Finite Group Databasebeta

← #29853204

Subgroups — #29853204

Order 15616.

2042 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454129C211yesmaximal, characteristic, complemented
2#21454127C212yesmaximal, complemented
3#21451791C214yesmaximal, complemented
4#21451042C214yesmaximal, complemented
5#21451041C214yesmaximal, complemented
6#21449546C212yesmaximal, complemented
7#21449545C212yesmaximal, complemented
8#21449541C212yesmaximal, complemented
9#21449539C212yesmaximal, complemented
10#21448430C214yesmaximal
11#380×D61C212yesmaximal
12C2×C4×#11758198C211yesmaximal, characteristic
13C61×#2851C211yesmaximal, characteristic, complemented
14C23×C4×D61C2211yescharacteristic
15C2×#14519769C2211yescharacteristic, complemented
16C2×#14519768C2212yescomplemented
17C2×C4⋊C4×D61C2212yes—
18C2×C4⋊C4×D61C2212yes—
19C22×#11758198C2212yes—
20C2×C22⋊C4×D61C2214yes—
21C22×C4×Dic61C2211yescharacteristic
22#16839703C2214yescomplemented
23#16839703C2214yescomplemented
24#16839702C2214yescomplemented
25#16839700C2214yescomplemented
26#16839696C2214yescomplemented
27#16839695C2214yescomplemented
28#16839693C2214yescomplemented
29#16839692C2214yescomplemented
30#16839661C2212yescomplemented
31#16839660C2212yescomplemented
32#16839658C2212yescomplemented
33#16839649C2214yescomplemented
34#16839646C2214yescomplemented
35#16839645C2214yescomplemented
36#16839637C2214yes—
37#16839636C2214yes—
38#16839636C2214yes—
39#16839635C2214yes—
40#16839634C2214yes—
41#16839633C2214yes—
42#16839633C2214yes—
43#16839628C2211yescharacteristic, complemented
44#16839627C2212yescomplemented
45#16839622C2212yes—
46#16839621C2214yes—
47#16839621C2214yes—
48#16839621C2214yes—
49#16839620C2214yes—
50#16839619C2214yes—
51#16839619C2214yes—
52#16839619C2212yes—
53#16839619C2212yes—
54C4×#11758198C2212yes—
55C4×#11758198C2212yes—
56C4×#11758198C2212yes—
57C4×#11758198C2212yes—
58C2×#14519684C2212yes—
59#95×D61C2214yes—
60C2×#125×C61C2211yescharacteristic, complemented
61C22×C42×C61C2211yescharacteristic
62C61×#389C2212yescomplemented
63C61×#387C2214yescomplemented
64C61×#385C2214yes—
65C61×#382C2211yescharacteristic, complemented
66C61×#380C2212yes—
67C24×D61C2311yescharacteristic
68C23×Dic61C2312yes—
69C23×Dic61C2311yescharacteristic
70C22×C4×D61—42no—
71C22×C4×D61—42no—
72C22×C4×D61Q818yes—
73C22×C4×D61C2312yes—
74C22×C4×D61C2314yes—
75C22×C4×D61C2314yes—
76C22×C4×D61C2312yes—
77#14519769—42no—
78#14519768—44no—
79C2×C4⋊Dic61C2312yes—
80C2×C4⋊Dic61C2312yes—
81C2×C4⋊Dic61C2314yes—
82C2×C4⋊Dic61C2314yes—
83C2×#11758196C2314yes—
84C2×#11758196C2314yes—
85C2×#11758196C2314yes—
86C2×#11758196C2314yes—
87C2×#11758196C2314yes—
88C2×#11758196C2314yes—
89C4⋊C4×D61—44no—
90C4⋊C4×D61—44no—
91C2×C22⋊Dic61C2314yes—
92C2×#11758198—28no—
93C2×#11758198C2314yes—
94C2×#11758198C2314yes—
95C2×#11758198C2312yes—
96C2×#11758198C2312yes—
97C2×#11758198C2312yes—
98C2×#11758198C2312yes—
99C22⋊C4×D61—216no—
100C2×C4×Dic61—42no—
101C2×C4×Dic61C2312yes—
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104#14519684C2312yes—
105#14519684C2314yes—
106#14519684C2314yes—
107#14519684C2312yes—
108#14519684C2314yes—
109#14519684C2314yes—
110#14519684C2312yes—
111#14519684C2314yes—
112#14519684C2312yes—
113#14519684C2314yes—
114#14519684C2314yes—
115#14519684C2314yes—
116#14519684C2314yes—
117C23×C4×C61C2311yescharacteristic
118C23×C4×C61C2312yes—
119#125×C61—42no—
120C2×C4⋊C4×C61C2312yes—
121C2×C4⋊C4×C61C2312yes—
122C2×C4⋊C4×C61C2314yes—
123C2×C4⋊C4×C61C2314yes—
124C2×C22⋊C4×C61C2314yes—
125C2×C42×C61—42no—
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2312yes—
129#95×C61C2314yes—
130#95×C61C2314yes—
131#95×C61C2314yes—
132C24×C61C2411yescharacteristic
133C23×D61—41no—
134C23×D61—41no—
135C23×D61—42no—
136C23×D61C2×Q814yes—
137C23×D61#4118yes—
138C23×D61C2412yes—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—24no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61C2×Q814yes—
145C22×C4×C61C2414yes—
146C22×C4×C61C2414yes—
147C22×C4×C61C2412yes—
148C22×C4×C61C2412yes—
149C22×C4×C61C2411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—24no—
152C22×Dic61—41no—
153C22×Dic61—41no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61C2×Q814yes—
157C22×Dic61C2414yes—
158C22×Dic61C2414yes—
159C22×Dic61C2412yes—
160C22×Dic61C2412yes—
161C22×Dic61C2412yes—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—28no—
165C2×C4×D61—28no—
166C2×C4×D61—416no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61—44no—
173C2×C4×D61C2×Q818yes—
174C2×C4×D61C2×Q818yes—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—44no—
178C4⋊C4×C61—44no—
179C22⋊C4×C61—28no—
180C42×C61—22no—
181C42×C61—22no—
182C42×C61—42no—
183C22⋊Dic61—28no—
184#11758198—28no—
185#11758198—28no—
186#11758198—24no—
187#11758198—24no—
188#11758198—24no—
189#11758198—24no—
190C4⋊Dic61—24no—
191C4⋊Dic61—24no—
192C4⋊Dic61—44no—
193C4⋊Dic61—44no—
194#11758196—44no—
195#11758196—44no—
196#11758196—44no—
197#11758196—44no—
198#11758196—44no—
199#11758196—44no—
200C4×Dic61—24no—
201C4×Dic61—41no—
202C4×Dic61—41no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61C22×Q812yes—
207C23×C61C2×#4114yes—
208C23×C61C2511yescharacteristic
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—44no—
214C22×D61—44no—
215C22×D61—48no—
216C22×D61—48no—
217C22×D61#125116yescomplemented
218C22×D61C22×Q814yes—
219C22×D61C2×#4114yes—
220C22×D61C2×#4114yes—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—28no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—48no—
237C2×C4×C61C22×Q812yes—
238C2×C4×C61C22×Q812yes—
239C2×C4×C61H(5,𝔽2)12yes—
240C2×C4×C61H(5,𝔽2)12yes—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—22no—
253C2×Dic61—24no—
254C2×Dic61—22no—
255C2×Dic61—44no—
256C2×Dic61—48no—
257C2×Dic61C22×Q814yes—
258C4×D61—216no—
259C4×D61—48no—
260C4×D61—88no—
261C4×D61—88no—
262#35020—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61C2×#12514yes—
268C22×C61C23×Q811yescharacteristic
269C22×C61C22×#4112yes—
270C22×C61C2×H(5,𝔽2)11yescharacteristic
271C22×C61C2×H(5,𝔽2)11yescharacteristic
272C22×C61#58512yes—
273C2×D61—24no—
274C2×D61—24no—
275C2×D61—24no—
276C2×D61—24no—
277C2×D61—24no—
278C2×D61—24no—
279C2×D61—416no—
280C2×D61C2×#12518yes—
281C4×C61—22no—
282C4×C61—44no—
283C4×C61—44no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61#54314yes—
287Dic61—24no—
288Dic61—44no—
289Dic61—44no—
290Dic61—42no—
291Dic61—42no—
292#2851—611no—
293#2851—612no—
294#2851D6111yescharacteristic, complemented
295#2646—614no—
296#2569—614no—
297#2569—614no—
298#2442—612no—
299#2442—612no—
300#2442—612no—
301#2442—612no—
302#2315—614no—
303C2×#380—612no—
304C2×C4×C22⋊C4—611no—
305C2×C61—24no—
306C2×C61C22×#12511yescharacteristic
307C2×C61C2×#54311yescharacteristic
308C2×C61#345811yescharacteristic
309C2×C61#350512yes—
310C2×C61#350512yes—
311D61—28no—
312C24×C4—611no—
313C2×#125—612no—
314C2×#125—611no—
315C2×#125C2×D6111yescharacteristic, complemented
316C22×C4⋊C4—612no—
317C22×C4⋊C4—612no—
318C22×C22⋊C4—612no—
319C22×C22⋊C4—614no—
320C22×C42—611no—
321C22×C42C2×D6111yescharacteristic
322#400—614no—
323#400—614no—
324#400—614no—
325#400—614no—
326#398—614no—
327#398—614no—
328#398—614no—
329#398—614no—
330#389—612no—
331#389—612no—
332#389—612no—
333#389C2×D6112yescomplemented
334#387—614no—
335#387—614no—
336#387—614no—
337#387C2×D6114yescomplemented
338#385—614no—
339#385—614no—
340#385—614no—
341#385—614no—
342#385—614no—
343#385—614no—
344#385—614no—
345#385C2×D6114yes—
346#382—612no—
347#382—611no—
348#382C2×D6111yescharacteristic, complemented
349#380—612no—
350#380—614no—
351#380—614no—
352#380—612no—
353#380—612no—
354#380—614no—
355#380—614no—
356#380—614no—
357#380—614no—
358#380C2×D6112yes—
359C4×C22⋊C4—612no—
360C4×C22⋊C4—612no—
361C4×C22⋊C4—612no—
362C4×C22⋊C4—612no—
363C2×#95—612no—
364C2×#95—614no—
365C61#3502011yescharacteristic, complemented, Sylow
366C25—611no—
367C23×C4—611no—
368C23×C4—612no—
369C23×C4—618no—
370C23×C4—612no—
371C23×C4—612no—
372C23×C4—614no—
373C23×C4—614no—
374C23×C4—2442no—
375C23×C4—2442no—
376C23×C4C22×D6111yescharacteristic
377C23×C4C22×D6112yes—
378#125—42no—
379#125—2442no—
380#125—2444no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—614no—
384C2×C4⋊C4—614no—
385C2×C4⋊C4—614no—
386C2×C4⋊C4—614no—
387C2×C4⋊C4—614no—
388C2×C4⋊C4—614no—
389C2×C4⋊C4—614no—
390C2×C4⋊C4—614no—
391C2×C4⋊C4—2444no—
392C2×C4⋊C4—2444no—
393C2×C4⋊C4C22×D6112yes—
394C2×C4⋊C4C22×D6112yes—
395C2×C4⋊C4C22×D6114yes—
396C2×C4⋊C4C22×D6114yes—
397C2×C22⋊C4—614no—
398C2×C22⋊C4—612no—
399C2×C22⋊C4—612no—
400C2×C22⋊C4—612no—
401C2×C22⋊C4—612no—
402C2×C22⋊C4—614no—
403C2×C22⋊C4—614no—
404C2×C22⋊C4—12216no—
405C2×C22⋊C4—1228no—
406C2×C22⋊C4C22×D6114yes—
407C2×C42—42no—
408C2×C42—612no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—2442no—
412C2×C42C22×D6111yescharacteristic
413C2×C42C22×D6111yescharacteristic
414C2×C42C22×D6112yes—
415#95—612no—
416#95—612no—
417#95—612no—
418#95—612no—
419#95—614no—
420#95—614no—
421#95—614no—
422#95—614no—
423#95—614no—
424#95—614no—
425#95—614no—
426#95—614no—
427#95—614no—
428#95C22×D6114yes—
429#95C22×D6114yes—
430#95C22×D6114yes—
431C24—612no—
432C24—614no—
433C24—618no—
434C24—2441no—
435C24—2441no—
436C24—2442no—
437C24C23×D6111yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—612no—
444C22×C4—614no—
445C22×C4—618no—
446C22×C4—618no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—614no—
450C22×C4—614no—
451C22×C4—1228no—
452C22×C4—1228no—
453C22×C4—1224no—
454C22×C4—1224no—
455C22×C4—1224no—
456C22×C4—24416no—
457C22×C4—2444no—
458C22×C4—2444no—
459C22×C4—2444no—
460C22×C4—2444no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4—2442no—
467C22×C4Q8×D6114yes—
468C22×C4C23×D6112yes—
469C22×C4C23×D6112yes—
470C22×C4C23×D6114yes—
471C22×C4C23×D6114yes—
472C22×C4C23×D6111yescharacteristic
473C22×C4C23×D6111yescharacteristic
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4—2444no—
481C4⋊C4—2444no—
482C4⋊C4—2444no—
483C4⋊C4—2444no—
484C4⋊C4—2444no—
485C4⋊C4—2444no—
486C4⋊C4—2444no—
487C4⋊C4—2444no—
488C22⋊C4—28no—
489C22⋊C4—1228no—
490C22⋊C4—1228no—
491C22⋊C4—1224no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C22⋊C4—1228no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—1224no—
500C42—2441no—
501C42—2441no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—614no—
506C23—614no—
507C23—614no—
508C23—6116no—
509C23—1222no—
510C23—1222no—
511C23—1222no—
512C23—1222no—
513C23—2444no—
514C23—2444no—
515C23—2448no—
516C23—2448no—
517C23C2×Q8×D6112yes—
518C23#41×D6114yes—
519C23C24×D6111yescharacteristic
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—614no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—12216no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1224no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—4888no—
553C2×C4—4888no—
554C2×C4—2448no—
555C2×C4—2444no—
556C2×C4—2448no—
557C2×C4C2×Q8×D6112yes—
558C2×C4C2×Q8×D6112yes—
559C2×C4#1451982212yes—
560C2×C4#1451982312yes—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—618no—
566C22—1224no—
567C22—1224no—
568C22—1224no—
569C22—1224no—
570C22—1224no—
571C22—1224no—
572C22—24416no—
573C22#125×D6114yes—
574C22C22×Q8×D6111yescharacteristic
575C22C2×#41×D6112yes—
576C22C2×#1451982211yescharacteristic
577C22C2×#1451982311yescharacteristic
578C22#1684059812yes—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—1224no—
585C4—2444no—
586C4—2444no—
587C4—2442no—
588C4—2442no—
589C4#1684038714yes—
590C2—24no—
591C2—1228no—
592C2C2×#125×D6111yescharacteristic
593C2C2×#1684038711yescharacteristic
594C2#2145922211yescharacteristic
595C2#2145964712yes—
596C2#2145965412yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2851C2
prime radicalC24×D61C23
hypercenter#2851D61
hypocenterC61#35020
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. #21454129C2semidirect
2. #21454127C2semidirect
3. #21451791C2semidirect
4. #21451042C2semidirect
5. #21451041C2semidirect
6. #21449546C2semidirect
7. #21449545C2semidirect
8. #21449541C2semidirect
9. #21449539C2semidirect
13. C61×#2851C2semidirect
15. C2×#14519769C22semidirect
16. C2×#14519768C22semidirect
22. #16839703C22semidirect
23. #16839703C22semidirect
24. #16839702C22semidirect
25. #16839700C22semidirect
26. #16839696C22semidirect
27. #16839695C22semidirect
28. #16839693C22semidirect
29. #16839692C22semidirect
30. #16839661C22semidirect
31. #16839660C22semidirect
32. #16839658C22semidirect
33. #16839649C22semidirect
34. #16839646C22semidirect
35. #16839645C22semidirect
43. #16839628C22semidirect
44. #16839627C22semidirect
60. C2×#125×C61C22semidirect
62. C61×#389C22semidirect
63. C61×#387C22semidirect
65. C61×#382C22semidirect
217. C22×D61#125semidirect
294. #2851D61semidirect
315. C2×#125C2×D61semidirect
333. #389C2×D61semidirect
337. #387C2×D61semidirect
348. #382C2×D61semidirect
365. C61#35020semidirect

Maximal subgroups

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

subgroupindex
#214541292
#214541272
#214517912
#214510422
#214510412
#214495462
#214495452
#214495412
#214495392
#214484302
#380×D612
C2×C4×#117581982
C61×#28512
#3502061