Finite Group Databasebeta

← #29887178

Subgroups — #29887178

Order 15616.

1854 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454106C212yesmaximal
2#21453014C214yesmaximal, complemented
3#21452724C214yesmaximal, complemented
4#21452477C214yesmaximal, complemented
5#21452285C214yesmaximal, complemented
6#21451933C211yesmaximal, characteristic, complemented
7#21451789C211yesmaximal, characteristic, complemented
8#21450059C214yesmaximal
9#21449954C214yesmaximal, complemented
10#21448552C212yesmaximal, complemented
11C2×C61×#380C211yesmaximal, characteristic
12C2×#14519770C2214yes—
13C2×#14519767C2214yescomplemented
14C2×#14519756C2212yes—
15C2×#14519756C2212yes—
16C2×#14519756C2212yes—
17#16839704C2214yescomplemented
18#16839700C2214yescomplemented
19#16839696C2214yescomplemented
20#16839695C2214yescomplemented
21#16839694C2214yescomplemented
22#16839693C2214yescomplemented
23#16839692C2214yescomplemented
24#16839690C2214yes—
25#16839690C2214yes—
26#16839687C2214yes—
27#16839685C2214yes—
28#16839683C2214yescomplemented
29#16839666C2214yescomplemented
30#16839659C2214yescomplemented
31#16839657C2214yescomplemented
32#16839656C2214yes—
33#16839655C2214yes—
34#16839650C2214yes—
35#16839649C2214yescomplemented
36#16839648C2214yescomplemented
37#16839645C2214yescomplemented
38#16839643C2214yescomplemented
39#16839635C2214yes—
40#16839635C2214yes—
41#16839635C2214yes—
42#16839635C2214yes—
43#16839635C2214yes—
44#16839622C2212yes—
45#16839622C2212yes—
46#16839622C2212yes—
47#16839622C2214yes—
48#16839618C2214yes—
49C4×C22⋊Dic61C2214yes—
50C24×C4×C61C2211yescharacteristic
51C22×C4⋊C4×C61C2212yes—
52C22×C22⋊C4×C61C2212yes—
53C61×#380C2214yes—
54C61×#380C2214yes—
55C2×#95×C61C2211yescharacteristic
56C2×#95×C61C2211yescharacteristic
57C22×Dic122C2312yes—
58#14519770—48no—
59#14519767—48no—
60#14519756—28no—
61#14519756—44no—
62#14519756—44no—
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2314yes—
65C2×C4⋊Dic61C2314yes—
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2312yes—
68C2×#11758196C2314yes—
69C2×#11758196C2314yes—
70C2×#11758196C2314yes—
71C2×#11758196C2314yes—
72C2×#11758196C2314yes—
73C2×#11758196C2314yes—
74C2×#11758196C2314yes—
75C2×C22⋊Dic61C2314yes—
76C2×C22⋊Dic61C2314yes—
77C2×C22⋊Dic61C2314yes—
78C2×C22⋊Dic61C2314yes—
79C2×C22⋊Dic61C2314yes—
80C2×C22⋊Dic61C2314yes—
81C2×C4×Dic61C2314yes—
82C2×C4×Dic61C2314yes—
83#14519684C2314yes—
84#14519684C2314yes—
85#14519684C2314yes—
86#14519684C2314yes—
87#14519684C2314yes—
88#14519684C2314yes—
89#14519684C2314yes—
90#14519684C2314yes—
91#14519684C2314yes—
92#14519684C2314yes—
93#14519684C2314yes—
94C25×C61C2311yescharacteristic
95C23×C4×C61—24no—
96C23×C4×C61—44no—
97C23×C4×C61C2311yescharacteristic
98C23×C4×C61C2312yes—
99C23×C4×C61C2312yes—
100C23×C4×C61C2312yes—
101C23×C4×C61C2312yes—
102C23×C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61—44no—
104C2×C4⋊C4×C61C2314yes—
105C2×C4⋊C4×C61C2314yes—
106C2×C22⋊C4×C61—44no—
107C2×C22⋊C4×C61C2314yes—
108C2×C22⋊C4×C61C2314yes—
109#95×C61C2314yes—
110#95×C61C2314yes—
111C24×C61—22no—
112C24×C61—22no—
113C24×C61—22no—
114C24×C61—24no—
115C24×C61—42no—
116C24×C61C2411yescharacteristic
117C24×C61C2411yescharacteristic
118C24×C61C2411yescharacteristic
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61—24no—
122C22×C4×C61—24no—
123C22×C4×C61—24no—
124C22×C4×C61—24no—
125C22×C4×C61—24no—
126C22×C4×C61—24no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—416no—
135C22×C4×C61#4114yes—
136C22×C4×C61C2414yes—
137C22×C4×C61C2414yes—
138C22×C4×C61C2412yes—
139C22×C4×C61C2412yes—
140C22×Dic61C2414yes—
141C22×Dic61C2414yes—
142C22×Dic61C2414yes—
143C22×Dic61C2414yes—
144C2×Dic122—24no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147C4⋊C4×C61—28no—
148C4⋊C4×C61—44no—
149C22⋊C4×C61—28no—
150C22⋊C4×C61—44no—
151C22⋊Dic61—28no—
152C22⋊Dic61—28no—
153C22⋊Dic61—28no—
154C22⋊Dic61—28no—
155C22⋊Dic61—44no—
156C22⋊Dic61—44no—
157C4⋊Dic61—24no—
158C4⋊Dic61—42no—
159C4⋊Dic61—42no—
160C4⋊Dic61—44no—
161C4⋊Dic61—44no—
162#11758196—28no—
163#11758196—28no—
164#11758196—28no—
165#11758196—44no—
166#11758196—44no—
167#11758196—44no—
168#11758196—44no—
169C4×Dic61—44no—
170C4×Dic61—44no—
171C23×C61—22no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—28no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—48no—
184C23×C61—48no—
185C23×C61—44no—
186C23×C61—44no—
187C23×C61C2×#4112yes—
188C23×C61H(5,𝔽2)12yes—
189C23×C612−1+412yes—
190C23×C612−1+414yes—
191C23×C61C2511yescharacteristic
192C2×C4×C61—28no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—44no—
204C2×C4×C61—48no—
205C2×C4×C61—48no—
206C2×C4×C61—84no—
207C2×C4×C61—84no—
208C2×C4×C61C2×#4114yes—
209C2×C4×C61H(5,𝔽2)14yes—
210C2×Dic61—24no—
211C2×Dic61—24no—
212C2×Dic61—24no—
213C2×Dic61—24no—
214C2×Dic61—24no—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—24no—
218C2×Dic61—24no—
219C2×Dic61—24no—
220C2×Dic61—24no—
221C2×Dic61—24no—
222Dic122—84no—
223#36728—611nomaximal, Sylow
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—24no—
227C22×C61—24no—
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—24no—
231C22×C61—24no—
232C22×C61—416no—
233C22×C61—41no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61#54114yes—
240C22×C61C22×#4111yescharacteristic
241C22×C61C2×H(5,𝔽2)12yes—
242C22×C61C2×H(5,𝔽2)11yescharacteristic
243C22×C61C2×2−1+412yes—
244C22×C61C2×2−1+411yescharacteristic
245C4×C61—24no—
246C4×C61—24no—
247C4×C61—44no—
248C4×C61—44no—
249Dic61—44no—
250Dic61—44no—
251Dic61—44no—
252Dic61—44no—
253#2848—612no—
254#2747—614no—
255#2722—614no—
256#2704—614no—
257#2690—614no—
258#2659—611no—
259#2646—611no—
260#2480—614no—
261#2473—614no—
262#2329—612no—
263C2×#380D6111yescharacteristic
264C2×C61—24no—
265C2×C61—42no—
266C2×C61—42no—
267C2×C61C2×#54111yescharacteristic
268C2×C61#349011yescharacteristic
269C2×C61#349211yescharacteristic
270C2×C61#353612yes—
271C2×C61#353812yes—
272C24×C4C2×D6111yescharacteristic
273C2×#126—614no—
274C2×#125—614no—
275C2×C22⋊Q8—612no—
276C2×C22⋊Q8—612no—
277C2×C22⋊Q8—612no—
278C22×C4⋊C4C2×D6112yes—
279C22×C22⋊C4C2×D6112yes—
280#400—614no—
281#400—614no—
282#398—614no—
283#398—614no—
284#398—614no—
285#398—614no—
286#398—614no—
287#397—614no—
288#397—614no—
289#397—614no—
290#396—614no—
291#395—614no—
292#391—614no—
293#389—614no—
294#389—614no—
295#388—614no—
296#388—614no—
297#388—614no—
298#387—614no—
299#387—614no—
300#387—614no—
301#387—614no—
302#385—614no—
303#385—614no—
304#385—614no—
305#385—614no—
306#385—614no—
307#380—612no—
308#380—612no—
309#380—612no—
310#380—614no—
311#380C2×D6114yes—
312#380C2×D6114yes—
313#379—614no—
314C4×C22⋊C4—614no—
315C2×#95C2×D6111yescharacteristic
316C2×#95C2×D6111yescharacteristic
317C61#3672811yescharacteristic, complemented, Sylow
318C25C22×D6111yescharacteristic
319C22×Q8—612no—
320C23×C4—24no—
321C23×C4—44no—
322C23×C4C22×D6111yescharacteristic
323C23×C4C22×D6112yes—
324C23×C4C22×D6112yes—
325C23×C4C22×D6112yes—
326C23×C4C22×D6112yes—
327C23×C4C22×D6111yescharacteristic
328#126—2448no—
329#125—2448no—
330C22⋊Q8—1228no—
331C22⋊Q8—2444no—
332C22⋊Q8—2444no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—614no—
335C2×C4⋊C4—614no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—614no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—614no—
341C2×C4⋊C4—614no—
342C2×C4⋊C4—614no—
343C2×C4⋊C4—614no—
344C2×C4⋊C4—614no—
345C2×C4⋊C4—614no—
346C2×C4⋊C4C22×D6114yes—
347C2×C4⋊C4C22×D6114yes—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—614no—
350C2×C22⋊C4—614no—
351C2×C22⋊C4—614no—
352C2×C22⋊C4—614no—
353C2×C22⋊C4—614no—
354C2×C22⋊C4—614no—
355C2×C22⋊C4C22×D6114yes—
356C2×C22⋊C4C22×D6114yes—
357C2×C42—614no—
358C2×C42—614no—
359#95—614no—
360#95—614no—
361#95—614no—
362#95—614no—
363#95—614no—
364#95—614no—
365#95—614no—
366#95—614no—
367#95—614no—
368#95—614no—
369#95—614no—
370#95C22×D6114yes—
371#95C22×D6114yes—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—42no—
377C24C23×D6111yescharacteristic
378C24C23×D6111yescharacteristic
379C24C23×D6111yescharacteristic
380C2×Q8—1224no—
381C2×Q8—2442no—
382C2×Q8—2442no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—614no—
400C22×C4—614no—
401C22×C4—614no—
402C22×C4—614no—
403C22×C4#1175822214yes—
404C22×C4C23×D6114yes—
405C22×C4C23×D6114yes—
406C22×C4C23×D6112yes—
407C22×C4C23×D6112yes—
408C4⋊C4—28no—
409C4⋊C4—44no—
410C4⋊C4—1228no—
411C4⋊C4—1224no—
412C4⋊C4—1228no—
413C4⋊C4—1228no—
414C4⋊C4—2444no—
415C4⋊C4—2444no—
416C4⋊C4—2444no—
417C4⋊C4—2444no—
418C4⋊C4—2442no—
419C4⋊C4—2442no—
420C4⋊C4—2444no—
421C4⋊C4—2444no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—1228no—
425C22⋊C4—1228no—
426C22⋊C4—1228no—
427C22⋊C4—1228no—
428C22⋊C4—2444no—
429C22⋊C4—2444no—
430C42—2444no—
431C42—2444no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—28no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—48no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23C2×#1175822212yes—
449C23#1451982212yes—
450C23#1451982414yes—
451C23#1451982512yes—
452C23C24×D6111yescharacteristic
453Q8—4884no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—1224no—
471C2×C4—1224no—
472C2×C4—1224no—
473C2×C4—1224no—
474C2×C4—1224no—
475C2×C4—1224no—
476C2×C4—1224no—
477C2×C4—1224no—
478C2×C4—1224no—
479C2×C4—1224no—
480C2×C4—1224no—
481C2×C4—1224no—
482C2×C4C2×#1175822214yes—
483C2×C4#1451982214yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—416no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22#1684037414yes—
500C22C22×#1175822211yescharacteristic
501C22C2×#1451982212yes—
502C22C2×#1451982211yescharacteristic
503C22C2×#1451982412yes—
504C22C2×#1451982511yescharacteristic
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—2444no—
510C4—2444no—
511C4—2444no—
512C4—2444no—
513C2—24no—
514C2—42no—
515C2—42no—
516C2C2×#1684037411yescharacteristic
517C2#2145950811yescharacteristic
518C2#2145952811yescharacteristic
519C2#2146004112yes—
520C2#2146006412yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC2×C61×#380C2
prime radicalC25×C61C23
hypercenterC2×#380D61
hypocenterC61#36728
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #21453014C2semidirect
3. #21452724C2semidirect
4. #21452477C2semidirect
5. #21452285C2semidirect
6. #21451933C2semidirect
7. #21451789C2semidirect
9. #21449954C2semidirect
10. #21448552C2semidirect
13. C2×#14519767C22semidirect
17. #16839704C22semidirect
18. #16839700C22semidirect
19. #16839696C22semidirect
20. #16839695C22semidirect
21. #16839694C22semidirect
22. #16839693C22semidirect
23. #16839692C22semidirect
28. #16839683C22semidirect
29. #16839666C22semidirect
30. #16839659C22semidirect
31. #16839657C22semidirect
35. #16839649C22semidirect
36. #16839648C22semidirect
37. #16839645C22semidirect
38. #16839643C22semidirect
317. C61#36728semidirect

Maximal subgroups

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

subgroupindex
#214541062
#214530142
#214527242
#214524772
#214522852
#214519332
#214517892
#214500592
#214499542
#214485522
C2×C61×#3802
#3672861