Finite Group Databasebeta

← #29667895

Subgroups — #29667895

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456966C211yesmaximal, characteristic, complemented
2#21456733C211yesmaximal, characteristic
3#21456377C211yesmaximal, characteristic, complemented
4#492×D61C211yesmaximal, characteristic, complemented
5#21456209C211yesmaximal, characteristic, complemented
6#21456190C211yesmaximal, characteristic, complemented
7#21445472C211yesmaximal, characteristic, complemented
8#21445470C211yesmaximal, characteristic, complemented
9#21445460C211yesmaximal, characteristic, complemented
10#21445422C211yesmaximal, characteristic, complemented
11#21445419C211yesmaximal, characteristic, complemented
12#21445410C211yesmaximal, characteristic, complemented
13#21445408C211yesmaximal, characteristic, complemented
14#21444098C211yesmaximal, characteristic, complemented
15C61×#1717C211yesmaximal, characteristic, complemented
16C2×SD16×D61—22no—
17#16840317C2211yescharacteristic
18#16840310C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#127×D61C2211yescharacteristic
21#16840167C2211yescharacteristic, complemented
22#16840165C2211yescharacteristic, complemented
23#16840158C2211yescharacteristic, complemented
24#16840138C2211yescharacteristic
25#16840136C2211yescharacteristic
26#16840135C2211yescharacteristic
27#16840134C2211yescharacteristic
28#16840134C2211yescharacteristic, complemented
29#16840133—22no—
30#16840125—22no—
31#16840120—22no—
32#16840114C2211yescharacteristic, complemented
33#16840113C2211yescharacteristic, complemented
34#16840113C2211yescharacteristic, complemented
35#16840112C2211yescharacteristic, complemented
36#16840110C2211yescharacteristic
37#16840109C2211yescharacteristic
38#16840109C2211yescharacteristic
39#16840108C2211yescharacteristic
40#16840108C2211yescharacteristic
41#16840108C2211yescharacteristic, complemented
42#16840096—22no—
43#16840055—22no—
44#16839990—22no—
45#16839955—22no—
46#16839894—22no—
47#16839853—22no—
48#16839813C2211yescharacteristic
49#16839765—22no—
50#16839760—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic, complemented
54C8×#92418C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#493C2211yescharacteristic
57C61×#492C2211yescharacteristic
58C61×#487C2211yescharacteristic, complemented
59C61×#486C2211yescharacteristic
60C61×#486C2211yescharacteristic, complemented
61C61×#486C2211yescharacteristic, complemented
62C61×#322C2211yescharacteristic, complemented
63C2×#11758224—21no—
64C2×Q8×D61—21no—
65C2×D4×D61—21no—
66C2×D4×D61—21no—
67C2×D4×D61—21no—
68C2×#11758190—22no—
69C2×#11758200—22no—
70C2×#11758200—22no—
71C2×#11758201—22no—
72SD16×D61—48no—
73C2×#11758199—22no—
74C2×#11758189D412yes—
75C2×C8×D61D412yes—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519776—21no—
80#14519776—21no—
81#14519775—21no—
82#14519774C2311yescharacteristic
83#14519773—21no—
84#14519773—21no—
85#14519773—21no—
86#14519768—21no—
87#14519764—21no—
88#14519757—21no—
89#14519753—21no—
90#14519748—21no—
91#14519748—21no—
92Q8×Dic61—21no—
93#14519742—21no—
94D4×Dic61—21no—
95D4×Dic61—21no—
96#14519735—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C42×D61C2311yescharacteristic
100#14519711—22no—
101#14519710—22no—
102#14519708—22no—
103#14519706D412yes—
104#14519702—22no—
105#14519702—22no—
106#14519701—22no—
107#14519700—22no—
108#14519699—22no—
109#14519699—22no—
110#14519698—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×D8×C61—22no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#127×C61C2311yescharacteristic
125#127×C61C2311yescharacteristic
126#103×C61—22no—
127#102×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—44no—
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—44no—
164D8×C61—82no—
165C2×C8×C61C2×D412yes—
166C2×C8×C61C2×D412yes—
167C2×C8×C61C2×D412yes—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C42×C61C2411yescharacteristic
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175#11758201—44no—
176#11758200—44no—
177#11758200—44no—
178#11758199—82no—
179#11758198—41no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187#11758196—41no—
188#11758196—41no—
189#11758196—41no—
190C4×Dic61—41no—
191C4×Dic61C2×D411yescharacteristic
192C4×Dic61C2×D411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C2×#92418C2×D412yes—
195C2×#92418C2×D412yes—
196C2×#92418C2×D412yes—
197#11758190—44no—
198#11758189—24no—
199C8×D61—24no—
200C23×C61—41no—
201C23×C61—41no—
202C22×D61—41no—
203C22×D61—82no—
204C22×D61—82no—
205C22×D61—82no—
206C22×D61#12711yescharacteristic
207Q8×C61—42no—
208Q8×C61—42no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—42no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C22⋊D61—82no—
221C22⋊D61—82no—
222C22⋊D61—82no—
223C22⋊D61—82no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—82no—
228C2×Dic61#12711yescharacteristic
229C2×Dic61#12711yescharacteristic
230C2×Dic61#12711yescharacteristic
231D244—42no—
232D244—42no—
233D244—42no—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—82no—
237C4×D61#13514yes—
238Dic122—42no—
239Dic122—42no—
240Dic122—42no—
241Dic122—82no—
242C8×C61—22no—
243C8×C61—22no—
244C8×C61—22no—
245#92418—22no—
246#92418—22no—
247#92418AGL(1,ℤ/8)14yes—
248#19625—611nomaximal, Sylow
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61C2×#12711yescharacteristic
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—164no—
262C2×D61#49512yes—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61C2×#13512yes—
266C4×C61C2×AGL(1,ℤ/8)12yes—
267C4×C61#57712yes—
268Dic61—21no—
269Dic61—21no—
270Dic61—81no—
271Dic61—81no—
272Dic61—81no—
273Dic61#49512yes—
274#3230—611no—
275#3205—611no—
276#3167—611no—
277C2×#492—611no—
278#3141—611no—
279#3138—611no—
280#1719—611no—
281#1719—611no—
282#1717—611no—
283#1717D6111yescharacteristic, complemented
284#1708—611no—
285#1708—611no—
286#1706—611no—
287#1706—611no—
288C8×M16—611no—
289C2×C61—161no—
290C2×C61—161no—
291C2×C61C2×#49511yescharacteristic
292C2×C61#315111yescharacteristic
293C2×C61#315411yescharacteristic
294D61—22no—
295D61—161no—
296C22×SD16—1222no—
297#533—611no—
298#532—611no—
299C2×C4⋊Q8—611no—
300C2×#127—611no—
301#500—611no—
302#500—611no—
303#498—611no—
304#493—611no—
305#493C2×D6111yescharacteristic
306#492—611no—
307#492—611no—
308#492—611no—
309#492—611no—
310#492—1222no—
311#492C2×D6111yescharacteristic
312#490—1222no—
313#489—1222no—
314#487—611no—
315#487—611no—
316#487—611no—
317#487—611no—
318#487C2×D6111yescharacteristic, complemented
319#486—611no—
320#486—611no—
321#486—611no—
322#486—611no—
323#486—611no—
324#486—611no—
325#486C2×D6111yescharacteristic
326#486C2×D6111yescharacteristic, complemented
327#486C2×D6111yescharacteristic, complemented
328#484—1222no—
329#477—1222no—
330#465—1222no—
331#459—1222no—
332#447—1222no—
333C4×SD16—1222no—
334#431—611no—
335#418—1222no—
336#417—1222no—
337C4×M16—611no—
338C2×C4×C8—611no—
339#322—611no—
340#322C2×D6111yescharacteristic, complemented
341C82—611no—
342C82—611no—
343C61#1962511yescharacteristic, complemented, Sylow
344C2×#41—1221no—
345C22×Q8—1221no—
346C22×D4—1221no—
347C22×D4—1221no—
348C22×D4—1221no—
349C2×SD16—22no—
350C2×SD16—1222no—
351C2×SD16—1222no—
352C2×SD16—1222no—
353C2×SD16—1222no—
354C2×SD16—2448no—
355C2×D8—22no—
356C2×D8—1222no—
357C2×M16—612no—
358C22×C8—612no—
359C4⋊Q8—611no—
360C4⋊Q8—611no—
361C4⋊Q8—611no—
362C4⋊Q8—1221no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365C4⋊Q8C22×D6111yescharacteristic
366C4⋊Q8C22×D6111yescharacteristic
367#127—611no—
368#127—1221no—
369#127—1221no—
370#127—1221no—
371#127C22×D6111yescharacteristic
372#127C22×D6111yescharacteristic
373#125—1221no—
374#124—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377#121—1221no—
378#121—1221no—
379C4×Q8—1221no—
380C4×Q8—1221no—
381C4×D4—1221no—
382C4×D4—1221no—
383#117—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C42—611no—
387#107—1222no—
388#107—1222no—
389#106—1222no—
390C4⋊C8—612no—
391#103—22no—
392#103—1222no—
393#103—1222no—
394#103—1222no—
395#103—1222no—
396#102—22no—
397#102—22no—
398#102—22no—
399#102—1222no—
400#102—1222no—
401#102—1222no—
402#102—1222no—
403#102—1222no—
404#102—1222no—
405C22⋊C8—612no—
406#97—612no—
407C4×C8—612no—
408C4×C8—611no—
409C4×C8—611no—
410C4×C8—611no—
411C4×C8C22×D6111yescharacteristic
412C4×C8C22×D6111yescharacteristic
413C4×C8C22×D6111yescharacteristic
414C24—2441no—
415#41—2444no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—2441no—
422C2×Q8—2444no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—4882no—
433C2×D4—2444no—
434C2×D4—2441no—
435C2×D4—2441no—
436C2×D4—2441no—
437C2×D4—2444no—
438C2×D4—2441no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—2441no—
443C22×C4—2441no—
444SD16—44no—
445SD16—2444no—
446SD16—2444no—
447SD16—2444no—
448SD16—2444no—
449D8—82no—
450D8—4882no—
451M16—1224no—
452C2×C8—612no—
453C2×C8—612no—
454C2×C8—612no—
455C2×C8—1224no—
456C2×C8D4×D6112yes—
457C2×C8D4×D6112yes—
458C2×C8D4×D6112yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C42—611no—
477C42—611no—
478C42—611no—
479C42—2441no—
480C42C23×D6111yescharacteristic
481C23—41no—
482C23—41no—
483C23—611no—
484C23—4882no—
485C23—4882no—
486C23—4882no—
487C23—2441no—
488Q8—42no—
489Q8—42no—
490Q8—4882no—
491Q8—2442no—
492Q8—2442no—
493Q8—2442no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—4882no—
501D4—4882no—
502D4—4882no—
503D4—4882no—
504D4—2442no—
505D4—2442no—
506D4—2442no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—611no—
510C2×C4—614no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—4882no—
516C2×C4—4882no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C2×D4×D6111yescharacteristic
522C2×C4C2×D4×D6111yescharacteristic
523C8—22no—
524C8—22no—
525C8—22no—
526C8—614no—
527C8—1222no—
528C8—1222no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—612no—
536C22—1221no—
537C22—1221no—
538C22—9764no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22#127×D6111yescharacteristic
543C4—81no—
544C4—81no—
545C4—612no—
546C4—1221no—
547C4—1221no—
548C4—4881no—
549C4—4881no—
550C4—4881no—
551C4#135×D6112yes—
552C4AGL(1,ℤ/8)×D6112yes—
553C4#1684057212yes—
554C2—161no—
555C2—161no—
556C2—1222no—
557C2—9761no—
558C2#495×D6111yescharacteristic
559C2#2145626811yescharacteristic
560C2#2145629211yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1717C2
socleC22×C61C2×#127
prime radical#21456733C2
hypercenter#1717D61
hypocenterC61#19625
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1717

Complements

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

NH ≅ G/Ntype
1. #21456966C2semidirect
3. #21456377C2semidirect
4. #492×D61C2semidirect
5. #21456209C2semidirect
6. #21456190C2semidirect
7. #21445472C2semidirect
8. #21445470C2semidirect
9. #21445460C2semidirect
10. #21445422C2semidirect
11. #21445419C2semidirect
12. #21445410C2semidirect
13. #21445408C2semidirect
14. #21444098C2semidirect
15. C61×#1717C2semidirect
21. #16840167C22semidirect
22. #16840165C22semidirect
23. #16840158C22semidirect
28. #16840134C22semidirect
32. #16840114C22semidirect
33. #16840113C22semidirect
34. #16840113C22semidirect
35. #16840112C22semidirect
41. #16840108C22semidirect
53. #16839484C22semidirect
54. C8×#92418C22semidirect
55. C8×#92418C22semidirect
58. C61×#487C22semidirect
60. C61×#486C22semidirect
61. C61×#486C22semidirect
62. C61×#322C22semidirect
283. #1717D61semidirect
318. #487C2×D61semidirect
326. #486C2×D61semidirect
327. #486C2×D61semidirect
340. #322C2×D61semidirect
343. C61#19625semidirect

Maximal subgroups

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

subgroupindex
#214569662
#214567332
#214563772
#492×D612
#214562092
#214561902
#214454722
#214454702
#214454602
#214454222
#214454192
#214454102
#214454082
#214440982
C61×#17172
#1962561