmaximal, characteristic, complemented
11#21445432C211yesmaximal, characteristic, complemented
12#21445420C211yesmaximal, characteristic, complemented
13#21445419C211yesmaximal, characteristic, complemented
14C8×#11758189C211yesmaximal, characteristic, complemented
15C61×#1711C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840317C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19#127×D61C2211yescharacteristic
20#16840165C2211yescharacteristic, complemented
21#16840160C2211yescharacteristic, complemented
22#16840158C2211yescharacteristic, complemented
23#16840141C2211yescharacteristic, complemented
24#16840135C2211yescharacteristic
25#16840134C2211yescharacteristic
26#16840129—22no—
27#16840125—22no—
28#16840114C2211yescharacteristic, complemented
29#16840113C2211yescharacteristic, complemented
30#16840113C2211yescharacteristic, complemented
31#16840112C2211yescharacteristic, complemented
32#16840112C2211yescharacteristic, complemented
33#16840110C2211yescharacteristic
34#16840110C2211yescharacteristic
35#16840110C2211yescharacteristic
36#16840109C2211yescharacteristic
37#16840109C2211yescharacteristic
38#16840108C2211yescharacteristic
39#16840108C2211yescharacteristic
40#16840107—22no—
41#16840091—22no—
42#16840045—22no—
43#16840002—22no—
44#16839961—22no—
45#16839914—22no—
46#16839854—22no—
47#16839813C2211yescharacteristic
48#16839769—22no—
49#16839765—22no—
50#102×D61—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic, complemented
54#16839483C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#487C2211yescharacteristic, complemented
57C61×#487C2211yescharacteristic, complemented
58C61×#487C2211yescharacteristic, complemented
59C61×#486C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C82×C61C2211yescharacteristic, complemented
63C2×Q8×D61—21no—
64C2×D4×D61—21no—
65C2×D4×D61—21no—
66C2×D4×D61—21no—
67C2×#11758202—22no—
68C2×#11758190—22no—
69C2×#11758200—22no—
70C2×#11758189D412yes—
71C2×C8×D61D412yes—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519776—21no—
77#14519775—21no—
78#14519774C2311yescharacteristic
79#14519773—21no—
80#14519773—21no—
81#14519773—21no—
82#14519768—21no—
83#14519768—21no—
84#14519757—21no—
85#14519757—21no—
86#14519753—21no—
87#14519753—21no—
88Q8×Dic61—21no—
89Q8×Dic61—21no—
90#14519742—21no—
91#14519742—21no—
92#14519735—21no—
93#14519735—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C42×D61C2311yescharacteristic
97#14519710—22no—
98#14519709—22no—
99#14519708—22no—
100#14519706D412yes—
101#14519702—22no—
102#14519702—22no—
103#14519701—22no—
104#14519701—22no—
105#14519700—22no—
106#14519699—22no—
107#14519699—22no—
108#14519698—22no—
109#14519698—22no—
110#14519697—22no—
111#14519697—22no—
112#14519697—22no—
113#14519689D412yes—
114#14519687D412yes—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C8×Dic61D412yes—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#103×C61—22no—
124#103×C61—22no—
125#103×C61—22no—
126#102×C61—22no—
127#102×C61—22no—
128#102×C61—22no—
129C4×C8×C61C2311yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C23×D61—41no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142Q8×D61—44no—
143D4×D61—44no—
144D4×D61—44no—
145D4×D61—82no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×C4×D61—41no—
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157C2×C8×C61C2×D412yes—
158C2×C8×C61C2×D412yes—
159C2×C8×C61C2×D412yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C42×C61C2411yescharacteristic
167#11758202—44no—
168#11758200—82no—
169#11758198—41no—
170#11758198—41no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—41no—
178C4⋊Dic61—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196—41no—
182#11758196—41no—
183C4×Dic61—41no—
184C4×Dic61—41no—
185C4×Dic61C2×D411yescharacteristic
186C4×Dic61C2×D411yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D412yes—
189C2×#92418C2×D412yes—
190C2×#92418C2×D412yes—
191#11758190—82no—
192#11758189—24no—
193C8×D61—42no—
194C23×C61—41no—
195C22×D61—41no—
196C22×D61—82no—
197C22×D61—82no—
198C22×D61—82no—
199C22×D61#12711yescharacteristic
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—42no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C22⋊D61—82no—
213C22⋊D61—82no—
214C22⋊D61—82no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61#12711yescharacteristic
219C2×Dic61#12711yescharacteristic
220C2×Dic61#12711yescharacteristic
221D244—42no—
222D244—42no—
223D244—42no—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—82no—
227C4×D61#13514yes—
228Dic122—42no—
229Dic122—42no—
230Dic122—42no—
231Dic122—82no—
232C8×C61—22no—
233C8×C61—22no—
234C8×C61—22no—
235#92418—22no—
236#92418—22no—
237#92418—22no—
238#19637—611nomaximal, Sylow
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61C2×#12711yescharacteristic
243C2×D61—21no—
244C2×D61—21no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—164no—
249C2×D61#49512yes—
250C4×C61—81no—
251C4×C61—81no—
252C4×C61—81no—
253C4×C61C2×#13512yes—
254C4×C61#57512yes—
255C4×C61#57712yes—
256Dic61—21no—
257Dic61—21no—
258Dic61—81no—
259Dic61—81no—
260Dic61—81no—
261Dic61#49512yes—
262#3229—611no—
263#3208—611no—
264#3170—611no—
265#3139—611no—
266#3138—611no—
267C2×#486—611no—
268#1720—611no—
269#1720—611no—
270#1719—611no—
271#1719—611no—
272#1711—611no—
273#1711D6111yescharacteristic, complemented
274#1708—611no—
275#1708—611no—
276C8×M16—611no—
277C2×C61—161no—
278C2×C61C2×#49511yescharacteristic
279C2×C61#315311yescharacteristic
280C2×C61#315411yescharacteristic
281D61—22no—
282D61—161no—
283#533—611no—
284#533—611no—
285C2×C4⋊Q8—611no—
286C2×#127—611no—
287#500—611no—
288#498—611no—
289#498—611no—
290#494—611no—
291#492—611no—
292#492—611no—
293#491—1222no—
294#490—1222no—
295#487—611no—
296#487—611no—
297#487—611no—
298#487—611no—
299#487—611no—
300#487C2×D6111yescharacteristic, complemented
301#487C2×D6111yescharacteristic, complemented
302#487C2×D6111yescharacteristic, complemented
303#486—611no—
304#486—611no—
305#486—611no—
306#486—611no—
307#486—611no—
308#486—611no—
309#486—611no—
310#486—1222no—
311#486C2×D6111yescharacteristic
312#486C2×D6111yescharacteristic
313#486C2×D6111yescharacteristic
314#483—1222no—
315#475—1222no—
316#467—1222no—
317#460—1222no—
318#450—1222no—
319C4×Q16—1222no—
320#431—611no—
321#419—1222no—
322#418—1222no—
323C2×#102—1222no—
324C4×M16—611no—
325C2×C4×C8—611no—
326#322—611no—
327#322—611no—
328C82—611no—
329C82C2×D6111yescharacteristic, complemented
330C61#1963711yescharacteristic, complemented, Sylow
331C22×Q8—1221no—
332C22×D4—1221no—
333C22×D4—1221no—
334C22×D4—1221no—
335C2×Q16—1222no—
336C2×SD16—1222no—
337C2×SD16—1222no—
338C2×M16—612no—
339C22×C8—612no—
340C4⋊Q8—611no—
341C4⋊Q8—611no—
342C4⋊Q8—611no—
343C4⋊Q8—1221no—
344C4⋊Q8—1221no—
345C4⋊Q8—1221no—
346C4⋊Q8C22×D6111yescharacteristic
347C4⋊Q8C22×D6111yescharacteristic
348C4⋊Q8C22×D6111yescharacteristic
349#127—611no—
350#127—1221no—
351#127—1221no—
352#127—1221no—
353#127C22×D6111yescharacteristic
354#125—1221no—
355#125—1221no—
356C22⋊Q8—1221no—
357C22⋊Q8—1221no—
358C22⋊Q8—1221no—
359C22⋊Q8—1221no—
360C4×Q8—1221no—
361C4×Q8—1221no—
362C4×Q8—1221no—
363C4×Q8—1221no—
364#117—1221no—
365#117—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C42—611no—
369#107—1222no—
370#106—1222no—
371#106—1222no—
372C4⋊C8—612no—
373#103—22no—
374#103—22no—
375#103—22no—
376#103—1222no—
377#103—1222no—
378#103—1222no—
379#103—1222no—
380#103—1222no—
381#102—22no—
382#102—22no—
383#102—22no—
384#102—1222no—
385#102—1222no—
386#102—1222no—
387#102—1222no—
388#102—1222no—
389#102—1222no—
390#102—1222no—
391C22⋊C8—612no—
392#97—612no—
393C4×C8—612no—
394C4×C8—611no—
395C4×C8—611no—
396C4×C8—611no—
397C4×C8C22×D6111yescharacteristic
398C4×C8C22×D6111yescharacteristic
399C4×C8C22×D6111yescharacteristic
400C24—2441no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8—1221no—
407C2×Q8—2441no—
408C2×Q8—2444no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—4882no—
416C2×D4—2444no—
417C2×D4—2441no—
418C2×D4—2441no—
419C2×D4—2441no—
420C2×D4—2444no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—2441no—
425Q16—2444no—
426SD16—4882no—
427SD16—4882no—
428M16—1224no—
429C2×C8—612no—
430C2×C8—612no—
431C2×C8—612no—
432C2×C8—2442no—
433C2×C8D4×D6112yes—
434C2×C8D4×D6112yes—
435C2×C8D4×D6112yes—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4—1221no—
447C4⋊C4—1221no—
448C4⋊C4—2441no—
449C4⋊C4—2441no—
450C4⋊C4—2441no—
451C4⋊C4—2441no—
452C4⋊C4—2441no—
453C4⋊C4—2441no—
454C22⋊C4—2441no—
455C22⋊C4—2441no—
456C42—611no—
457C42—611no—
458C42—611no—
459C42—2441no—
460C42—2441no—
461C42C23×D6111yescharacteristic
462C23—41no—
463C23—611no—
464C23—4882no—
465C23—4882no—
466C23—4882no—
467C23—2441no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—4882no—
472Q8—2442no—
473Q8—2442no—
474Q8—2442no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—4882no—
479D4—4882no—
480D4—4882no—
481D4—2442no—
482D4—2442no—
483D4—2442no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—611no—
488C2×C4—614no—
489C2×C4—611no—
490C2×C4—611no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—4882no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4C2×D4×D6111yescharacteristic
499C2×C4C2×D4×D6111yescharacteristic
500C8—22no—
501C8—22no—
502C8—22no—
503C8—1222no—
504C8—1222no—
505C8—1222no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—612no—
510C22—1221no—
511C22—1221no—
512C22—9764no—
513C22—4881no—
514C22—4881no—
515C22—4881no—
516C22#127×D6111yescharacteristic
517C4—81no—
518C4—81no—
519C4—81no—
520C4—612no—
521C4—1221no—
522C4—1221no—
523C4—4881no—
524C4—4881no—
525C4—4881no—
526C4#135×D6112yes—
527C4#1684055912yes—
528C4#1684056912yes—
529C2—161no—
530C2—1222no—
531C2—9761no—
532C2#495×D6111yescharacteristic
533C2#2145628011yescharacteristic
534C2#2145629011yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1711C2
socleC22×C61C2×#127
prime radical#127×D61C22
hypercenter#1711D61
hypocenterC61#19637
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1711

Complements

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

NH ≅ G/Ntype
1. #21456959C2semidirect
3. #21456404C2semidirect
4. #21456196C2semidirect
7. #21445481C2semidirect
8. #21445476C2semidirect
9. #21445474C2semidirect
10. #21445469C2semidirect
11. #21445432C2semidirect
12. #21445420C2semidirect
13. #21445419C2semidirect
14. C8×#11758189C2semidirect
15. C61×#1711C2semidirect
20. #16840165C22semidirect
21. #16840160C22semidirect
22. #16840158C22semidirect
23. #16840141C22semidirect
28. #16840114C22semidirect
29. #16840113C22semidirect
30. #16840113C22semidirect
31. #16840112C22semidirect
32. #16840112C22semidirect
53. #16839484C22semidirect
54. #16839483C22semidirect
55. C8×#92418C22semidirect
56. C61×#487C22semidirect
57. C61×#487C22semidirect
58. C61×#487C22semidirect
62. C82×C61C22semidirect
273. #1711D61semidirect
300. #487C2×D61semidirect
301. #487C2×D61semidirect
302. #487C2×D61semidirect
329. C82C2×D61semidirect
330. C61#19637semidirect

Maximal subgroups

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

subgroupindex
#214569592
#214567612
#214564042
#214561962
#214561902
#486×D612
#214454812
#214454762
#214454742
#214454692
#214454322
#214454202
#214454192
C8×#117581892
C61×#17112
#1963761