Finite Group Databasebeta

← #29840490

Subgroups — #29840490

Order 15616.

2094 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D61C211yesmaximal, characteristic
2#21448918C214yesmaximal, complemented
3#21448912C214yesmaximal, complemented
4#21448695C214yesmaximal
5#21448693C214yesmaximal
6#21448561C211yesmaximal, characteristic, complemented
7#21448560C211yesmaximal, characteristic, complemented
8#21448558C211yesmaximal, characteristic, complemented
9#21448434C212yesmaximal
10#21448378C212yesmaximal, complemented
11C4×#14519768C212yesmaximal, complemented
12#21448224C211yesmaximal, characteristic
13#21448173C212yesmaximal
14#21448158C211yesmaximal, characteristic
15C61×#2330C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×#14519768C2212yes—
18C2×#14519767C2214yes—
19#125×D61C4116yescomplemented
20C2×#14519735C2212yes—
21C2×#14519734C2211yescharacteristic
22C2×#14519734C2211yescharacteristic
23C2×C4⋊C4×D61C2212yes—
24C2×C4⋊C4×D61C2214yes—
25C2×C42×D61C2211yescharacteristic
26#16839661C2212yes—
27#16839661C2214yes—
28#16839660C2212yes—
29#16839659C2214yes—
30#16839659C2214yes—
31#16839658C2212yes—
32#16839657C2214yes—
33#16839657C2214yes—
34#16839649C2214yes—
35#16839649C2214yes—
36#16839648C2214yes—
37#16839647C2214yes—
38#16839645C2214yes—
39#16839643C2214yes—
40#16839643C2214yes—
41#16839632C2211yescharacteristic
42#16839627C2212yes—
43#16839621C2214yes—
44#16839619C2214yes—
45#16839619C2214yes—
46C4⋊C4×Dic61C2214yes—
47C4⋊C4×Dic61C2212yes—
48C4⋊C4×Dic61C2212yes—
49C4⋊C4×Dic61C2214yes—
50C4×#11758196C2214yes—
51C4×#11758196C2214yes—
52C4×C4⋊Dic61C2212yes—
53C4×C4⋊Dic61C2214yes—
54#16839613C2214yes—
55#16839613C2212yes—
56#16839613C2214yes—
57#16839613C2212yes—
58C4×#11758198C2214yes—
59#16839609C2212yes—
60#16839609C2212yes—
61#16839609C2214yes—
62#16839608C2211yescharacteristic
63C42×Dic61C2211yescharacteristic
64C42×Dic61C2211yescharacteristic
65C2×#125×C61C2211yescharacteristic
66C61×#389C2212yes—
67C61×#389C2214yes—
68C61×#387C2214yes—
69C61×#384C2211yescharacteristic
70C4×C4⋊C4×C61C2212yes—
71C61×#376C2211yescharacteristic
72C22×C4×D61C2312yes—
73C22×C4×D61C2314yes—
74C22×C4×D61C2311yescharacteristic
75#14519769C2×C418yes—
76#14519768C2×C4116yes—
77#14519767C2×C4132yes—
78#14519735—28no—
79#14519734—24no—
80#14519734Q818yes—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2314yes—
83C2×C4⋊Dic61C2314yes—
84C2×C4⋊Dic61C2314yes—
85C2×C4⋊Dic61C2312yes—
86C2×#11758196C2311yescharacteristic
87C2×#11758196C2311yescharacteristic
88C2×#11758196C2314yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C2×#11758196C2314yes—
93C2×#11758196C2312yes—
94C4⋊C4×D61C2×C4116yes—
95C4⋊C4×D61C2×C4132yes—
96C2×#11758198C2311yescharacteristic
97C2×#11758198C2311yescharacteristic
98C2×#11758198C2314yes—
99C2×#11758198C2312yes—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2314yes—
102C2×C4×Dic61C2312yes—
103C2×C4×Dic61C2312yes—
104C2×C4×Dic61C2312yes—
105C2×C4×Dic61C2312yes—
106C2×C4×Dic61C2314yes—
107C2×C4×Dic61C2314yes—
108C2×C4×Dic61C2314yes—
109C2×C4×Dic61C2312yes—
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C42×D61C2×C418yes—
113#14519684C2312yes—
114#14519684C2312yes—
115#14519684C2314yes—
116#14519684C2314yes—
117#14519684C2314yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#125×C61C2×C418yes—
124C2×C4⋊C4×C61C2312yes—
125C2×C4⋊C4×C61C2314yes—
126C2×C4⋊C4×C61C2314yes—
127C2×C4⋊C4×C61C2314yes—
128C2×C4⋊C4×C61C2312yes—
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2314yes—
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2311yescharacteristic
133#95×C61C2312yes—
134#95×C61C2312yes—
135#95×C61C2314yes—
136#95×C61C2314yes—
137C23×D61C2411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×C4×C61C2414yes—
140C22×C4×C61C2411yescharacteristic
141C22×C4×C61C2414yes—
142C22×C4×C61C2412yes—
143C22×C4×C61C2412yes—
144C22×C4×C61C2411yescharacteristic
145C22×Dic61C2411yescharacteristic
146C22×Dic61C2411yescharacteristic
147C22×Dic61C2414yes—
148C22×Dic61C2412yes—
149C22×Dic61C2414yes—
150C22×Dic61C2412yes—
151C22×Dic61C2411yescharacteristic
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61C22×C418yes—
157C2×C4×D61C22×C4116yes—
158C2×C4×D61C22×C414yes—
159C2×C4×D61C2×Q814yes—
160C2×C4×D61#4114yes—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—44no—
163C4⋊C4×C61—44no—
164C4⋊C4×C61C22×C4116yes—
165C4⋊C4×C61C22×C418yes—
166C42×C61—22no—
167C42×C61—44no—
168C42×C61C22×C414yes—
169C42×C61C2×Q814yes—
170#11758198—22no—
171#11758198—22no—
172#11758198—28no—
173#11758198C2×Q818yes—
174C4⋊Dic61—24no—
175C4⋊Dic61—44no—
176C4⋊Dic61—44no—
177C4⋊Dic61C22×C4116yes—
178C4⋊Dic61C22×C418yes—
179#11758196—22no—
180#11758196—22no—
181#11758196—44no—
182#11758196—44no—
183#11758196C22×C4116yes—
184#11758196C22×C4116yes—
185#11758196C22×C4116yes—
186#11758196C2×Q818yes—
187C4×Dic61—22no—
188C4×Dic61—28no—
189C4×Dic61—42no—
190C4×Dic61—42no—
191C4×Dic61—44no—
192C4×Dic61—44no—
193C4×Dic61—42no—
194C4×Dic61—42no—
195C4×Dic61—44no—
196C4×Dic61C22×C418yes—
197C4×Dic61C22×C414yes—
198C4×Dic61C2×Q814yes—
199C23×C61C2511yescharacteristic
200C22×D61—22no—
201C22×D61—22no—
202C22×D61C23×C412yes—
203C22×D61C22×Q812yes—
204C22×D61C2×#4112yes—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61C23×C414yes—
219C2×C4×C61C23×C418yes—
220C2×C4×C61C23×C412yes—
221C2×C4×C61C22×Q814yes—
222C2×C4×C61C22×Q812yes—
223C2×C4×C61C2×#4112yes—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61C23×C414yes—
238C2×Dic61C23×C418yes—
239C2×Dic61C23×C412yes—
240C2×Dic61C22×Q814yes—
241C2×Dic61C22×Q812yes—
242C2×Dic61C2×#4112yes—
243C4×D61—48no—
244C4×D61—416no—
245C4×D61C4×Q8116yes—
246#34217—611nomaximal, Sylow
247C22×C61C24×C411yescharacteristic
248C22×C61C23×Q811yescharacteristic
249C22×C61C22×#4111yescharacteristic
250C22×C61#58512yes—
251C22×C61#58512yes—
252C2×D61—24no—
253C2×D61—24no—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61C2×C4×Q814yes—
257C4×C61—28no—
258C4×C61—24no—
259C4×C61—24no—
260C4×C61—44no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61C2×C4×Q814yes—
264Dic61—24no—
265Dic61—28no—
266Dic61—24no—
267Dic61—41no—
268Dic61—41no—
269Dic61—44no—
270Dic61C2×C4×Q814yes—
271C22×#125—611no—
272#2370—614no—
273#2370—614no—
274#2345—614no—
275#2345—614no—
276#2330—611no—
277#2330—611no—
278#2330—611no—
279#2330D6111yescharacteristic, complemented
280#2316—612no—
281C4×#125—612no—
282C4×#125—612no—
283#2285—611no—
284#2274—612no—
285C4×#117—611no—
286C2×C61C22×C4×Q811yescharacteristic
287C2×C61#342812yes—
288C2×C61#350412yes—
289C2×C61#351211yescharacteristic
290C2×C61#351211yescharacteristic
291D61—24no—
292C2×#125—6116no—
293C2×#125—612no—
294C2×#125—614no—
295C2×#125—611no—
296C2×#125C2×D6111yescharacteristic
297C2×#117—612no—
298C2×#117—611no—
299C2×#117—611no—
300C22×C4⋊C4—612no—
301C22×C4⋊C4—614no—
302C22×C42—611no—
303#389—612no—
304#389—612no—
305#389—614no—
306#389—614no—
307#389—614no—
308#389—612no—
309#389—614no—
310#389—614no—
311#389C2×D6112yes—
312#389C2×D6114yes—
313#387—614no—
314#387—614no—
315#387—614no—
316#387—614no—
317#387—614no—
318#387—614no—
319#387—614no—
320#387C2×D6114yes—
321#384—611no—
322#384C2×D6111yescharacteristic
323#382—612no—
324#380—614no—
325#380—614no—
326#380—614no—
327C4×C4⋊C4—612no—
328C4×C4⋊C4—612no—
329C4×C4⋊C4—614no—
330C4×C4⋊C4—614no—
331C4×C4⋊C4—612no—
332C4×C4⋊C4—614no—
333C4×C4⋊C4—614no—
334C4×C4⋊C4—614no—
335C4×C4⋊C4—614no—
336C4×C4⋊C4—612no—
337C4×C4⋊C4—612no—
338C4×C4⋊C4—614no—
339C4×C4⋊C4C2×D6112yes—
340C4×C22⋊C4—614no—
341C4×C22⋊C4—612no—
342C4×C22⋊C4—612no—
343C4×C22⋊C4—614no—
344#376—611no—
345#376C2×D6111yescharacteristic
346C43—611no—
347C43—611no—
348C61#3421711yescharacteristic, complemented, Sylow
349C23×C4—611no—
350C23×C4—614no—
351C23×C4—612no—
352#125—6132no—
353#125—6116no—
354#125—618no—
355#125C4×D6118yes—
356#117—618no—
357#117—1224no—
358#117—1228no—
359C2×C4⋊C4—614no—
360C2×C4⋊C4—614no—
361C2×C4⋊C4—614no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—614no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—6116no—
366C2×C4⋊C4—6132no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—614no—
371C2×C4⋊C4—614no—
372C2×C4⋊C4—614no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4C22×D6112yes—
375C2×C4⋊C4C22×D6114yes—
376C2×C4⋊C4C22×D6114yes—
377C2×C4⋊C4C22×D6112yes—
378C2×C4⋊C4C22×D6114yes—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—614no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—612no—
386C2×C42—618no—
387C2×C42—611no—
388C2×C42—614no—
389C2×C42—612no—
390C2×C42—612no—
391C2×C42—612no—
392C2×C42—612no—
393C2×C42—614no—
394C2×C42—614no—
395C2×C42—614no—
396C2×C42C22×D6111yescharacteristic
397C2×C42C22×D6114yes—
398C2×C42C22×D6111yescharacteristic
399C2×C42C22×D6111yescharacteristic
400#95—612no—
401#95—612no—
402#95—614no—
403#95—614no—
404#95—614no—
405#95—614no—
406#95—614no—
407#95—614no—
408#95—614no—
409#95—614no—
410#95C22×D6112yes—
411#95C22×D6112yes—
412#95C22×D6114yes—
413#95C22×D6114yes—
414C24—611no—
415C22×C4—611no—
416C22×C4—614no—
417C22×C4—614no—
418C22×C4—614no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—614no—
422C22×C4—618no—
423C22×C4—6116no—
424C22×C4—611no—
425C22×C4—614no—
426C22×C4—611no—
427C22×C4—2444no—
428C22×C4—2442no—
429C22×C4—2444no—
430C22×C4—2442no—
431C22×C4C23×D6111yescharacteristic
432C22×C4C23×D6114yes—
433C22×C4C23×D6111yescharacteristic
434C22×C4C23×D6114yes—
435C22×C4C23×D6112yes—
436C22×C4C23×D6112yes—
437C22×C4C23×D6111yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—6116no—
442C4⋊C4—6116no—
443C4⋊C4—6116no—
444C4⋊C4—6116no—
445C4⋊C4—618no—
446C4⋊C4—618no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—1224no—
450C4⋊C4—2444no—
451C4⋊C4—2444no—
452C4⋊C4—2444no—
453C4⋊C4—2444no—
454C4⋊C4C2×C4×D61116yes—
455C4⋊C4C2×C4×D6118yes—
456C22⋊C4—618no—
457C22⋊C4—1222no—
458C22⋊C4—1222no—
459C22⋊C4—1228no—
460C42—22no—
461C42—44no—
462C42—618no—
463C42—614no—
464C42—614no—
465C42—1228no—
466C42—1222no—
467C42—2442no—
468C42—2442no—
469C42—2444no—
470C42—2444no—
471C42—2444no—
472C42—2442no—
473C42—2442no—
474C42C2×C4×D6114yes—
475C42Q8×D6114yes—
476C23—612no—
477C23—612no—
478C23—612no—
479C23—1222no—
480C23—1222no—
481C23C24×D6111yescharacteristic
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—6116no—
499C2×C4—614no—
500C2×C4—618no—
501C2×C4—614no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1224no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1224no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1224no—
511C2×C4—1222no—
512C2×C4—1224no—
513C2×C4—1222no—
514C2×C4—1224no—
515C2×C4—2448no—
516C2×C4—24416no—
517C2×C4C22×C4×D6114yes—
518C2×C4C22×C4×D6118yes—
519C2×C4C22×C4×D6112yes—
520C2×C4C2×Q8×D6114yes—
521C2×C4C2×Q8×D6112yes—
522C2×C4#41×D6112yes—
523C22—614no—
524C22—1222no—
525C22—1222no—
526C22—1224no—
527C22—1224no—
528C22C23×C4×D6111yescharacteristic
529C22C22×Q8×D6111yescharacteristic
530C22C2×#41×D6111yescharacteristic
531C22#1684059712yes—
532C22#1684059912yes—
533C4—28no—
534C4—24no—
535C4—24no—
536C4—41no—
537C4—44no—
538C4—41no—
539C4—614no—
540C4—1224no—
541C4—1228no—
542C4—1224no—
543C4—2441no—
544C4—2444no—
545C4—2441no—
546C4C4×Q8×D6114yes—
547C2—1224no—
548C2C2×C4×Q8×D6111yescharacteristic
549C2#2145901912yes—
550C2#2145964212yes—
551C2#2145973811yescharacteristic
552C2#2145974211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2330C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2330D61
hypocenterC61#34217
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #21448918C2semidirect
3. #21448912C2semidirect
6. #21448561C2semidirect
7. #21448560C2semidirect
8. #21448558C2semidirect
10. #21448378C2semidirect
11. C4×#14519768C2semidirect
15. C61×#2330C2semidirect
19. #125×D61C4semidirect
279. #2330D61semidirect
348. C61#34217semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D612
#214489182
#214489122
#214486952
#214486932
#214485612
#214485602
#214485582
#214484342
#214483782
C4×#145197682
#214482242
#214481732
#214481582
C61×#23302
#3421761