Finite Group Databasebeta

← #29667875

Subgroups — #29667875

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456756C211yesmaximal, characteristic, complemented
2#21456740C211yesmaximal, characteristic, complemented
3#21456377C211yesmaximal, characteristic, complemented
4#21456362C211yesmaximal, characteristic, complemented
5#21456270C211yesmaximal, characteristic
6#486×D61C211yesmaximal, characteristic, complemented
7#21445446C211yesmaximal, characteristic, complemented
8#21445445C211yesmaximal, characteristic, complemented
9#21445442C211yesmaximal, characteristic, complemented
10#21445439C211yesmaximal, characteristic, complemented
11#21445420C211yesmaximal, characteristic, complemented
12#21445419C211yesmaximal, characteristic, complemented
13#21445410C211yesmaximal, characteristic, complemented
14C8×#11758189C211yesmaximal, characteristic, complemented
15C61×#1706C211yesmaximal, characteristic, complemented
16C2×#14519801—22no—
17#16840317C2211yescharacteristic
18#16840310C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#127×D61C2211yescharacteristic
21#16840170—22no—
22#16840165C2211yescharacteristic, complemented
23#16840160C2211yescharacteristic, complemented
24#16840158C2211yescharacteristic, complemented
25#16840158C2211yescharacteristic, complemented
26#16840150—22no—
27#16840138C2211yescharacteristic
28#16840136C2211yescharacteristic
29#16840135C2211yescharacteristic
30#16840134C2211yescharacteristic
31#16840134C2211yescharacteristic, complemented
32#16840134C2211yescharacteristic, complemented
33#16840113C2211yescharacteristic, complemented
34#16840113C2211yescharacteristic, complemented
35#16840112C2211yescharacteristic, complemented
36#16840110C2211yescharacteristic
37#16840109C2211yescharacteristic
38#16840109C2211yescharacteristic
39#16840108C2211yescharacteristic
40#16840108C2211yescharacteristic
41#16840107—22no—
42#16839990—22no—
43#16839965—22no—
44#16839960—22no—
45#16839911—22no—
46#16839899—22no—
47#16839853—22no—
48#16839813C2211yescharacteristic
49#16839803—22no—
50#102×D61—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic, complemented
54#16839483C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#500C2211yescharacteristic, complemented
57C61×#493C2211yescharacteristic
58C61×#492C2211yescharacteristic
59C61×#492C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic
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—
68#14519801—48no—
69C2×#11758190—22no—
70C2×#11758200—22no—
71C2×#11758200—22no—
72C2×#11758200—22no—
73C2×#11758201—22no—
74C2×#11758199—22no—
75C2×#11758189D412yes—
76C2×C8×D61D412yes—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519776—21no—
81#14519776—21no—
82#14519775—21no—
83#14519774C2311yescharacteristic
84#14519773—21no—
85#14519773—21no—
86#14519773—21no—
87#14519768—21no—
88#14519764—21no—
89#14519757—21no—
90#14519753—21no—
91#14519748—21no—
92#14519748—21no—
93Q8×Dic61—21no—
94#14519742—21no—
95D4×Dic61—21no—
96D4×Dic61—21no—
97#14519735—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—21no—
100C42×D61C2311yescharacteristic
101#14519710—22no—
102#14519709—22no—
103#14519708—22no—
104#14519708—22no—
105#14519706D412yes—
106#14519702—22no—
107#14519702—22no—
108#14519701—22no—
109#14519699—22no—
110#14519699—22no—
111#14519698—22no—
112#14519698—22no—
113#14519697—22no—
114#14519689D412yes—
115#14519687D412yes—
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C4×#92418C2311yescharacteristic
119C8×Dic61D412yes—
120C2×SD16×C61—22no—
121C2×SD16×C61—22no—
122C2×D8×C61—22no—
123C4⋊Q8×C61C2311yescharacteristic
124C4⋊Q8×C61C2311yescharacteristic
125#127×C61C2311yescharacteristic
126#127×C61C2311yescharacteristic
127#107×C61—22no—
128#102×C61—22no—
129#102×C61—22no—
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C4×C8×C61C2311yescharacteristic
133C23×D61—41no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C22×Dic61—41no—
147Q8×D61—82no—
148#11758224—44no—
149D4×D61—44no—
150D4×D61—44no—
151D4×D61—82no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×C4×D61—41no—
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—41no—
163SD16×C61—82no—
164SD16×C61—82no—
165D8×C61—44no—
166C2×C8×C61C2×D412yes—
167C2×C8×C61C2×D412yes—
168C2×C8×C61C2×D412yes—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C42×C61C2411yescharacteristic
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176#11758201—82no—
177#11758200—44no—
178#11758200—44no—
179#11758200—82no—
180#11758199—44no—
181#11758198—41no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—41no—
189#11758196—41no—
190#11758196—41no—
191#11758196—41no—
192C4×Dic61—41no—
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61C2×D411yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196C2×#92418C2×D412yes—
197C2×#92418C2×D412yes—
198C2×#92418C2×D412yes—
199#11758190—44no—
200#11758189SD1618yescomplemented
201C8×D61—42no—
202C23×C61—41no—
203C23×C61—41no—
204C22×D61—41no—
205C22×D61—82no—
206C22×D61—82no—
207C22×D61—82no—
208C22×D61#12711yescharacteristic
209Q8×C61—42no—
210Q8×C61—42no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—42no—
215D4×C61—42no—
216D4×C61—42no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C22⋊D61—82no—
223C22⋊D61—82no—
224C22⋊D61—82no—
225C22⋊D61—82no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—82no—
230C2×Dic61#12711yescharacteristic
231C2×Dic61#12711yescharacteristic
232C2×Dic61#12711yescharacteristic
233D244—42no—
234D244—42no—
235D244—42no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—82no—
239C4×D61C2×SD1614yes—
240Dic122—42no—
241Dic122—42no—
242Dic122—42no—
243Dic122—82no—
244C8×C61—22no—
245C8×C61—22no—
246C8×C61C2×SD1614yes—
247#92418—22no—
248#92418—22no—
249#92418C2×SD1614yes—
250#19623—611nomaximal, Sylow
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C2×#12711yescharacteristic
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—164no—
264C2×D61#49212yes—
265C4×C61—81no—
266C4×C61—81no—
267C4×C61C22×SD1612yes—
268C4×C61#57512yes—
269C4×C61#57612yes—
270Dic61—21no—
271Dic61—21no—
272Dic61—81no—
273Dic61—81no—
274Dic61—81no—
275Dic61#49212yes—
276#3208—611no—
277#3206—611no—
278#3167—611no—
279#3165—611no—
280#3151—611no—
281C2×#486—611no—
282#1715—611no—
283#1715—611no—
284#1714—611no—
285#1714—611no—
286#1708—611no—
287#1708—611no—
288#1706—611no—
289#1706D6111yescharacteristic, complemented
290C8×M16—611no—
291C2×C61—161no—
292C2×C61—161no—
293C2×C61C2×#49211yescharacteristic
294C2×C61#315311yescharacteristic
295C2×C61#315511yescharacteristic
296D61—22no—
297D61—161no—
298C2×AGL(1,ℤ/8)—1222no—
299#533—611no—
300#532—611no—
301C2×C4⋊Q8—611no—
302C2×#127—611no—
303#501—1222no—
304#500—611no—
305#500C2×D6111yescharacteristic, complemented
306#498—611no—
307#498—611no—
308#498—611no—
309#496—1222no—
310#493—611no—
311#493C2×D6111yescharacteristic
312#492—611no—
313#492—611no—
314#492—611no—
315#492—611no—
316#492—611no—
317#492C2×D6111yescharacteristic
318#492C2×D6111yescharacteristic, complemented
319#487—611no—
320#487—611no—
321#487—611no—
322#486—611no—
323#486—611no—
324#486—611no—
325#486—611no—
326#486—611no—
327#486—1222no—
328#486C2×D6111yescharacteristic
329#486C2×D6111yescharacteristic, complemented
330#465—1222no—
331#461—1222no—
332#460—1222no—
333#450—1222no—
334#448—1222no—
335C4×SD16—1222no—
336#431—611no—
337#428—1222no—
338C2×#102—1222no—
339C4×M16—611no—
340C2×C4×C8—611no—
341#322—611no—
342#322—611no—
343C82—611no—
344C82C2×D6111yescharacteristic, complemented
345C61#1962311yescharacteristic, complemented, Sylow
346C2×#41—1221no—
347C22×Q8—1221no—
348C22×D4—1221no—
349C22×D4—1221no—
350C22×D4—1221no—
351AGL(1,ℤ/8)—2448no—
352C2×SD16—22no—
353C2×SD16—22no—
354C2×SD16—1222no—
355C2×SD16—1222no—
356C2×SD16—1222no—
357C2×SD16—1222no—
358C2×SD16—1222no—
359C2×D8—22no—
360C2×D8—1222no—
361C2×M16—612no—
362C22×C8—612no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8C22×D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371#127—611no—
372#127—1221no—
373#127—1221no—
374#127—1221no—
375#127C22×D6111yescharacteristic
376#127C22×D6111yescharacteristic
377#125—1221no—
378#124—1221no—
379C22⋊Q8—1221no—
380C22⋊Q8—1221no—
381#121—1221no—
382#121—1221no—
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×D4—1221no—
386C4×D4—1221no—
387#117—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C42—611no—
391#107—22no—
392#107—1222no—
393#106—1222no—
394#106—1222no—
395#106—1222no—
396C4⋊C8—612no—
397#103—1222no—
398#103—1222no—
399#103—1222no—
400#102—22no—
401#102—22no—
402#102—1222no—
403#102—1222no—
404#102—1222no—
405#102—1222no—
406#102—1222no—
407C22⋊C8—612no—
408#97—612no—
409C4×C8—612no—
410C4×C8—611no—
411C4×C8—611no—
412C4×C8—611no—
413C4×C8C22×D6111yescharacteristic
414C4×C8C22×D6111yescharacteristic
415C4×C8C22×D6111yescharacteristic
416C24—2441no—
417#41—2444no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—4882no—
424C2×Q8—2441no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—4882no—
435C2×D4—2444no—
436C2×D4—2441no—
437C2×D4—2441no—
438C2×D4—2441no—
439C2×D4—2444no—
440C2×D4—2441no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—2441no—
445C22×C4—2441no—
446SD16—82no—
447SD16—82no—
448SD16—4882no—
449SD16—4882no—
450SD16—2444no—
451SD16—2444no—
452SD16—2444no—
453D8—44no—
454D8—2444no—
455M16—618no—
456C2×C8—612no—
457C2×C8—612no—
458C2×C8—612no—
459C2×C8—2442no—
460C2×C8D4×D6112yes—
461C2×C8D4×D6112yes—
462C2×C8D4×D6112yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C22⋊C4—2441no—
478C22⋊C4—2441no—
479C22⋊C4—2441no—
480C42—611no—
481C42—611no—
482C42—611no—
483C42—2441no—
484C42C23×D6111yescharacteristic
485C23—41no—
486C23—41no—
487C23—611no—
488C23—4882no—
489C23—4882no—
490C23—4882no—
491C23—2441no—
492Q8—42no—
493Q8—42no—
494Q8—4882no—
495Q8—2442no—
496Q8—2442no—
497Q8—2442no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—4882no—
505D4—4882no—
506D4—4882no—
507D4—4882no—
508D4—2442no—
509D4—2442no—
510D4—2442no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—611no—
514C2×C4—614no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—4882no—
520C2×C4—4882no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4C2×D4×D6111yescharacteristic
526C2×C4C2×D4×D6111yescharacteristic
527C8—22no—
528C8—22no—
529C8—614no—
530C8—1222no—
531C8—1222no—
532C8SD16×D6114yes—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—612no—
540C22—1221no—
541C22—1221no—
542C22—9764no—
543C22—4881no—
544C22—4881no—
545C22—4881no—
546C22#127×D6111yescharacteristic
547C4—81no—
548C4—81no—
549C4—612no—
550C4—1221no—
551C4—1221no—
552C4—4881no—
553C4—4881no—
554C4—4881no—
555C4C2×SD16×D6112yes—
556C4#1684055912yes—
557C4#1684056312yes—
558C2—161no—
559C2—161no—
560C2—1222no—
561C2—9761no—
562C2#492×D6111yescharacteristic
563C2#2145628011yescharacteristic
564C2#2145629911yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1706C2
socleC22×C61C2×#127
prime radical#21456270C2
hypercenter#1706D61
hypocenterC61#19623
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1706

Complements

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

NH ≅ G/Ntype
1. #21456756C2semidirect
2. #21456740C2semidirect
3. #21456377C2semidirect
4. #21456362C2semidirect
6. #486×D61C2semidirect
7. #21445446C2semidirect
8. #21445445C2semidirect
9. #21445442C2semidirect
10. #21445439C2semidirect
11. #21445420C2semidirect
12. #21445419C2semidirect
13. #21445410C2semidirect
14. C8×#11758189C2semidirect
15. C61×#1706C2semidirect
22. #16840165C22semidirect
23. #16840160C22semidirect
24. #16840158C22semidirect
25. #16840158C22semidirect
31. #16840134C22semidirect
32. #16840134C22semidirect
33. #16840113C22semidirect
34. #16840113C22semidirect
35. #16840112C22semidirect
53. #16839484C22semidirect
54. #16839483C22semidirect
55. C8×#92418C22semidirect
56. C61×#500C22semidirect
59. C61×#492C22semidirect
61. C61×#486C22semidirect
62. C82×C61C22semidirect
200. #11758189SD16semidirect
289. #1706D61semidirect
305. #500C2×D61semidirect
318. #492C2×D61semidirect
329. #486C2×D61semidirect
344. C82C2×D61semidirect
345. C61#19623semidirect

Maximal subgroups

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

subgroupindex
#214567562
#214567402
#214563772
#214563622
#214562702
#486×D612
#214454462
#214454452
#214454422
#214454392
#214454202
#214454192
#214454102
C8×#117581892
C61×#17062
#1962361