Finite Group Databasebeta

← #29840467

Subgroups — #29840467

Order 15616.

2110 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×#16840318C211yesmaximal, characteristic
2#21448903C214yesmaximal
3#21448896C214yesmaximal
4#21448773C214yesmaximal, complemented
5#21448772C214yesmaximal, complemented
6#21448566C212yesmaximal, complemented
7#21448563C212yesmaximal, complemented
8#21448441C212yesmaximal
9#125×Dic61C211yesmaximal, characteristic, complemented
10C4×#14519769C211yesmaximal, characteristic, complemented
11#21448374C211yesmaximal, characteristic, complemented
12#382×D61C211yesmaximal, characteristic
13#21448173C212yesmaximal
14C4×#14519734C211yesmaximal, characteristic
15C4×#125×C61C211yesmaximal, characteristic, complemented
16#16840318C4116yescomplemented
17C2×#14519776C2212yes—
18C2×#14519769C2211yescharacteristic
19C2×#14519755C2214yes—
20C2×#14519742C2214yes—
21C2×#14519735C2212yes—
22C2×#14519734C2211yescharacteristic
23C2×#14519734C2211yescharacteristic
24C2×C4⋊C4×D61C2212yes—
25C2×C42×D61C2211yescharacteristic
26#16839668C2214yes—
27#16839667C2214yes—
28#16839666C2214yes—
29#16839666C2214yes—
30#16839661C2212yes—
31#16839660C2212yes—
32#16839658C2212yes—
33#16839649C2214yes—
34#16839648C2214yes—
35#16839646C2214yes—
36#16839646C2214yes—
37#16839644C2214yes—
38#16839644C2214yes—
39#16839643C2214yes—
40#16839636C2214yes—
41#16839633C2214yes—
42#16839631C2214yes—
43#16839628C2211yescharacteristic
44#16839627C2212yes—
45#16839626C2214yes—
46#16839626C2214yes—
47C4⋊C4×Dic61C2214yes—
48C4⋊C4×Dic61C2212yes—
49C4⋊C4×Dic61C2212yes—
50C4×#11758196C2212yes—
51C4×#11758196C2214yes—
52C4×#11758196C2212yes—
53C4×#11758196C2214yes—
54C4×C4⋊Dic61C2212yes—
55#16839613C2214yes—
56#16839613C2214yes—
57C4×#11758198C2212yes—
58C4×#11758198C2212yes—
59C4×#11758198C2214yes—
60#16839609C2214yes—
61#16839608C2211yescharacteristic
62#16839608C2211yescharacteristic
63#95×D61C2214yes—
64C42×Dic61C2211yescharacteristic
65C2×#125×C61C2211yescharacteristic
66C61×#389C2212yes—
67C61×#387C2214yes—
68C61×#382C2211yescharacteristic
69C4×C4⋊C4×C61C2214yes—
70C4×C4⋊C4×C61C2212yes—
71C43×C61C2211yescharacteristic
72C22×Dic122C2314yes—
73C22×C4×D61C2314yes—
74C22×C4×D61C2312yes—
75C22×C4×D61C2311yescharacteristic
76#14519776C2×C4116yescomplemented
77#14519769C2×C418yescomplemented
78#14519755C2×C4132yes—
79#14519742C2×C4132yescomplemented
80#14519735C2×C4116yes—
81#14519734—24no—
82#14519734Q818yescomplemented
83C2×C4⋊Dic61C2312yes—
84C2×C4⋊Dic61C2314yes—
85C2×C4⋊Dic61C2312yes—
86C2×#11758196C2314yes—
87C2×#11758196C2312yes—
88C2×#11758196C2314yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C2×#11758196C2311yescharacteristic
93C2×#11758196C2311yescharacteristic
94C4⋊C4×D61—28no—
95C2×#11758198C2312yes—
96C2×#11758198C2314yes—
97C2×#11758198C2311yescharacteristic
98C2×#11758198C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2312yes—
101C2×C4×Dic61C2312yes—
102C2×C4×Dic61C2314yes—
103C2×C4×Dic61C2314yes—
104C2×C4×Dic61C2314yes—
105C2×C4×Dic61C2312yes—
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2314yes—
108C2×C4×Dic61C2311yescharacteristic
109C42×D61C2×C418yes—
110#14519684C2314yes—
111#14519684C2314yes—
112#14519684C2314yes—
113#14519684C2312yes—
114#14519684C2312yes—
115#14519684C2312yes—
116#14519684C2312yes—
117#14519684C2314yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121#14519684C2314yes—
122#14519684C2314yes—
123#125×C61C2×C418yescomplemented
124C2×C4⋊C4×C61C2312yes—
125C2×C4⋊C4×C61C2314yes—
126C2×C4⋊C4×C61C2312yes—
127C2×C4⋊C4×C61C2314yes—
128C2×C42×C61C2311yescharacteristic
129C2×C42×C61C2312yes—
130C2×C42×C61C2312yes—
131C2×C42×C61C2314yes—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2311yescharacteristic
134#95×C61C2314yes—
135#95×C61C2314yes—
136#95×C61C2314yes—
137C23×D61C2411yescharacteristic
138C22×C4×C61C2414yes—
139C22×C4×C61C2412yes—
140C22×C4×C61C2412yes—
141C22×C4×C61C2414yes—
142C22×C4×C61C2411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61C2414yes—
146C22×Dic61C2412yes—
147C22×Dic61C2412yes—
148C22×Dic61C2414yes—
149C22×Dic61C2411yescharacteristic
150C22×Dic61C2411yescharacteristic
151C22×Dic61C2411yescharacteristic
152C2×C4×D61—28no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×C4×D61C22×C418yes—
158C2×C4×D61C22×C414yes—
159C2×C4×D61C2×Q814yes—
160C2×C4×D61#4114yes—
161C2×Dic122—44no—
162C2×Dic122—44no—
163C2×Dic122C22×C4116yes—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—44no—
166C4⋊C4×C61C22×C418yes—
167C4⋊C4×C61C22×C4116yes—
168C42×C61—22no—
169C42×C61—42no—
170C42×C61—44no—
171C42×C61—42no—
172C42×C61C22×C414yes—
173C42×C61C2×Q814yes—
174#11758198—24no—
175#11758198C22×C4116yes—
176#11758198C2×Q814yes—
177#11758198C2×Q814yes—
178C4⋊Dic61—24no—
179C4⋊Dic61—44no—
180C4⋊Dic61C22×C418yes—
181#11758196—28no—
182#11758196—24no—
183#11758196—44no—
184#11758196—44no—
185#11758196C22×C4116yes—
186#11758196C22×C4116yes—
187#11758196C2×Q814yes—
188#11758196C2×Q814yes—
189C4×Dic61—22no—
190C4×Dic61—44no—
191C4×Dic61—42no—
192C4×Dic61—44no—
193C4×Dic61—42no—
194C4×Dic61—44no—
195C4×Dic61C22×C418yes—
196C4×Dic61C22×C414yes—
197C4×Dic61C22×C4116yes—
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—22no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61C23×C414yes—
218C2×C4×C61C23×C418yes—
219C2×C4×C61C23×C412yes—
220C2×C4×C61C22×Q812yes—
221C2×C4×C61C22×Q812yes—
222C2×C4×C61C22×Q812yes—
223C2×C4×C61C2×#4112yes—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61C23×C414yes—
237C2×Dic61C23×C418yes—
238C2×Dic61C23×C412yes—
239C2×Dic61C22×Q812yes—
240C2×Dic61C22×Q812yes—
241C2×Dic61C22×Q812yes—
242C2×Dic61C2×#4112yes—
243C4×D61—48no—
244C4×D61—88no—
245C4×D61C4×Q8116yes—
246Dic122—416no—
247#34215—611nomaximal, Sylow
248C22×C61C24×C411yescharacteristic
249C22×C61C23×Q811yescharacteristic
250C22×C61C22×#4111yescharacteristic
251C22×C61#58512yes—
252C22×C61#58512yes—
253C2×D61—24no—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—24no—
257C2×D61C2×C4×Q814yes—
258C4×C61—28no—
259C4×C61—24no—
260C4×C61—22no—
261C4×C61—22no—
262C4×C61—44no—
263C4×C61—42no—
264C4×C61C2×C4×Q814yes—
265Dic61—24no—
266Dic61—28no—
267Dic61—22no—
268Dic61—22no—
269Dic61—44no—
270Dic61—42no—
271Dic61C2×C4×Q814yes—
272C2×#533—611no—
273#2368—614no—
274#2368—614no—
275#2355—614no—
276#2355—614no—
277#2331—612no—
278#2331—612no—
279#2317—612no—
280C4×#125—611no—
281C4×#125—611no—
282C4×#125—611no—
283C4×#125D6111yescharacteristic, complemented
284C2×#382—611no—
285#2274—612no—
286C4×#117—611no—
287C2×C61C22×C4×Q811yescharacteristic
288C2×C61#342812yes—
289C2×C61#350411yescharacteristic
290C2×C61#350411yescharacteristic
291C2×C61#351012yes—
292D61—24no—
293#533—6116no—
294C2×C4⋊Q8—612no—
295C2×#125—611no—
296C2×#125C2×D6111yescharacteristic
297C2×C22⋊Q8—614no—
298C2×C4×Q8—614no—
299C2×#117—612no—
300C2×#117—611no—
301C2×#117—611no—
302C22×C4⋊C4—612no—
303C22×C42—611no—
304#391—614no—
305#391—614no—
306#391—614no—
307#391—614no—
308#389—612no—
309#389—612no—
310#389—612no—
311#389C2×D6112yes—
312#387—614no—
313#387—614no—
314#387—614no—
315#387—614no—
316#387—614no—
317#387—614no—
318#387—614no—
319#387C2×D6114yes—
320#385—614no—
321#385—614no—
322#384—614no—
323#382—612no—
324#382—611no—
325#382—614no—
326#382—614no—
327#382C2×D6111yescharacteristic
328C4×C4⋊C4—612no—
329C4×C4⋊C4—612no—
330C4×C4⋊C4—614no—
331C4×C4⋊C4—612no—
332C4×C4⋊C4—614no—
333C4×C4⋊C4—612no—
334C4×C4⋊C4—612no—
335C4×C4⋊C4—614no—
336C4×C4⋊C4—614no—
337C4×C4⋊C4—614no—
338C4×C4⋊C4C2×D6112yes—
339C4×C4⋊C4C2×D6114yes—
340C4×C22⋊C4—612no—
341C4×C22⋊C4—612no—
342C4×C22⋊C4—614no—
343C4×C22⋊C4—614no—
344#376—611no—
345#376—611no—
346C2×#95—614no—
347C43—611no—
348C43C2×D6111yescharacteristic
349C61#3421511yescharacteristic, complemented, Sylow
350C22×Q8—614no—
351C23×C4—611no—
352C23×C4—612no—
353C23×C4—614no—
354C4⋊Q8—6116no—
355#125—618no—
356#125C4×D6118yescomplemented
357C22⋊Q8—6132no—
358C4×Q8—6132no—
359#117—6116no—
360#117—618no—
361#117—1224no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
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⋊C4—1228no—
374C2×C4⋊C4C22×D6112yes—
375C2×C4⋊C4C22×D6114yes—
376C2×C4⋊C4C22×D6112yes—
377C2×C4⋊C4C22×D6114yes—
378C2×C22⋊C4—614no—
379C2×C22⋊C4—611no—
380C2×C22⋊C4—611no—
381C2×C22⋊C4—612no—
382C2×C42—614no—
383C2×C42—611no—
384C2×C42—612no—
385C2×C42—611no—
386C2×C42—618no—
387C2×C42—611no—
388C2×C42—612no—
389C2×C42—612no—
390C2×C42—614no—
391C2×C42—614no—
392C2×C42—614no—
393C2×C42C22×D6111yescharacteristic
394C2×C42C22×D6112yes—
395C2×C42C22×D6112yes—
396C2×C42C22×D6114yes—
397C2×C42C22×D6111yescharacteristic
398C2×C42C22×D6111yescharacteristic
399#95—614no—
400#95—614no—
401#95—614no—
402#95—612no—
403#95—612no—
404#95—612no—
405#95—612no—
406#95—614no—
407#95—614no—
408#95—614no—
409#95—614no—
410#95—614no—
411#95—614no—
412#95C22×D6114yes—
413#95C22×D6114yes—
414#95C22×D6114yes—
415C24—611no—
416C2×Q8—6116no—
417C2×Q8—2444no—
418C2×Q8—2444no—
419C22×C4—611no—
420C22×C4—614no—
421C22×C4—614no—
422C22×C4—614no—
423C22×C4—612no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—614no—
427C22×C4—618no—
428C22×C4—614no—
429C22×C4—612no—
430C22×C4—1228no—
431C22×C4—2444no—
432C22×C4—2444no—
433C22×C4—2442no—
434C22×C4—2442no—
435C22×C4C23×D6114yes—
436C22×C4C23×D6112yes—
437C22×C4C23×D6112yes—
438C22×C4C23×D6114yes—
439C22×C4C23×D6111yescharacteristic
440C22×C4C23×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442C4⋊C4—24no—
443C4⋊C4—44no—
444C4⋊C4—6116no—
445C4⋊C4—6116no—
446C4⋊C4—618no—
447C4⋊C4—614no—
448C4⋊C4—614no—
449C4⋊C4—1228no—
450C4⋊C4—1224no—
451C4⋊C4—1224no—
452C4⋊C4—2444no—
453C4⋊C4—2444no—
454C4⋊C4—2444no—
455C4⋊C4C2×C4×D6118yes—
456C4⋊C4C2×C4×D61116yes—
457C22⋊C4—6116no—
458C22⋊C4—614no—
459C22⋊C4—614no—
460C22⋊C4—1224no—
461C42—22no—
462C42—42no—
463C42—44no—
464C42—42no—
465C42—618no—
466C42—614no—
467C42—6116no—
468C42—614no—
469C42—1222no—
470C42—2444no—
471C42—2442no—
472C42—2442no—
473C42—2444no—
474C42—2444no—
475C42C2×C4×D6114yes—
476C42Q8×D6114yes—
477C23—612no—
478C23—612no—
479C23—612no—
480C23—1222no—
481C23—1222no—
482C23C24×D6111yescharacteristic
483Q8—24416no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—6116no—
500C2×C4—614no—
501C2×C4—618no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1224no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1224no—
515C2×C4—1224no—
516C2×C4—4888no—
517C2×C4—2448no—
518C2×C4C22×C4×D6114yes—
519C2×C4C22×C4×D6118yes—
520C2×C4C22×C4×D6112yes—
521C2×C4C2×Q8×D6112yes—
522C2×C4C2×Q8×D6112yes—
523C2×C4C2×Q8×D6112yes—
524C2×C4#41×D6112yes—
525C22—614no—
526C22—1224no—
527C22—1222no—
528C22—1222no—
529C22—1224no—
530C22C23×C4×D6111yescharacteristic
531C22C22×Q8×D6111yescharacteristic
532C22C2×#41×D6111yescharacteristic
533C22#1684059812yes—
534C22#1684059812yes—
535C4—24no—
536C4—28no—
537C4—22no—
538C4—22no—
539C4—42no—
540C4—44no—
541C4—614no—
542C4—1224no—
543C4—1228no—
544C4—1222no—
545C4—1222no—
546C4—2442no—
547C4—2444no—
548C4C4×Q8×D6114yes—
549C2—1224no—
550C2C2×C4×Q8×D6111yescharacteristic
551C2#2145901812yes—
552C2#2145964011yescharacteristic
553C2#2145964311yescharacteristic
554C2#2145970812yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC4×#125×C61C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenterC4×#125D61
hypocenterC61#34215
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
4. #21448773C2semidirect
5. #21448772C2semidirect
6. #21448566C2semidirect
7. #21448563C2semidirect
9. #125×Dic61C2semidirect
10. C4×#14519769C2semidirect
11. #21448374C2semidirect
15. C4×#125×C61C2semidirect
16. #16840318C4semidirect
76. #14519776C2×C4semidirect
77. #14519769C2×C4semidirect
79. #14519742C2×C4semidirect
82. #14519734Q8semidirect
123. #125×C61C2×C4semidirect
283. C4×#125D61semidirect
349. C61#34215semidirect
356. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
#214489032
#214488962
#214487732
#214487722
#214485662
#214485632
#214484412
#125×Dic612
C4×#145197692
#214483742
#382×D612
#214481732
C4×#145197342
C4×#125×C612
#3421561