Finite Group Databasebeta

← #29840460

Subgroups — #29840460

Order 15616.

2310 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840318C211yesmaximal, characteristic
2#21448827C214yesmaximal
3#21448821C214yesmaximal
4#21448773C214yesmaximal, complemented
5#21448733C214yesmaximal, complemented
6#21448561C211yesmaximal, characteristic, complemented
7#21448560C211yesmaximal, characteristic, complemented
8#21448558C211yesmaximal, characteristic, complemented
9#21448441C212yesmaximal
10#21448390C212yesmaximal, complemented
11C4×#14519776C212yesmaximal, complemented
12#384×D61C211yesmaximal, characteristic
13#21448173C212yesmaximal
14#21448158C211yesmaximal, characteristic
15C61×#2330C211yesmaximal, characteristic, complemented
16#16840318C4116yescomplemented
17C2×#14519776C2212yes—
18C2×#14519769C2211yescharacteristic
19C2×#14519755C2214yes—
20C2×C4×Dic122C2214yes—
21C2×#14519742C2214yes—
22C2×#14519742C2214yes—
23C2×#14519735C2212yes—
24C2×#14519734C2211yescharacteristic
25C2×#14519734C2211yescharacteristic
26C2×C4⋊C4×D61C2212yes—
27C2×C4⋊C4×D61C2214yes—
28C2×C42×D61C2211yescharacteristic
29#16839666C2214yes—
30#16839666C2214yes—
31#16839661C2214yes—
32#16839661C2212yes—
33#16839660C2212yes—
34#16839659C2214yes—
35#16839659C2214yes—
36#16839658C2212yes—
37#16839657C2214yes—
38#16839657C2214yes—
39#16839649C2214yes—
40#16839648C2214yes—
41#16839643C2214yes—
42#16839636C2214yes—
43#16839633C2214yes—
44#16839632C2211yescharacteristic
45#16839631C2214yes—
46#16839631C2212yes—
47#16839631C2214yes—
48#16839626C2214yes—
49C4⋊C4×Dic61C2212yes—
50C4⋊C4×Dic61C2212yes—
51C4×#11758196C2214yes—
52C4×#11758196C2214yes—
53C4×C4⋊Dic61C2212yes—
54#16839613C2212yes—
55#16839613C2214yes—
56#16839613C2212yes—
57#16839613C2214yes—
58C4×#11758198C2214yes—
59#16839609C2212yes—
60#16839609C2214yes—
61#16839609C2212yes—
62#16839608C2211yescharacteristic
63C42×Dic61C2211yescharacteristic
64C42×Dic61C2211yescharacteristic
65C2×#125×C61C2211yescharacteristic
66C61×#389C2214yes—
67C61×#389C2212yes—
68C61×#387C2214yes—
69C61×#384C2211yescharacteristic
70C4×C4⋊C4×C61C2212yes—
71C61×#376C2211yescharacteristic
72C22×Dic122C2314yes—
73C22×C4×D61C2312yes—
74C22×C4×D61C2314yes—
75C22×C4×D61C2311yescharacteristic
76#14519776C2×C4116yescomplemented
77#14519769C2×C418yescomplemented
78#14519755C2×C4132yes—
79C4×Dic122—48no—
80#14519742—48no—
81#14519742C2×C4132yescomplemented
82#14519735C2×C4116yes—
83#14519734—24no—
84#14519734Q818yescomplemented
85C2×C4⋊Dic61C2312yes—
86C2×C4⋊Dic61C2314yes—
87C2×C4⋊Dic61C2314yes—
88C2×C4⋊Dic61C2312yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C2×#11758196C2311yescharacteristic
93C2×#11758196C2311yescharacteristic
94C2×#11758196C2314yes—
95C2×#11758196C2314yes—
96C2×#11758196C2314yes—
97C2×#11758196C2314yes—
98C2×#11758196C2312yes—
99C4⋊C4×D61—28no—
100C4⋊C4×D61—48no—
101C2×#11758198C2311yescharacteristic
102C2×#11758198C2311yescharacteristic
103C2×#11758198C2314yes—
104C2×#11758198C2312yes—
105C2×C4×Dic61C2311yescharacteristic
106C2×C4×Dic61C2312yes—
107C2×C4×Dic61C2312yes—
108C2×C4×Dic61C2312yes—
109C2×C4×Dic61C2312yes—
110C2×C4×Dic61C2314yes—
111C2×C4×Dic61C2314yes—
112C2×C4×Dic61C2314yes—
113C2×C4×Dic61C2312yes—
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2314yes—
116C2×C4×Dic61C2311yescharacteristic
117C42×D61C2×C418yes—
118#14519684C2312yes—
119#14519684C2312yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#14519684C2314yes—
124#14519684C2314yes—
125#14519684C2314yes—
126#125×C61C2×C418yescomplemented
127C2×C4⋊C4×C61C2312yes—
128C2×C4⋊C4×C61C2314yes—
129C2×C4⋊C4×C61C2314yes—
130C2×C4⋊C4×C61C2312yes—
131C2×C4⋊C4×C61C2314yes—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2314yes—
134C2×C42×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136#95×C61C2312yes—
137#95×C61C2312yes—
138#95×C61C2314yes—
139#95×C61C2314yes—
140C23×D61C2411yescharacteristic
141C22×C4×C61C2414yes—
142C22×C4×C61C2411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×C4×C61C2412yes—
145C22×C4×C61C2414yes—
146C22×C4×C61C2412yes—
147C22×C4×C61C2411yescharacteristic
148C22×Dic61C2414yes—
149C22×Dic61C2411yescharacteristic
150C22×Dic61C2411yescharacteristic
151C22×Dic61C2414yes—
152C22×Dic61C2412yes—
153C22×Dic61C2411yescharacteristic
154C22×Dic61C2412yes—
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—28no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61C22×C418yes—
161C2×C4×D61C22×C414yes—
162C2×C4×D61C2×Q814yes—
163C2×C4×D61#4114yes—
164C2×Dic122—28no—
165C2×Dic122—28no—
166C2×Dic122C22×C4116yes—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—44no—
169C4⋊C4×C61—44no—
170C4⋊C4×C61C22×C418yes—
171C4⋊C4×C61C22×C4116yes—
172C42×C61—22no—
173C42×C61—44no—
174C42×C61C22×C414yes—
175C42×C61C2×Q814yes—
176#11758198—22no—
177#11758198—22no—
178#11758198C22×C4116yes—
179#11758198C2×Q818yes—
180C4⋊Dic61—24no—
181C4⋊Dic61—44no—
182C4⋊Dic61—44no—
183C4⋊Dic61C22×C418yes—
184#11758196—28no—
185#11758196—22no—
186#11758196—22no—
187#11758196—28no—
188#11758196—28no—
189#11758196—44no—
190#11758196—44no—
191#11758196C22×C4116yes—
192#11758196C22×C4116yes—
193#11758196C2×Q818yes—
194C4×Dic61—22no—
195C4×Dic61—24no—
196C4×Dic61—24no—
197C4×Dic61—24no—
198C4×Dic61—24no—
199C4×Dic61—44no—
200C4×Dic61—44no—
201C4×Dic61—44no—
202C4×Dic61C22×C418yes—
203C4×Dic61C22×C414yes—
204C4×Dic61C22×C4116yes—
205C4×Dic61C2×Q814yes—
206C23×C61C2511yescharacteristic
207C22×D61—22no—
208C22×D61—22no—
209C22×D61C23×C412yes—
210C22×D61C22×Q812yes—
211C22×D61C2×#4112yes—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61C23×C414yes—
226C2×C4×C61C23×C418yes—
227C2×C4×C61C23×C412yes—
228C2×C4×C61C22×Q814yes—
229C2×C4×C61C22×Q812yes—
230C2×C4×C61C2×#4112yes—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—24no—
238C2×Dic61—22no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—22no—
242C2×Dic61C23×C418yes—
243C2×Dic61C23×C412yes—
244C2×Dic61C23×C414yes—
245C2×Dic61C22×Q814yes—
246C2×Dic61C22×Q812yes—
247C2×Dic61C2×#4112yes—
248C2×Dic61H(5,𝔽2)14yes—
249C2×Dic612−1+414yes—
250C4×D61—216no—
251C4×D61—88no—
252C4×D61C4×Q8116yescomplemented
253Dic122—232no—
254#34214—611nomaximal, Sylow
255C22×C61C24×C411yescharacteristic
256C22×C61C23×Q811yescharacteristic
257C22×C61C22×#4111yescharacteristic
258C22×C61C2×H(5,𝔽2)12yes—
259C22×C61C2×2−1+412yes—
260C2×D61—24no—
261C2×D61—24no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61C2×C4×Q814yes—
265C4×C61—28no—
266C4×C61—24no—
267C4×C61—24no—
268C4×C61—44no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61C2×C4×Q814yes—
272Dic61—28no—
273Dic61—24no—
274Dic61—44no—
275Dic61—41no—
276Dic61—41no—
277Dic61C2×C4×Q814yes—
278Dic61#52018yes—
279C2×#533—611no—
280#2360—614no—
281#2360—614no—
282#2355—614no—
283Q8×C4⋊C4—614no—
284#2330—611no—
285#2330—611no—
286#2330—611no—
287#2330D6111yescharacteristic, complemented
288#2317—612no—
289C4×C4⋊Q8—612no—
290C4×C4⋊Q8—612no—
291C2×#384—611no—
292#2274—612no—
293C4×#117—611no—
294C2×C61C22×C4×Q811yescharacteristic
295C2×C61C2×#52012yes—
296C2×C61#350312yes—
297C2×C61#350811yescharacteristic
298C2×C61#350911yescharacteristic
299D61—24no—
300#533—6116no—
301C2×C4⋊Q8—612no—
302C2×#125—611no—
303C2×#125C2×D6111yescharacteristic
304C2×C22⋊Q8—614no—
305C2×C4×Q8—614no—
306C2×C4×Q8—614no—
307C2×C4×Q8—614no—
308C2×#117—612no—
309C2×#117—611no—
310C2×#117—611no—
311C22×C4⋊C4—612no—
312C22×C4⋊C4—614no—
313C22×C42—611no—
314#391—614no—
315#391—614no—
316#389—612no—
317#389—612no—
318#389—614no—
319#389—614no—
320#389—614no—
321#389—612no—
322#389—614no—
323#389—614no—
324#389C2×D6112yes—
325#389C2×D6114yes—
326#387—614no—
327#387—614no—
328#387—614no—
329#387C2×D6114yes—
330#385—614no—
331#385—614no—
332#384—612no—
333#384—614no—
334#384—614no—
335#384—611no—
336#384C2×D6111yescharacteristic
337#382—614no—
338C4×C4⋊C4—612no—
339C4×C4⋊C4—612no—
340C4×C4⋊C4—612no—
341C4×C4⋊C4—614no—
342C4×C4⋊C4—614no—
343C4×C4⋊C4—614no—
344C4×C4⋊C4—612no—
345C4×C4⋊C4—614no—
346C4×C4⋊C4—612no—
347C4×C4⋊C4C2×D6112yes—
348C4×C22⋊C4—614no—
349C4×C22⋊C4—612no—
350C4×C22⋊C4—612no—
351C4×C22⋊C4—614no—
352#376—611no—
353#376C2×D6111yescharacteristic
354C43—611no—
355C43—611no—
356C61#3421411yescharacteristic, complemented, Sylow
357C22×Q8—614no—
358C23×C4—611no—
359C23×C4—612no—
360C23×C4—614no—
361C4⋊Q8—6116no—
362#125—618no—
363#125C4×D6118yescomplemented
364C22⋊Q8—6132no—
365C4×Q8—6132no—
366C4×Q8—2448no—
367C4×Q8—2448no—
368#117—6116no—
369#117—618no—
370#117—1224no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4—614no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—614no—
377C2×C4⋊C4—614no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—614no—
382C2×C4⋊C4—614no—
383C2×C4⋊C4—614no—
384C2×C4⋊C4—614no—
385C2×C4⋊C4—1228no—
386C2×C4⋊C4—2448no—
387C2×C4⋊C4C22×D6112yes—
388C2×C4⋊C4C22×D6114yes—
389C2×C4⋊C4C22×D6114yes—
390C2×C4⋊C4C22×D6112yes—
391C2×C4⋊C4C22×D6114yes—
392C2×C22⋊C4—614no—
393C2×C22⋊C4—612no—
394C2×C22⋊C4—611no—
395C2×C22⋊C4—611no—
396C2×C42—614no—
397C2×C42—611no—
398C2×C42—612no—
399C2×C42—611no—
400C2×C42—618no—
401C2×C42—611no—
402C2×C42—612no—
403C2×C42—612no—
404C2×C42—612no—
405C2×C42—612no—
406C2×C42—614no—
407C2×C42—614no—
408C2×C42—614no—
409C2×C42C22×D6111yescharacteristic
410C2×C42C22×D6114yes—
411C2×C42C22×D6111yescharacteristic
412C2×C42C22×D6111yescharacteristic
413#95—612no—
414#95—612no—
415#95—614no—
416#95—614no—
417#95—614no—
418#95—614no—
419#95—614no—
420#95—614no—
421#95C22×D6112yes—
422#95C22×D6112yes—
423#95C22×D6114yes—
424#95C22×D6114yes—
425C24—611no—
426C2×Q8—6116no—
427C2×Q8—1228no—
428C2×Q8—1228no—
429C22×C4—611no—
430C22×C4—612no—
431C22×C4—614no—
432C22×C4—614no—
433C22×C4—614no—
434C22×C4—612no—
435C22×C4—614no—
436C22×C4—618no—
437C22×C4—611no—
438C22×C4—614no—
439C22×C4—611no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1228no—
443C22×C4—2444no—
444C22×C4—2444no—
445C22×C4C23×D6111yescharacteristic
446C22×C4C23×D6114yes—
447C22×C4C23×D6111yescharacteristic
448C22×C4C23×D6114yes—
449C22×C4C23×D6112yes—
450C22×C4C23×D6112yes—
451C22×C4C23×D6111yescharacteristic
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—6116no—
456C4⋊C4—6116no—
457C4⋊C4—618no—
458C4⋊C4—618no—
459C4⋊C4—1228no—
460C4⋊C4—1228no—
461C4⋊C4—1228no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—1224no—
465C4⋊C4—2444no—
466C4⋊C4—2444no—
467C4⋊C4—2444no—
468C4⋊C4—2444no—
469C4⋊C4C2×C4×D6118yes—
470C4⋊C4C2×C4×D61116yes—
471C22⋊C4—6116no—
472C22⋊C4—618no—
473C22⋊C4—1222no—
474C22⋊C4—1222no—
475C42—22no—
476C42—44no—
477C42—618no—
478C42—614no—
479C42—6116no—
480C42—614no—
481C42—1224no—
482C42—1224no—
483C42—1224no—
484C42—1224no—
485C42—1222no—
486C42—2444no—
487C42—2444no—
488C42—2444no—
489C42C2×C4×D6114yes—
490C42Q8×D6114yes—
491C23—612no—
492C23—612no—
493C23—612no—
494C23—1222no—
495C23—1222no—
496C23C24×D6111yescharacteristic
497Q8—12232no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—612no—
512C2×C4—614no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—614no—
516C2×C4—614no—
517C2×C4—6116no—
518C2×C4—618no—
519C2×C4—614no—
520C2×C4—12216no—
521C2×C4—1222no—
522C2×C4—1224no—
523C2×C4—1222no—
524C2×C4—1224no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1224no—
528C2×C4—1222no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—1222no—
532C2×C4—4888no—
533C2×C4C22×C4×D6114yes—
534C2×C4C22×C4×D6118yes—
535C2×C4C22×C4×D6112yes—
536C2×C4C2×Q8×D6114yes—
537C2×C4C2×Q8×D6112yes—
538C2×C4#41×D6112yes—
539C22—614no—
540C22—1224no—
541C22—1224no—
542C22—1222no—
543C22—1222no—
544C22C23×C4×D6111yescharacteristic
545C22C22×Q8×D6111yescharacteristic
546C22C2×#41×D6111yescharacteristic
547C22H(5,𝔽2)×D6112yes—
548C222−1+4×D6112yes—
549C4—24no—
550C4—28no—
551C4—24no—
552C4—41no—
553C4—44no—
554C4—41no—
555C4—614no—
556C4—618no—
557C4—1228no—
558C4—1224no—
559C4—2441no—
560C4—2444no—
561C4—2441no—
562C4C4×Q8×D6114yes—
563C2—1224no—
564C2C2×C4×Q8×D6111yescharacteristic
565C2#520×D6112yes—
566C2#2145963412yes—
567C2#2145969111yescharacteristic
568C2#2145970111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2330C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2330D61
hypocenterC61#34214
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
4. #21448773C2semidirect
5. #21448733C2semidirect
6. #21448561C2semidirect
7. #21448560C2semidirect
8. #21448558C2semidirect
10. #21448390C2semidirect
11. C4×#14519776C2semidirect
15. C61×#2330C2semidirect
16. #16840318C4semidirect
76. #14519776C2×C4semidirect
77. #14519769C2×C4semidirect
81. #14519742C2×C4semidirect
84. #14519734Q8semidirect
126. #125×C61C2×C4semidirect
252. C4×D61C4×Q8semidirect
287. #2330D61semidirect
356. C61#34214semidirect
363. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
#214488272
#214488212
#214487732
#214487332
#214485612
#214485602
#214485582
#214484412
#214483902
C4×#145197762
#384×D612
#214481732
#214481582
C61×#23302
#3421461