Finite Group Databasebeta

← #29667915

Subgroups — #29667915

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456763C211yesmaximal, characteristic, complemented
2#21456733C211yesmaximal, characteristic
3#21456377C211yesmaximal, characteristic, complemented
4#21456209C211yesmaximal, characteristic, complemented
5#21456190C211yesmaximal, characteristic, complemented
6#486×D61C211yesmaximal, characteristic, complemented
7#21445482C211yesmaximal, characteristic, complemented
8#21445480C211yesmaximal, characteristic, complemented
9#21445457C211yesmaximal, characteristic, complemented
10#21445455C211yesmaximal, characteristic, complemented
11#21445431C211yesmaximal, characteristic, complemented
12#21445430C211yesmaximal, characteristic, complemented
13#21445423C211yesmaximal, characteristic, complemented
14C8×#11758189C211yesmaximal, characteristic, complemented
15C61×#1709C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840310C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19#127×D61C2211yescharacteristic
20#16840158C2211yescharacteristic, complemented
21#16840141C2211yescharacteristic, complemented
22#16840138C2211yescharacteristic
23#16840134C2211yescharacteristic, complemented
24#16840125—22no—
25#16840120—22no—
26#16840114C2211yescharacteristic, complemented
27#16840113C2211yescharacteristic, complemented
28#16840113C2211yescharacteristic, complemented
29#16840113C2211yescharacteristic, complemented
30#16840112C2211yescharacteristic, complemented
31#16840110C2211yescharacteristic
32#16840110C2211yescharacteristic
33#16840109C2211yescharacteristic
34#16840109C2211yescharacteristic
35#16840109C2211yescharacteristic
36#16840108C2211yescharacteristic
37#16840108C2211yescharacteristic
38#16840108C2211yescharacteristic
39#16840108C2211yescharacteristic, complemented
40#16840107—22no—
41#16839990—22no—
42#16839970—22no—
43#16839955—22no—
44#16839916—22no—
45#16839894—22no—
46#16839853—22no—
47#16839813C2211yescharacteristic
48#16839765—22no—
49#16839760—22no—
50#102×D61—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic, complemented
54#16839483C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#494C2211yescharacteristic, complemented
57C61×#493C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#486C2211yescharacteristic
60C61×#486C2211yescharacteristic, complemented
61C61×#486C2211yescharacteristic, complemented
62C82×C61C2211yescharacteristic, complemented
63C2×#11758224—21no—
64C2×Q8×D61—21no—
65C2×D4×D61—21no—
66C2×D4×D61—21no—
67C2×D4×D61—21no—
68C2×#11758202—22no—
69C2×#11758200—22no—
70C2×#11758199—22no—
71C2×#11758189D412yes—
72C2×C8×D61D412yes—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519776—21no—
78#14519775—21no—
79#14519774C2311yescharacteristic
80#14519773—21no—
81#14519773—21no—
82#14519773—21no—
83#14519768—21no—
84#14519764—21no—
85#14519757—21no—
86#14519753—21no—
87#14519748—21no—
88#14519748—21no—
89Q8×Dic61—21no—
90#14519742—21no—
91D4×Dic61—21no—
92D4×Dic61—21no—
93#14519735—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C42×D61C2311yescharacteristic
97#14519708—22no—
98#14519706D412yes—
99#14519702—22no—
100#14519702—22no—
101#14519702—22no—
102#14519701—22no—
103#14519700—22no—
104#14519699—22no—
105#14519699—22no—
106#14519699—22no—
107#14519698—22no—
108#14519698—22no—
109#14519698—22no—
110#14519698—22no—
111#14519697—22no—
112#14519697—22no—
113#14519689D412yes—
114#14519687D412yes—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C8×Dic61D412yes—
119C2×Q16×C61—22no—
120C2×D8×C61—22no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#127×C61C2311yescharacteristic
124#127×C61C2311yescharacteristic
125#102×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×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C22×Dic61—41no—
146Q8×D61—82no—
147#11758224—44no—
148D4×D61—44no—
149D4×D61—44no—
150D4×D61—82no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×C4×D61—41no—
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162Q16×C61—82no—
163D8×C61—82no—
164C2×C8×C61C2×D412yes—
165C2×C8×C61C2×D412yes—
166C2×C8×C61C2×D412yes—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C42×C61C2411yescharacteristic
172C22⋊Dic61—41no—
173C22⋊Dic61—41no—
174#11758202—82no—
175#11758200—44no—
176#11758199—82no—
177#11758198—41no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—41no—
185#11758196—41no—
186#11758196—41no—
187#11758196—41no—
188C4×Dic61—41no—
189C4×Dic61C2×D411yescharacteristic
190C4×Dic61C2×D411yescharacteristic
191C4×Dic61C2×D411yescharacteristic
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C2×#92418C2×D412yes—
195#11758189—24no—
196C8×D61—42no—
197C23×C61—41no—
198C23×C61—41no—
199C22×D61—41no—
200C22×D61—82no—
201C22×D61—82no—
202C22×D61—82no—
203C22×D61#12711yescharacteristic
204Q8×C61—42no—
205Q8×C61—42no—
206D4×C61—42no—
207D4×C61—42no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C22⋊D61—82no—
218C22⋊D61—82no—
219C22⋊D61—82no—
220C22⋊D61—82no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—82no—
225C2×Dic61#12711yescharacteristic
226C2×Dic61#12711yescharacteristic
227C2×Dic61#12711yescharacteristic
228D244—42no—
229D244—42no—
230D244—42no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—82no—
234C4×D61#13514yes—
235Dic122—42no—
236Dic122—42no—
237Dic122—42no—
238Dic122—82no—
239C8×C61—22no—
240C8×C61—22no—
241C8×C61—22no—
242#92418—22no—
243#92418—22no—
244#92418—22no—
245#19627—611nomaximal, Sylow
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61C2×#12711yescharacteristic
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—164no—
259C2×D61#49512yes—
260C4×C61—81no—
261C4×C61—81no—
262C4×C61C2×#13512yes—
263C4×C61#57512yes—
264C4×C61#57612yes—
265Dic61—21no—
266Dic61—21no—
267Dic61—81no—
268Dic61—81no—
269Dic61—81no—
270Dic61#49512yes—
271#3209—611no—
272#3205—611no—
273#3167—611no—
274#3141—611no—
275#3138—611no—
276C2×#486—611no—
277#1720—611no—
278#1720—611no—
279#1717—611no—
280#1717—611no—
281#1711—611no—
282#1711—611no—
283#1709—611no—
284#1709D6111yescharacteristic, complemented
285C8×M16—611no—
286C2×C61—161no—
287C2×C61—161no—
288C2×C61C2×#49511yescharacteristic
289C2×C61#315311yescharacteristic
290C2×C61#315511yescharacteristic
291D61—22no—
292D61—161no—
293#533—611no—
294#532—611no—
295C2×C4⋊Q8—611no—
296C2×#127—611no—
297#498—611no—
298#494—611no—
299#494C2×D6111yescharacteristic, complemented
300#493—611no—
301#493C2×D6111yescharacteristic
302#492—611no—
303#490—1222no—
304#489—1222no—
305#487—611no—
306#487—611no—
307#487—611no—
308#487—611no—
309#487—611no—
310#486—611no—
311#486—611no—
312#486—611no—
313#486—611no—
314#486—611no—
315#486—611no—
316#486—611no—
317#486—611no—
318#486—611no—
319#486—1222no—
320#486C2×D6111yescharacteristic
321#486C2×D6111yescharacteristic
322#486C2×D6111yescharacteristic, complemented
323#486C2×D6111yescharacteristic, complemented
324#465—1222no—
325#462—1222no—
326#459—1222no—
327#451—1222no—
328#447—1222no—
329C4×SD16—1222no—
330#431—611no—
331#418—1222no—
332#417—1222no—
333C2×#102—1222no—
334C4×M16—611no—
335C2×C4×C8—611no—
336#322—611no—
337#322—611no—
338C82—611no—
339C82C2×D6111yescharacteristic, complemented
340C61#1962711yescharacteristic, complemented, Sylow
341C2×#41—1221no—
342C22×Q8—1221no—
343C22×D4—1221no—
344C22×D4—1221no—
345C22×D4—1221no—
346C2×Q16—22no—
347C2×Q16—1222no—
348C2×SD16—1222no—
349C2×D8—22no—
350C2×D8—1222no—
351C2×M16—612no—
352C22×C8—612no—
353C4⋊Q8—611no—
354C4⋊Q8—611no—
355C4⋊Q8—611no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8C22×D6111yescharacteristic
360C4⋊Q8C22×D6111yescharacteristic
361#127—611no—
362#127—1221no—
363#127—1221no—
364#127—1221no—
365#127C22×D6111yescharacteristic
366#127C22×D6111yescharacteristic
367#125—1221no—
368#124—1221no—
369C22⋊Q8—1221no—
370C22⋊Q8—1221no—
371#121—1221no—
372#121—1221no—
373C4×Q8—1221no—
374C4×Q8—1221no—
375C4×D4—1221no—
376C4×D4—1221no—
377#117—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C42—611no—
381#106—1222no—
382C4⋊C8—612no—
383#103—1222no—
384#103—1222no—
385#103—1222no—
386#103—1222no—
387#103—1222no—
388#102—22no—
389#102—22no—
390#102—22no—
391#102—22no—
392#102—1222no—
393#102—1222no—
394#102—1222no—
395#102—1222no—
396#102—1222no—
397#102—1222no—
398#102—1222no—
399#102—1222no—
400#102—1222no—
401C22⋊C8—612no—
402#97—612no—
403C4×C8—612no—
404C4×C8—611no—
405C4×C8—611no—
406C4×C8—611no—
407C4×C8C22×D6111yescharacteristic
408C4×C8C22×D6111yescharacteristic
409C4×C8C22×D6111yescharacteristic
410C24—2441no—
411#41—2444no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×Q8—4882no—
418C2×Q8—2441no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—4882no—
429C2×D4—2444no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4—2441no—
433C2×D4—2444no—
434C2×D4—2441no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—2441no—
439C22×C4—2441no—
440Q16—82no—
441Q16—4882no—
442SD16—2444no—
443D8—82no—
444D8—4882no—
445M16—1224no—
446C2×C8—612no—
447C2×C8—612no—
448C2×C8—612no—
449C2×C8—2442no—
450C2×C8D4×D6112yes—
451C2×C8D4×D6112yes—
452C2×C8D4×D6112yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C22⋊C4—2441no—
470C42—611no—
471C42—611no—
472C42—611no—
473C42—2441no—
474C42C23×D6111yescharacteristic
475C23—41no—
476C23—41no—
477C23—611no—
478C23—4882no—
479C23—4882no—
480C23—4882no—
481C23—2441no—
482Q8—42no—
483Q8—42no—
484Q8—4882no—
485Q8—2442no—
486Q8—2442no—
487Q8—2442no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—4882no—
495D4—4882no—
496D4—4882no—
497D4—4882no—
498D4—2442no—
499D4—2442no—
500D4—2442no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—611no—
504C2×C4—614no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—4882no—
510C2×C4—4882no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4C2×D4×D6111yescharacteristic
517C8—22no—
518C8—22no—
519C8—22no—
520C8—1222no—
521C8—1222no—
522C8—1222no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—612no—
530C22—1221no—
531C22—1221no—
532C22—9764no—
533C22—4881no—
534C22—4881no—
535C22—4881no—
536C22#127×D6111yescharacteristic
537C4—81no—
538C4—81no—
539C4—612no—
540C4—1221no—
541C4—1221no—
542C4—4881no—
543C4—4881no—
544C4—4881no—
545C4#135×D6112yes—
546C4#1684055912yes—
547C4#1684056312yes—
548C2—161no—
549C2—161no—
550C2—1222no—
551C2—9761no—
552C2#495×D6111yescharacteristic
553C2#2145628011yescharacteristic
554C2#2145629911yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1709C2
socleC22×C61C2×#127
prime radical#21456733C2
hypercenter#1709D61
hypocenterC61#19627
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1709

Complements

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

NH ≅ G/Ntype
1. #21456763C2semidirect
3. #21456377C2semidirect
4. #21456209C2semidirect
5. #21456190C2semidirect
6. #486×D61C2semidirect
7. #21445482C2semidirect
8. #21445480C2semidirect
9. #21445457C2semidirect
10. #21445455C2semidirect
11. #21445431C2semidirect
12. #21445430C2semidirect
13. #21445423C2semidirect
14. C8×#11758189C2semidirect
15. C61×#1709C2semidirect
20. #16840158C22semidirect
21. #16840141C22semidirect
23. #16840134C22semidirect
26. #16840114C22semidirect
27. #16840113C22semidirect
28. #16840113C22semidirect
29. #16840113C22semidirect
30. #16840112C22semidirect
39. #16840108C22semidirect
53. #16839484C22semidirect
54. #16839483C22semidirect
55. C8×#92418C22semidirect
56. C61×#494C22semidirect
60. C61×#486C22semidirect
61. C61×#486C22semidirect
62. C82×C61C22semidirect
284. #1709D61semidirect
299. #494C2×D61semidirect
322. #486C2×D61semidirect
323. #486C2×D61semidirect
339. C82C2×D61semidirect
340. C61#19627semidirect

Maximal subgroups

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

subgroupindex
#214567632
#214567332
#214563772
#214562092
#214561902
#486×D612
#214454822
#214454802
#214454572
#214454552
#214454312
#214454302
#214454232
C8×#117581892
C61×#17092
#1962761