Finite Group Databasebeta

← #29886098

Subgroups — #29886098

Order 15616.

2130 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#21454200C213yesmaximal, complemented
2#21454199C211yesmaximal, characteristic, complemented
3#21454185C213yesmaximal, complemented
4#21454184C211yesmaximal, characteristic, complemented
5#21454136C213yesmaximal, complemented
6#21454134C211yesmaximal, characteristic, complemented
7#21454132C213yesmaximal, complemented
8#21448373C213yesmaximal, complemented
9#126×Dic61C211yesmaximal, characteristic, complemented
10#21448369C213yesmaximal, complemented
11#21448367C211yesmaximal, characteristic, complemented
12#21448304C213yesmaximal
13#21448298C213yesmaximal, complemented
14C2×C42×Dic61C211yesmaximal, characteristic, complemented
15C61×#2852C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C2×#14519772C2213yes—
18C2×#14519771C2211yescharacteristic
19C2×#14519770C2213yes—
20C2×#14519733C2213yes—
21C22×C4×Dic61C2213yes—
22C22×C4×Dic61C2213yes—
23#16839666C2213yes—
24#16839666C2213yes—
25#16839662C2213yes—
26#16839662C2213yes—
27#16839659C2213yes—
28#16839659C2213yes—
29#16839657C2213yes—
30#16839657C2213yes—
31#16839656C2213yes—
32#16839655C2213yes—
33#16839654C2213yes—
34#16839653C2213yes—
35#16839652C2213yes—
36#16839651C2213yes—
37#16839650C2213yes—
38#16839649C2213yes—
39#16839648C2213yes—
40#16839647C2213yes—
41#16839646C2213yes—
42#16839645C2213yes—
43#16839644C2213yes—
44#16839643C2213yes—
45#16839642C2213yes—
46#16839641C2213yes—
47#16839639C2213yes—
48#16839637C2213yes—
49#16839635C2213yes—
50#16839634C2213yes—
51#16839630C2211yescharacteristic
52#16839629C2213yes—
53#16839629C2213yes—
54C4⋊C4×Dic61C2213yes—
55C4⋊C4×Dic61C2213yes—
56#16839613C2213yes—
57#16839613C2213yes—
58C22⋊C4×Dic61C2213yes—
59C22⋊C4×Dic61C2213yes—
60#16839609C2213yes—
61#16839609C2213yes—
62C42×Dic61C2211yescharacteristic
63C42×Dic61C2213yes—
64C42×Dic61C2211yescharacteristic
65C42×Dic61C2213yes—
66C2×#126×C61C2211yescharacteristic
67C22×C42×C61C2211yescharacteristic
68C61×#388C2213yes—
69C61×#387C2213yes—
70C61×#386C2213yes—
71C61×#385C2213yes—
72C61×#383C2211yescharacteristic
73C2×#11758224—212no—
74C22×Dic122C2311yescharacteristic
75C22×C22⋊D61C2311yescharacteristic
76C22×C22⋊D61C2311yescharacteristic
77C23×Dic61C2313yes—
78C23×Dic61C2311yescharacteristic
79C23×Dic61C2311yescharacteristic
80C22×C4×D61C2311yescharacteristic
81#14519772—212no—
82#14519771—24no—
83#14519770—212no—
84#14519733—212no—
85C2×C4⋊Dic61C2313yes—
86C2×C4⋊Dic61C2313yes—
87C2×#11758196C2313yes—
88C2×#11758196C2313yes—
89C2×#11758196C2313yes—
90C2×#11758196C2313yes—
91C2×#11758196C2313yes—
92C2×#11758196C2313yes—
93C2×C22⋊Dic61C2313yes—
94C2×C22⋊Dic61C2313yes—
95C2×C22⋊Dic61C2313yes—
96C2×#11758198C2313yes—
97C2×#11758198C2313yes—
98C2×#11758198C2313yes—
99C2×#11758198C2313yes—
100C2×C4×Dic61—212no—
101C2×C4×Dic61—212no—
102C2×C4×Dic61C2313yes—
103C2×C4×Dic61C2313yes—
104C2×C4×Dic61C2313yes—
105C2×C4×Dic61C2313yes—
106C2×C4×Dic61C2313yes—
107C2×C4×Dic61C2313yes—
108C2×C4×Dic61C2313yes—
109C2×C4×Dic61C2313yes—
110C2×C4×Dic61C2313yes—
111C2×C4×Dic61C2313yes—
112#14519684C2313yes—
113#14519684C2313yes—
114#14519684C2313yes—
115#14519684C2313yes—
116#14519684C2313yes—
117#14519684C2313yes—
118#14519684C2313yes—
119#14519684C2313yes—
120#14519684C2313yes—
121#14519684C2313yes—
122#14519684C2313yes—
123#14519684C2313yes—
124C22×D4×C61C2311yescharacteristic
125C23×C4×C61C2313yes—
126#126×C61—24no—
127C2×C4⋊C4×C61C2313yes—
128C2×C4⋊C4×C61C2313yes—
129C2×C22⋊C4×C61C2313yes—
130C2×C22⋊C4×C61C2313yes—
131C2×C22⋊C4×C61C2313yes—
132C2×C42×C61—24no—
133C2×C42×C61C2313yes—
134C2×C42×C61C2311yescharacteristic
135#95×C61C2313yes—
136#95×C61C2313yes—
137#95×C61C2313yes—
138#95×C61C2313yes—
139C24×C61C2411yescharacteristic
140C24×C61C2411yescharacteristic
141C23×D61C2411yescharacteristic
142C2×D4×C61—26no—
143C22×C4×C61—26no—
144C22×C4×C61—26no—
145C22×C4×C61—26no—
146C22×C4×C61C2411yescharacteristic
147C22×C4×C61C2413yes—
148C22×C4×C61C2413yes—
149C22×C4×C61C2413yes—
150C22×C4×C61C2413yes—
151C2×C22⋊D61—26no—
152C2×C22⋊D61—26no—
153C22×Dic61—26no—
154C22×Dic61—26no—
155C22×Dic61—26no—
156C22×Dic61—26no—
157C22×Dic61#41112yes—
158C22×Dic61C2411yescharacteristic
159C22×Dic61C2413yes—
160C22×Dic61C2413yes—
161C22×Dic61C2413yes—
162C22×Dic61C2413yes—
163C22×Dic61C2411yescharacteristic
164C22×Dic61C2411yescharacteristic
165#11758224—416no—
166C2×C4×D61—26no—
167C2×Dic122—26no—
168C4⋊C4×C61—26no—
169C4⋊C4×C61—26no—
170C22⋊C4×C61—26no—
171C22⋊C4×C61—26no—
172C22⋊C4×C61—26no—
173C42×C61—26no—
174C42×C61#4114yescomplemented
175C22⋊Dic61—26no—
176C22⋊Dic61—26no—
177C22⋊Dic61—26no—
178#11758198—26no—
179#11758198—26no—
180#11758198—26no—
181#11758198—26no—
182C4⋊Dic61—26no—
183C4⋊Dic61—26no—
184#11758196—26no—
185#11758196—26no—
186#11758196—26no—
187#11758196—26no—
188#11758196—26no—
189#11758196—26no—
190C4×Dic61—26no—
191C4×Dic61—26no—
192C4×Dic61—26no—
193C4×Dic61—26no—
194C4×Dic61—26no—
195C4×Dic61—26no—
196C4×Dic61—26no—
197C4×Dic61—26no—
198C4×Dic61—26no—
199C4×Dic61#41112yescomplemented
200C23×C61—23no—
201C23×C61—21no—
202C23×C61—23no—
203C23×C61—21no—
204C23×C61—23no—
205C23×C61C2×#4116yes—
206C23×C61C2511yescharacteristic
207C22×D61—23no—
208C22×D61—21no—
209C22×D61—23no—
210D4×C61—44no—
211C2×C4×C61—224no—
212C2×C4×C61—23no—
213C2×C4×C61—23no—
214C2×C4×C61—23no—
215C2×C4×C61—23no—
216C2×C4×C61—23no—
217C2×C4×C61—23no—
218C2×C4×C61—23no—
219C2×C4×C61—23no—
220C2×C4×C61—23no—
221C2×C4×C61—23no—
222C2×C4×C61—23no—
223C2×C4×C61C2×#4116yes—
224C2×C4×C61C2×#4116yes—
225C22⋊D61—44no—
226C22⋊D61—44no—
227C2×Dic61—224no—
228C2×Dic61—23no—
229C2×Dic61—23no—
230C2×Dic61—23no—
231C2×Dic61—23no—
232C2×Dic61—23no—
233C2×Dic61—23no—
234C2×Dic61—23no—
235C2×Dic61—23no—
236C2×Dic61—23no—
237C2×Dic61—23no—
238C2×Dic61—23no—
239C2×Dic61—44no—
240C2×Dic61#126116yescomplemented
241C2×Dic61C2×#4116yes—
242C2×Dic61C2×#4116yes—
243C2×Dic61C2×#4116yes—
244C2×Dic61C2×#4116yes—
245C4×D61—44no—
246Dic122—44no—
247#36641—611nomaximal, Sylow
248C22×C61—26no—
249C22×C61—26no—
250C22×C61—41no—
251C22×C61—43no—
252C22×C61—43no—
253C22×C61C2×#12614yes—
254C22×C61C22×#4113yes—
255C22×C61C22×#4113yes—
256C22×C61C22×#4111yescharacteristic
257C2×D61—41no—
258C2×D61—43no—
259C2×D61—43no—
260C4×C61—26no—
261C4×C61—26no—
262C4×C61—41no—
263C4×C61—43no—
264C4×C61—43no—
265Dic61—26no—
266Dic61—26no—
267Dic61—41no—
268Dic61—43no—
269Dic61—43no—
270Dic61C2×#12614yes—
271Dic61C2×#12614yes—
272#2863—613no—
273#2863—611no—
274#2860—613no—
275#2860—611no—
276#2852—611no—
277#2852—613no—
278#2852—613no—
279#2852D6111yescharacteristic, complemented
280C4×#126—613no—
281C4×#126—613no—
282C4×#126—611no—
283C4×#126—611no—
284#2297—613no—
285#2296—613no—
286C2×C43—611no—
287C2×C61—24no—
288C2×C61—81no—
289C2×C61C22×#12611yescharacteristic
290C2×C61#344413yes—
291C2×C61#344413yes—
292D61—81no—
293C22×#41—611no—
294C2×#126—613no—
295C2×#126—613no—
296C2×#126—611no—
297C2×#126C2×D6111yescharacteristic
298C2×#117—613no—
299C22×C42—613no—
300C22×C42—613no—
301C22×C42C2×D6111yescharacteristic
302#391—613no—
303#391—613no—
304#390—613no—
305#390—613no—
306#389—613no—
307#389—613no—
308#389—613no—
309#389—613no—
310#388—613no—
311#388—613no—
312#388—613no—
313#388—613no—
314#388—613no—
315#388—613no—
316#388—613no—
317#388C2×D6113yes—
318#387—613no—
319#387—613no—
320#387—613no—
321#387—613no—
322#387—613no—
323#387—613no—
324#387—613no—
325#387C2×D6113yes—
326#386—613no—
327#386—613no—
328#386—613no—
329#386C2×D6113yes—
330#385—613no—
331#385—613no—
332#385—613no—
333#385C2×D6113yes—
334#383—611no—
335#383—613no—
336#383—613no—
337#383C2×D6111yescharacteristic
338C4×C4⋊C4—613no—
339C4×C4⋊C4—613no—
340C4×C4⋊C4—613no—
341C4×C4⋊C4—613no—
342C4×C22⋊C4—613no—
343C4×C22⋊C4—613no—
344C4×C22⋊C4—613no—
345C4×C22⋊C4—613no—
346C43—613no—
347C43—611no—
348C43—613no—
349C43—611no—
350C61#3664111yescharacteristic, complemented, Sylow
351C2×#41—12212no—
352C22×Q8—611no—
353C22×D4—611no—
354C22×D4—611no—
355C22×D4C22×D6111yescharacteristic
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—613no—
359C23×C4—611no—
360C23×C4C22×D6113yes—
361#126—24no—
362#126—12212no—
363#126—12212no—
364#126—1224no—
365#117—12212no—
366C2×C4⋊C4—613no—
367C2×C4⋊C4—613no—
368C2×C4⋊C4—613no—
369C2×C4⋊C4—613no—
370C2×C4⋊C4—613no—
371C2×C4⋊C4—613no—
372C2×C4⋊C4—613no—
373C2×C4⋊C4—613no—
374C2×C4⋊C4C22×D6113yes—
375C2×C4⋊C4C22×D6113yes—
376C2×C22⋊C4—613no—
377C2×C22⋊C4—613no—
378C2×C22⋊C4—613no—
379C2×C22⋊C4—613no—
380C2×C22⋊C4—613no—
381C2×C22⋊C4—613no—
382C2×C22⋊C4—613no—
383C2×C22⋊C4C22×D6113yes—
384C2×C22⋊C4C22×D6113yes—
385C2×C22⋊C4C22×D6113yes—
386C2×C42—24no—
387C2×C42—613no—
388C2×C42—613no—
389C2×C42—613no—
390C2×C42—613no—
391C2×C42—613no—
392C2×C42—613no—
393C2×C42—613no—
394C2×C42—613no—
395C2×C42—613no—
396C2×C42—613no—
397C2×C42—12212no—
398C2×C42—12212no—
399C2×C42C22×D6113yes—
400C2×C42C22×D6111yescharacteristic
401#95—613no—
402#95—613no—
403#95—613no—
404#95—613no—
405#95—613no—
406#95—613no—
407#95—613no—
408#95—613no—
409#95—613no—
410#95—613no—
411#95—613no—
412#95—613no—
413#95C22×D6113yes—
414#95C22×D6113yes—
415#95C22×D6113yes—
416#95C22×D6113yes—
417C24—611no—
418C24C23×D6111yescharacteristic
419C24C23×D6111yescharacteristic
420#41—24416no—
421C2×Q8—1226no—
422C2×D4—26no—
423C2×D4—1226no—
424C2×D4—1226no—
425C22×C4—26no—
426C22×C4—26no—
427C22×C4—26no—
428C22×C4—6112no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—613no—
432C22×C4—613no—
433C22×C4—611no—
434C22×C4—613no—
435C22×C4—613no—
436C22×C4—1226no—
437C22×C4—1226no—
438C22×C4—1226no—
439C22×C4—1226no—
440C22×C4—1226no—
441C22×C4C23×D6111yescharacteristic
442C22×C4C23×D6113yes—
443C22×C4C23×D6113yes—
444C22×C4C23×D6113yes—
445C22×C4C23×D6113yes—
446C4⋊C4—26no—
447C4⋊C4—26no—
448C4⋊C4—1226no—
449C4⋊C4—1226no—
450C4⋊C4—1226no—
451C4⋊C4—1226no—
452C4⋊C4—1226no—
453C4⋊C4—1226no—
454C4⋊C4—1226no—
455C4⋊C4—1226no—
456C22⋊C4—26no—
457C22⋊C4—26no—
458C22⋊C4—26no—
459C22⋊C4—1226no—
460C22⋊C4—1226no—
461C22⋊C4—1226no—
462C22⋊C4—1226no—
463C22⋊C4—1226no—
464C22⋊C4—1226no—
465C22⋊C4—1226no—
466C42—26no—
467C42—6112no—
468C42—1226no—
469C42—1226no—
470C42—1226no—
471C42—1226no—
472C42—1226no—
473C42—1226no—
474C42—1226no—
475C42—1226no—
476C42—1226no—
477C42#1175822414yescomplemented
478C23—23no—
479C23—21no—
480C23—23no—
481C23—23no—
482C23—21no—
483C23—1223no—
484C23—1223no—
485C23—1221no—
486C23#41×D6116yes—
487C23C24×D6111yescharacteristic
488Q8—2444no—
489D4—44no—
490D4—2444no—
491D4—2444no—
492C2×C4—224no—
493C2×C4—23no—
494C2×C4—23no—
495C2×C4—23no—
496C2×C4—23no—
497C2×C4—23no—
498C2×C4—23no—
499C2×C4—23no—
500C2×C4—23no—
501C2×C4—23no—
502C2×C4—23no—
503C2×C4—23no—
504C2×C4—6116no—
505C2×C4—616no—
506C2×C4—616no—
507C2×C4—616no—
508C2×C4—616no—
509C2×C4—12224no—
510C2×C4—1223no—
511C2×C4—1223no—
512C2×C4—1223no—
513C2×C4—1223no—
514C2×C4—1223no—
515C2×C4—1223no—
516C2×C4—1223no—
517C2×C4—1223no—
518C2×C4—1223no—
519C2×C4—1223no—
520C2×C4—1223no—
521C2×C4—2444no—
522C2×C4—2444no—
523C2×C4#41×D6116yes—
524C2×C4C2×#1175822416yes—
525C22—26no—
526C22—26no—
527C22—41no—
528C22—43no—
529C22—43no—
530C22—2441no—
531C22—2443no—
532C22—2443no—
533C22#126×D6114yes—
534C22C2×#41×D6113yes—
535C22C2×#41×D6113yes—
536C22C22×#1175822411yescharacteristic
537C4—26no—
538C4—26no—
539C4—41no—
540C4—43no—
541C4—43no—
542C4—614no—
543C4—614no—
544C4—1226no—
545C4—1226no—
546C4—2441no—
547C4—2443no—
548C4—2443no—
549C2—24no—
550C2—81no—
551C2—4881no—
552C2C2×#126×D6111yescharacteristic
553C2#2145911113yes—
554C2#2145911313yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2852C2
prime radicalC22×#11758224C22
hypercenter#2852D61
hypocenterC61#36641
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2852

Complements

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

NH ≅ G/Ntype
1. #21454200C2semidirect
2. #21454199C2semidirect
3. #21454185C2semidirect
4. #21454184C2semidirect
5. #21454136C2semidirect
6. #21454134C2semidirect
7. #21454132C2semidirect
8. #21448373C2semidirect
9. #126×Dic61C2semidirect
10. #21448369C2semidirect
11. #21448367C2semidirect
13. #21448298C2semidirect
14. C2×C42×Dic61C2semidirect
15. C61×#2852C2semidirect
174. C42×C61#41semidirect
199. C4×Dic61#41semidirect
240. C2×Dic61#126semidirect
279. #2852D61semidirect
350. C61#36641semidirect
477. C42#11758224semidirect

Maximal subgroups

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

subgroupindex
#214542002
#214541992
#214541852
#214541842
#214541362
#214541342
#214541322
#214483732
#126×Dic612
#214483692
#214483672
#214483042
#214482982
C2×C42×Dic612
C61×#28522
#3664161