Finite Group Databasebeta

← #29887168

Subgroups — #29887168

Order 15616.

2214 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454102C212yesmaximal, complemented
2#21453031C214yesmaximal, complemented
3#21452941C214yesmaximal, complemented
4#21452461C214yesmaximal, complemented
5#21452062C214yesmaximal, complemented
6#21451921C211yesmaximal, characteristic, complemented
7#21449558C214yesmaximal, complemented
8#21449353C214yesmaximal, complemented
9#21448519C212yesmaximal, complemented
10C2×#16839621C211yesmaximal, characteristic, complemented
11C61×#2646C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C2×#14519762C2214yescomplemented
14C2×#14519752C2212yes—
15C2×#14519752C2212yes—
16C2×#14519752C2214yes—
17C2×#14519751C2212yes—
18C2×#14519751C2214yes—
19C2×#14519747C2214yes—
20C2×#14519737C2214yes—
21C22×#11758196C2212yes—
22C22×#11758198C2212yes—
23#16839699C2214yescomplemented
24#16839697C2214yescomplemented
25#16839695C2214yescomplemented
26#16839691C2214yescomplemented
27#16839684C2214yes—
28#16839680C2214yes—
29#16839680C2214yescomplemented
30#16839679C2214yes—
31#16839672C2214yes—
32#16839672C2214yescomplemented
33#16839663C2214yes—
34#16839654C2214yes—
35#16839652C2214yescomplemented
36#16839651C2214yescomplemented
37#16839649C2214yescomplemented
38#16839645C2214yescomplemented
39#16839640C2214yes—
40#16839640C2214yes—
41#16839640C2214yes—
42#16839640C2214yescomplemented
43#16839639C2214yes—
44#16839639C2214yes—
45#16839633C2214yescomplemented
46#16839633C2214yescomplemented
47#16839621C2214yes—
48#16839621C2214yes—
49#16839619C2212yes—
50#16839619C2212yes—
51C2×#14519684C2211yescharacteristic
52C61×#400C2214yescomplemented
53C61×#398C2214yescomplemented
54C61×#385C2214yes—
55C61×#380C2212yes—
56C2×#95×C61C2211yescharacteristic
57C24×D61C2311yescharacteristic
58C22×D244C2314yes—
59C22×D244C2314yes—
60C22×D244C2312yes—
61C22×D244C2314yes—
62C22×C22⋊D61C2314yes—
63C23×Dic61C2311yescharacteristic
64C23×Dic61C2312yes—
65C22×C4×D61—24no—
66C22×C4×D61—44no—
67C22×C4×D61C2312yes—
68C22×C4×D61C2312yes—
69C22×C4×D61C2314yes—
70#14519762—48no—
71#14519752—44no—
72#14519752—44no—
73#14519752—48no—
74#14519751—28no—
75#14519751—48no—
76#14519747—48no—
77#14519737—48no—
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2314yes—
80C2×#11758196—44no—
81C2×#11758196C2314yes—
82C2×#11758196C2314yes—
83C2×#11758196C2314yes—
84C2×#11758198—44no—
85C2×#11758198C2314yes—
86C2×#11758198C2314yes—
87C2×#11758198C2314yes—
88C2×#11758198C2314yes—
89C2×#11758198C2314yes—
90C2×#11758198C2314yes—
91C2×#11758198C2314yes—
92C2×#11758198C2314yes—
93C2×#11758198C2314yes—
94C2×#11758198C2314yes—
95C2×#11758198C2314yes—
96C2×#11758198C2314yes—
97C2×C4×Dic61C2314yes—
98#14519684C2314yes—
99#14519684C2314yes—
100#14519684C2314yes—
101#14519684C2314yes—
102#14519684C2314yes—
103C23×C4×C61C2312yes—
104C23×C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2314yes—
106C2×C4⋊C4×C61C2314yes—
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61C2314yes—
110#95×C61C2314yes—
111#95×C61C2314yes—
112#95×C61C2314yes—
113#95×C61C2314yes—
114C24×C61C2411yescharacteristic
115C23×D61—22no—
116C23×D61—22no—
117C23×D61—22no—
118C23×D61—24no—
119C23×D61—42no—
120C23×D61C2414yes—
121C23×D61C2411yescharacteristic
122C23×D61C2411yescharacteristic
123C22×C4×C61—24no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61#4114yes—
128C22×C4×C61C2414yes—
129C22×C4×C61C2414yes—
130C22×C4×C61C2414yes—
131C22×C4×C61C2412yes—
132C2×C22⋊D61—28no—
133C2×C22⋊D61—28no—
134C2×C22⋊D61—28no—
135C22×Dic61—22no—
136C22×Dic61—24no—
137C22×Dic61—24no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61C2414yes—
141C22×Dic61C2414yes—
142C22×Dic61C2412yes—
143C2×D244—28no—
144C2×D244—28no—
145C2×D244—24no—
146C2×D244—44no—
147C2×D244—44no—
148C2×D244—44no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244—44no—
152C2×D244—44no—
153C2×D244—44no—
154C2×D244—44no—
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—416no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61—44no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61—44no—
168C4⋊C4×C61—44no—
169C22⋊C4×C61—28no—
170#11758198—28no—
171#11758198—28no—
172#11758198—28no—
173#11758198—28no—
174#11758198—44no—
175#11758198—44no—
176#11758198—44no—
177#11758198—44no—
178#11758198—44no—
179#11758198—44no—
180#11758198—44no—
181#11758198—44no—
182C4⋊Dic61—24no—
183C4⋊Dic61—44no—
184#11758196—28no—
185#11758196—44no—
186#11758196—44no—
187C4×Dic61—44no—
188C23×C61—22no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61C2×#4112yes—
192C23×C61H(5,𝔽2)14yes—
193C23×C61C2511yescharacteristic
194C22×D61—28no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—21no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—24no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—48no—
211C22×D61—48no—
212C22×D61—44no—
213C22×D61—44no—
214C22×D61H(5,𝔽2)12yes—
215C22×D612−1+412yes—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—44no—
228C2×C4×C61—84no—
229C2×C4×C61C2×#4114yes—
230C22⋊D61—416no—
231C2×Dic61—28no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—84no—
241C2×Dic61H(5,𝔽2)14yes—
242D244—84no—
243D244—88no—
244D244—88no—
245D244—88no—
246C4×D61—48no—
247C4×D61—48no—
248C4×D61—88no—
249#36727—611nomaximal, Sylow
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—24no—
253C22×C61—24no—
254C22×C61#54014yes—
255C22×C61C22×#4111yescharacteristic
256C22×C61C2×H(5,𝔽2)12yes—
257C22×C61C2×H(5,𝔽2)11yescharacteristic
258C22×C61C2×H(5,𝔽2)12yes—
259C22×C61C2×2−1+411yescharacteristic
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—24no—
263C2×D61—24no—
264C2×D61—416no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—44no—
274C2×D61—44no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—44no—
278C4×C61—24no—
279C4×C61—44no—
280C4×C61—44no—
281C4×C61—44no—
282Dic61—24no—
283Dic61—44no—
284Dic61—44no—
285#2847—612no—
286#2748—614no—
287#2741—614no—
288#2703—614no—
289#2671—614no—
290#2656—611no—
291#2646D6111yescharacteristic, complemented
292#2443—614no—
293#2424—614no—
294#2325—612no—
295C2×#380—611no—
296C2×C61—24no—
297C2×C61C2×#54011yescharacteristic
298C2×C61#349011yescharacteristic
299C2×C61#349211yescharacteristic
300C2×C61#353612yes—
301C2×C61#353712yes—
302D61—42no—
303D61—42no—
304D61—84no—
305C24×C4—611no—
306C2×#123—614no—
307C2×#121—612no—
308C2×#121—614no—
309C2×#121—612no—
310C2×#121—612no—
311C2×#121—614no—
312C2×#120—614no—
313C2×C4×D4—614no—
314C22×C4⋊C4—612no—
315C22×C22⋊C4—612no—
316#400C2×D6114yescomplemented
317#399—614no—
318#399—614no—
319#398—614no—
320#398C2×D6114yescomplemented
321#397—614no—
322#396—614no—
323#394—614no—
324#394—614no—
325#394—614no—
326#392—614no—
327#392—614no—
328#390—614no—
329#388—614no—
330#388—614no—
331#388—614no—
332#387—614no—
333#387—614no—
334#386—614no—
335#386—614no—
336#386—614no—
337#386—614no—
338#386—614no—
339#386—614no—
340#385—614no—
341#385—614no—
342#385C2×D6114yes—
343#380—612no—
344#380—612no—
345#380—614no—
346#380—614no—
347#380C2×D6112yes—
348C2×#95—611no—
349C2×#95C2×D6111yescharacteristic
350C61#3672711yescharacteristic, complemented, Sylow
351C25—611no—
352C22×D4—614no—
353C22×D4—614no—
354C22×D4—614no—
355C22×D4—612no—
356C22×D4—614no—
357C23×C4—611no—
358C23×C4—612no—
359C23×C4—612no—
360C23×C4—612no—
361C23×C4—614no—
362C23×C4—1224no—
363C23×C4—2444no—
364C23×C4C22×D6112yes—
365C23×C4C22×D6111yescharacteristic
366#123—2448no—
367#121—1228no—
368#121—2444no—
369#121—2444no—
370#121—2448no—
371#121—2448no—
372#120—2448no—
373C4×D4—2448no—
374C2×C4⋊C4—614no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—614no—
377C2×C4⋊C4—614no—
378C2×C4⋊C4—614no—
379C2×C4⋊C4—2444no—
380C2×C4⋊C4C22×D6112yes—
381C2×C4⋊C4C22×D6112yes—
382C2×C4⋊C4C22×D6114yes—
383C2×C4⋊C4C22×D6114yes—
384C2×C22⋊C4—614no—
385C2×C22⋊C4—614no—
386C2×C22⋊C4—614no—
387C2×C22⋊C4—614no—
388C2×C22⋊C4—614no—
389C2×C22⋊C4—614no—
390C2×C22⋊C4—614no—
391C2×C22⋊C4—614no—
392C2×C22⋊C4—614no—
393C2×C22⋊C4—614no—
394C2×C22⋊C4—614no—
395C2×C22⋊C4—614no—
396C2×C22⋊C4—2444no—
397C2×C22⋊C4C22×D6114yes—
398C2×C42—614no—
399#95—614no—
400#95—614no—
401#95—614no—
402#95—614no—
403#95—614no—
404#95C22×D6114yes—
405#95C22×D6114yes—
406#95C22×D6114yes—
407#95C22×D6114yes—
408C24—611no—
409C24—611no—
410C24—614no—
411C24—1222no—
412C24—1222no—
413C24—1222no—
414C24—1224no—
415C24—2442no—
416C24C23×D6111yescharacteristic
417C2×D4—1228no—
418C2×D4—1228no—
419C2×D4—1228no—
420C2×D4—1228no—
421C2×D4—1228no—
422C2×D4—1224no—
423C2×D4—2444no—
424C2×D4—2444no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4—2444no—
428C2×D4—2444no—
429C2×D4—2444no—
430C2×D4—2444no—
431C2×D4—2444no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—612no—
437C22×C4—614no—
438C22×C4—614no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—2444no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4—24416no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4#1175822614yes—
455C22×C4C23×D6114yes—
456C22×C4C23×D6114yes—
457C22×C4C23×D6114yes—
458C22×C4C23×D6112yes—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—1228no—
464C4⋊C4—1224no—
465C4⋊C4—2444no—
466C4⋊C4—2444no—
467C4⋊C4—2444no—
468C22⋊C4—28no—
469C22⋊C4—1228no—
470C22⋊C4—1228no—
471C22⋊C4—1228no—
472C22⋊C4—1228no—
473C22⋊C4—2444no—
474C22⋊C4—2444no—
475C22⋊C4—2444no—
476C22⋊C4—2444no—
477C22⋊C4—2444no—
478C22⋊C4—2444no—
479C22⋊C4—2444no—
480C22⋊C4—2444no—
481C42—2444no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—612no—
486C23—612no—
487C23—1228no—
488C23—1221no—
489C23—1221no—
490C23—1221no—
491C23—1222no—
492C23—1222no—
493C23—1222no—
494C23—1222no—
495C23—1221no—
496C23—1224no—
497C23—1224no—
498C23—1224no—
499C23—1224no—
500C23—1224no—
501C23—1224no—
502C23—1224no—
503C23—2448no—
504C23—2448no—
505C23—2444no—
506C23—2444no—
507C23C2×#1175822612yes—
508C23#1451982314yes—
509C23C24×D6111yescharacteristic
510D4—4884no—
511D4—4888no—
512D4—4888no—
513D4—4888no—
514D4—24416no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—44no—
527C2×C4—84no—
528C2×C4—614no—
529C2×C4—1228no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—4888no—
539C2×C4—4884no—
540C2×C4—2448no—
541C2×C4—2448no—
542C2×C4C2×#1175822614yes—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—1222no—
548C22—1222no—
549C22—1224no—
550C22—1224no—
551C22—24416no—
552C22—2444no—
553C22—2444no—
554C22—2444no—
555C22—2444no—
556C22—2444no—
557C22—2444no—
558C22—2444no—
559C22—2441no—
560C22—2441no—
561C22—2442no—
562C22—2442no—
563C22—2441no—
564C22—2441no—
565C22#1684036814yes—
566C22C22×#1175822611yescharacteristic
567C22H(5,𝔽2)×D6112yes—
568C22H(5,𝔽2)×D6111yescharacteristic
569C22C2×#1451982312yes—
570C222−1+4×D6111yescharacteristic
571C4—24no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—1224no—
576C4—2444no—
577C4—2444no—
578C2—24no—
579C2—4884no—
580C2—2442no—
581C2—2442no—
582C2C2×#1684036811yescharacteristic
583C2#2145950911yescharacteristic
584C2#2145952911yescharacteristic
585C2#2146004312yes—
586C2#2146005112yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2646C2
hypercenter#2646D61
hypocenterC61#36727
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2646

Complements

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

NH ≅ G/Ntype
1. #21454102C2semidirect
2. #21453031C2semidirect
3. #21452941C2semidirect
4. #21452461C2semidirect
5. #21452062C2semidirect
6. #21451921C2semidirect
7. #21449558C2semidirect
8. #21449353C2semidirect
9. #21448519C2semidirect
10. C2×#16839621C2semidirect
11. C61×#2646C2semidirect
13. C2×#14519762C22semidirect
23. #16839699C22semidirect
24. #16839697C22semidirect
25. #16839695C22semidirect
26. #16839691C22semidirect
29. #16839680C22semidirect
32. #16839672C22semidirect
35. #16839652C22semidirect
36. #16839651C22semidirect
37. #16839649C22semidirect
38. #16839645C22semidirect
42. #16839640C22semidirect
45. #16839633C22semidirect
46. #16839633C22semidirect
52. C61×#400C22semidirect
53. C61×#398C22semidirect
291. #2646D61semidirect
316. #400C2×D61semidirect
320. #398C2×D61semidirect
350. C61#36727semidirect

Maximal subgroups

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

subgroupindex
#214541022
#214530312
#214529412
#214524612
#214520622
#214519212
#214495582
#214493532
#214485192
C2×#168396212
C61×#26462
#3672761