Finite Group Databasebeta

← #29839159

Subgroups — #29839159

Order 15616.

2070 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840375C211yesmaximal, characteristic
2#21448622C214yesmaximal, complemented
3#21448593C214yesmaximal
4#21448538C211yesmaximal, characteristic, complemented
5#21448502C212yesmaximal
6#21448466C212yesmaximal, complemented
7#21448427C211yesmaximal, characteristic
8#21448386C211yesmaximal, characteristic, complemented
9#125×Dic61C211yesmaximal, characteristic, complemented
10#21448378C214yesmaximal, complemented
11#21448369C214yesmaximal
12#21448349C212yesmaximal
13C4×#14519742C212yesmaximal, complemented
14#21448158C211yesmaximal, characteristic
15C61×#2330C211yesmaximal, characteristic, complemented
16#16840375C4116yescomplemented
17C2×#14519777C2211yescharacteristic
18C2×#14519772C2214yes—
19C2×#14519768C2214yes—
20C2×#14519757C2212yes—
21C2×C4×Dic122C2212yes—
22C2×#14519742C2212yes—
23C2×#14519742C2212yes—
24C2×#14519734C2211yescharacteristic
25C2×#14519734C2211yescharacteristic
26C2×C42×D61C2211yescharacteristic
27#16839668C2212yes—
28#16839667C2212yes—
29#16839661C2212yes—
30#16839660C2214yes—
31#16839658C2214yes—
32#16839654C2214yes—
33#16839653C2214yes—
34#16839652C2214yes—
35#16839649C2214yes—
36#16839648C2214yes—
37#16839647C2214yes—
38#16839647C2214yes—
39#16839647C2214yes—
40#16839645C2214yes—
41#16839643C2214yes—
42#16839643C2214yes—
43#16839633C2214yes—
44#16839629C2214yes—
45#16839628C2211yescharacteristic
46#16839627C2214yes—
47#16839619C2212yes—
48C4⋊C4×Dic61C2212yes—
49C4⋊C4×Dic61C2214yes—
50C4⋊C4×Dic61C2212yes—
51C4⋊C4×Dic61C2214yes—
52C4⋊C4×Dic61C2214yes—
53C4×#11758196C2214yes—
54#16839613C2214yes—
55#16839613C2212yes—
56#16839613C2214yes—
57#16839613C2212yes—
58#16839609C2212yes—
59#16839609C2214yes—
60#16839609C2212yes—
61#16839608C2211yescharacteristic
62C42×Dic61C2214yes—
63C42×Dic61C2211yescharacteristic
64C42×Dic61C2211yescharacteristic
65C2×#125×C61C2211yescharacteristic
66C61×#389C2212yes—
67C61×#389C2214yes—
68C61×#387C2214yes—
69C61×#384C2211yescharacteristic
70C4×C4⋊C4×C61C2212yes—
71C61×#376C2211yescharacteristic
72C22×Dic122C2312yes—
73C22×C4×D61C2311yescharacteristic
74C22×C4×D61C2311yescharacteristic
75C22×C4×D61C2311yescharacteristic
76#14519777C2×C418yescomplemented
77#14519772C2×C4132yes—
78#14519768C2×C4132yescomplemented
79#14519757C2×C4116yes—
80C4×Dic122—28no—
81#14519742—28no—
82#14519742C2×C4116yescomplemented
83#14519734—24no—
84#14519734C2×C418yes—
85C2×C4⋊Dic61C2312yes—
86C2×C4⋊Dic61C2314yes—
87C2×C4⋊Dic61C2314yes—
88C2×C4⋊Dic61C2314yes—
89C2×#11758196C2312yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C2×#11758196C2312yes—
93C2×#11758196C2314yes—
94C2×#11758198C2312yes—
95C2×#11758198C2314yes—
96C2×#11758198C2312yes—
97C2×#11758198C2314yes—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2314yes—
102C2×C4×Dic61C2314yes—
103C2×C4×Dic61C2312yes—
104C2×C4×Dic61C2314yes—
105C2×C4×Dic61C2312yes—
106C2×C4×Dic61C2312yes—
107C2×C4×Dic61C2312yes—
108C2×C4×Dic61C2314yes—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2314yes—
111C2×C4×Dic61C2311yescharacteristic
112C42×D61—24no—
113#14519684C2312yes—
114#14519684C2312yes—
115#14519684C2314yes—
116#14519684C2314yes—
117#14519684C2314yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#14519684C2314yes—
124#125×C61C2×C418yescomplemented
125C2×C4⋊C4×C61C2312yes—
126C2×C4⋊C4×C61C2314yes—
127C2×C4⋊C4×C61C2314yes—
128C2×C4⋊C4×C61C2312yes—
129C2×C4⋊C4×C61C2314yes—
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2314yes—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2311yescharacteristic
134#95×C61C2312yes—
135#95×C61C2312yes—
136#95×C61C2314yes—
137#95×C61C2314yes—
138C23×D61C2411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×C4×C61C2414yes—
142C22×C4×C61C2412yes—
143C22×C4×C61C2414yes—
144C22×C4×C61C2412yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61C2414yes—
147C22×Dic61C2412yes—
148C22×Dic61C2411yescharacteristic
149C22×Dic61C2414yes—
150C22×Dic61C2412yes—
151C22×Dic61C2411yescharacteristic
152C22×Dic61C2411yescharacteristic
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—24no—
158C2×C4×D61—42no—
159C2×C4×D61C22×C414yes—
160C2×Dic122—24no—
161C2×Dic122—24no—
162C2×Dic122C22×C418yes—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—44no—
165C4⋊C4×C61—44no—
166C4⋊C4×C61C22×C418yes—
167C4⋊C4×C61C22×C4116yes—
168C42×C61—22no—
169C42×C61—44no—
170C42×C61C22×C414yes—
171C42×C61#4114yes—
172#11758198—24no—
173#11758198—44no—
174#11758198C22×C418yes—
175#11758198C22×C4116yes—
176C4⋊Dic61—24no—
177C4⋊Dic61—44no—
178C4⋊Dic61C22×C4116yes—
179C4⋊Dic61C22×C4116yes—
180#11758196—24no—
181#11758196—28no—
182#11758196—28no—
183#11758196C22×C418yes—
184#11758196C22×C4116yes—
185C4×Dic61—22no—
186C4×Dic61—28no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61—28no—
190C4×Dic61—24no—
191C4×Dic61—28no—
192C4×Dic61—24no—
193C4×Dic61—42no—
194C4×Dic61—44no—
195C4×Dic61—42no—
196C4×Dic61C22×C414yes—
197C4×Dic61C22×C4116yes—
198C4×Dic61#4114yes—
199C23×C61C2511yescharacteristic
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61C23×C412yes—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61C23×C418yes—
219C2×C4×C61C23×C414yes—
220C2×C4×C61C23×C412yes—
221C2×C4×C61C2×#4114yes—
222C2×C4×C61C2×#4112yes—
223C2×C4×C61C2×#4112yes—
224C2×Dic61—21no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61C23×C414yes—
238C2×Dic61C23×C418yes—
239C2×Dic61C23×C412yes—
240C2×Dic61C2×#4112yes—
241C2×Dic61C2×#4112yes—
242C2×Dic612−1+414yes—
243C4×D61—28no—
244C4×D61—44no—
245C4×D61—44no—
246Dic122—216no—
247C4×#541—611nomaximal, Sylow
248C22×C61C24×C411yescharacteristic
249C22×C61C22×#4111yescharacteristic
250C22×C61C22×#4111yescharacteristic
251C22×C61C2×2−1+412yes—
252C22×C61#58512yes—
253C2×D61—22no—
254C2×D61—24no—
255C2×D61—41no—
256C2×D61—41no—
257C2×D61—42no—
258C4×C61—24no—
259C4×C61—28no—
260C4×C61—24no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—44no—
264C4×C61C4×#4114yes—
265Dic61—28no—
266Dic61—22no—
267Dic61—24no—
268Dic61—22no—
269Dic61—44no—
270Dic61—42no—
271Dic61#54114yescomplemented
272C2×#541—611no—
273#2336—614no—
274#2334—614no—
275#2330D6111yescharacteristic, complemented
276#2327—611no—
277#2323—612no—
278#2320—612no—
279#2314—611no—
280C4×C4⋊Q8—611no—
281C4×#125—614no—
282C4×#125—611no—
283C4×#126—614no—
284C4×C22⋊Q8—612no—
285C42×Q8—612no—
286C4×#117—611no—
287C2×C61C2×C4×#4111yescharacteristic
288C2×C61C4×2−1+412yes—
289C2×C61C2×#54111yescharacteristic
290C2×C61#345511yescharacteristic
291C2×C61#346112yes—
292D61—42no—
293#541—6116no—
294C2×C4⋊Q8—611no—
295C2×#126—614no—
296C2×#125—614no—
297C2×#125C2×D6111yescharacteristic
298C2×C22⋊Q8—612no—
299C2×C4×Q8—612no—
300C2×C4×Q8—612no—
301C2×C4×Q8—612no—
302C2×#117—611no—
303C2×#117—611no—
304C22×C42—611no—
305#391—612no—
306#391—612no—
307#389—612no—
308#389—614no—
309#389—614no—
310#389C2×D6114yes—
311#389C2×D6112yes—
312#388—614no—
313#388—614no—
314#388—614no—
315#387—614no—
316#387—614no—
317#387—614no—
318#387—614no—
319#387—614no—
320#387—614no—
321#387—614no—
322#387—614no—
323#387C2×D6114yes—
324#385—614no—
325#384C2×D6111yescharacteristic
326#383—614no—
327#382—614no—
328#382—611no—
329#380—612no—
330C4×C4⋊C4—614no—
331C4×C4⋊C4—614no—
332C4×C4⋊C4—614no—
333C4×C4⋊C4—612no—
334C4×C4⋊C4—612no—
335C4×C4⋊C4—614no—
336C4×C4⋊C4—614no—
337C4×C4⋊C4—612no—
338C4×C4⋊C4—614no—
339C4×C4⋊C4—612no—
340C4×C4⋊C4C2×D6112yes—
341C4×C22⋊C4—612no—
342C4×C22⋊C4—614no—
343C4×C22⋊C4—612no—
344#376—611no—
345#376C2×D6111yescharacteristic
346C43—614no—
347C43—611no—
348C43—611no—
349C61C4×#54111yescharacteristic, complemented, Sylow
350C22×Q8—612no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C4⋊Q8—618no—
355#126—6132no—
356#125—6132no—
357#125C4×D6118yescomplemented
358C22⋊Q8—6116no—
359C4×Q8—6116no—
360C4×Q8—1228no—
361C4×Q8—1228no—
362#117—618no—
363#117—1224no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—614no—
366C2×C4⋊C4—614no—
367C2×C4⋊C4—614no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—614no—
371C2×C4⋊C4—614no—
372C2×C4⋊C4—614no—
373C2×C4⋊C4C22×D6112yes—
374C2×C4⋊C4C22×D6114yes—
375C2×C4⋊C4C22×D6114yes—
376C2×C4⋊C4C22×D6112yes—
377C2×C4⋊C4C22×D6114yes—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—614no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—614no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—614no—
385C2×C42—611no—
386C2×C42—614no—
387C2×C42—614no—
388C2×C42—612no—
389C2×C42—614no—
390C2×C42—612no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—612no—
394C2×C42—612no—
395C2×C42—614no—
396C2×C42—1224no—
397C2×C42C22×D6111yescharacteristic
398C2×C42C22×D6114yes—
399C2×C42C22×D6111yescharacteristic
400C2×C42C22×D6111yescharacteristic
401#95—612no—
402#95—612no—
403#95—614no—
404#95—614no—
405#95—614no—
406#95—614no—
407#95—614no—
408#95—614no—
409#95—614no—
410#95—614no—
411#95—614no—
412#95C22×D6112yes—
413#95C22×D6112yes—
414#95C22×D6114yes—
415#95C22×D6114yes—
416C24—611no—
417C2×Q8—618no—
418C2×Q8—1224no—
419C2×Q8—1224no—
420C22×C4—611no—
421C22×C4—612no—
422C22×C4—611no—
423C22×C4—614no—
424C22×C4—611no—
425C22×C4—614no—
426C22×C4—612no—
427C22×C4—614no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1224no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—2442no—
434C22×C4C23×D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436C22×C4C23×D6114yes—
437C22×C4C23×D6114yes—
438C22×C4C23×D6112yes—
439C22×C4C23×D6112yes—
440C22×C4C23×D6111yescharacteristic
441C4⋊C4—24no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—618no—
445C4⋊C4—6116no—
446C4⋊C4—6116no—
447C4⋊C4—6116no—
448C4⋊C4—1228no—
449C4⋊C4—1224no—
450C4⋊C4—1228no—
451C4⋊C4—1224no—
452C4⋊C4—2444no—
453C4⋊C4C2×C4×D6118yes—
454C4⋊C4C2×C4×D61116yes—
455C22⋊C4—618no—
456C22⋊C4—6116no—
457C22⋊C4—1224no—
458C22⋊C4—2444no—
459C42—22no—
460C42—44no—
461C42—614no—
462C42—6116no—
463C42—614no—
464C42—1222no—
465C42—1222no—
466C42—1224no—
467C42—1228no—
468C42—1224no—
469C42—1228no—
470C42—1222no—
471C42—1228no—
472C42—2442no—
473C42—2442no—
474C42—2444no—
475C42C2×C4×D6114yes—
476C42#1175822614yes—
477C23—612no—
478C23—1221no—
479C23—1221no—
480C23—1222no—
481C23—1222no—
482C23C24×D6111yescharacteristic
483Q8—12216no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—612no—
498C2×C4—614no—
499C2×C4—614no—
500C2×C4—618no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—1221no—
504C2×C4—1224no—
505C2×C4—1224no—
506C2×C4—1221no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1228no—
511C2×C4—1222no—
512C2×C4—1224no—
513C2×C4—1222no—
514C2×C4—1224no—
515C2×C4—1224no—
516C2×C4—1222no—
517C2×C4—2444no—
518C2×C4—2444no—
519C2×C4C22×C4×D6118yes—
520C2×C4C22×C4×D6114yes—
521C2×C4C22×C4×D6112yes—
522C2×C4#41×D6112yes—
523C2×C4C2×#1175822614yes—
524C2×C4C2×#1175822612yes—
525C22—1222no—
526C22—1224no—
527C22—2441no—
528C22—2442no—
529C22—2441no—
530C22C23×C4×D6111yescharacteristic
531C22C2×#41×D6111yescharacteristic
532C22C22×#1175822611yescharacteristic
533C222−1+4×D6112yes—
534C22#1684059712yes—
535C4—24no—
536C4—28no—
537C4—24no—
538C4—44no—
539C4—41no—
540C4—41no—
541C4—614no—
542C4—1222no—
543C4—1222no—
544C4—1228no—
545C4—1224no—
546C4—2442no—
547C4—2444no—
548C4C4×#1175822614yes—
549C2—2442no—
550C2C2×C4×#1175822611yescharacteristic
551C2#2145903012yes—
552C2#541×D6111yescharacteristic
553C2#2145920311yescharacteristic
554C2#2145925312yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2330C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2330D61
hypocenterC61C4×#541
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #21448622C2semidirect
4. #21448538C2semidirect
6. #21448466C2semidirect
8. #21448386C2semidirect
9. #125×Dic61C2semidirect
10. #21448378C2semidirect
13. C4×#14519742C2semidirect
15. C61×#2330C2semidirect
16. #16840375C4semidirect
76. #14519777C2×C4semidirect
78. #14519768C2×C4semidirect
82. #14519742C2×C4semidirect
124. #125×C61C2×C4semidirect
271. Dic61#541semidirect
275. #2330D61semidirect
349. C61C4×#541semidirect
357. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403752
#214486222
#214485932
#214485382
#214485022
#214484662
#214484272
#214483862
#125×Dic612
#214483782
#214483692
#214483492
C4×#145197422
#214481582
C61×#23302
C4×#54161