Finite Group Databasebeta

← #29843112

Subgroups — #29843112

Order 15616.

2438 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840313C211yesmaximal, characteristic
2#21449000C211yesmaximal, characteristic, complemented
3#21448995C211yesmaximal, characteristic, complemented
4#21448995C211yesmaximal, characteristic, complemented
5#21448865C214yesmaximal, complemented
6#21448758C214yesmaximal, complemented
7#21448648C214yesmaximal, complemented
8#21448631C214yesmaximal, complemented
9#21448518C214yesmaximal, complemented
10#21448451C214yesmaximal, complemented
11#21448216C211yesmaximal, characteristic, complemented
12#383×D61C211yesmaximal, characteristic, complemented
13C61×#2381C211yesmaximal, characteristic, complemented
14C22×#11758226C2212yes—
15#16840313C4116yescomplemented
16C2×#14519774C2211yescharacteristic
17C2×#14519766C2212yes—
18C2×#14519752C2212yes—
19C2×#14519752C2212yes—
20C2×C4×D244C2214yes—
21C2×#14519739C2214yes—
22C2×#14519739C2212yes—
23C2×#14519739C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839669C2214yes—
26#16839654C2214yes—
27#16839653C2214yes—
28#16839652C2214yes—
29#16839651C2214yes—
30#16839648C2214yes—
31#16839648C2214yes—
32#16839640C2214yes—
33#16839640C2214yes—
34#16839640C2214yes—
35#16839640C2214yes—
36#16839640C2214yes—
37#16839640C2214yes—
38#16839640C2214yes—
39#16839640C2214yes—
40#16839636C2214yes—
41#16839633C2214yes—
42#16839630C2211yescharacteristic
43#16839630C2211yescharacteristic
44#16839629C2212yes—
45#16839629C2212yes—
46#16839629C2212yes—
47#16839626C2212yes—
48#16839624C2214yes—
49#16839624C2212yes—
50#16839621C2212yes—
51#16839619C2214yes—
52#16839613C2212yes—
53C4×#11758198C2214yes—
54C4×#11758198C2214yes—
55#16839609C2212yes—
56#16839609C2214yes—
57#16839607C2212yes—
58#95×D61C2214yes—
59#95×D61C2212yes—
60#95×D61C2212yes—
61#95×D61C2212yes—
62C2×C4⋊Q8×C61C2211yescharacteristic
63C61×#391C2214yes—
64C61×#387C2214yes—
65C61×#387C2214yes—
66C61×#383C2211yescharacteristic
67C61×#383C2211yescharacteristic
68C2×#11758226—44no—
69C2×#11758226—44no—
70C2×#11758226C2×C4116yescomplemented
71C22×D244C2314yes—
72C22×D244C2312yes—
73C22×D244C2312yes—
74C22×D244C2312yes—
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2314yes—
77C22×C4×D61C2314yes—
78C22×C4×D61C2314yes—
79C22×C4×D61C2311yescharacteristic
80C22×C4×D61C2311yescharacteristic
81C22×C4×D61C2314yes—
82#14519774C2×C418yescomplemented
83#14519766C2×C4116yescomplemented
84#14519752C2×C4116yescomplemented
85#14519752C2×C4116yescomplemented
86C4×D244—48no—
87#14519739—48no—
88#14519739C2×C4116yescomplemented
89#14519739C2×C4116yescomplemented
90C2×C4⋊Dic61C2314yes—
91C2×#11758196C2314yes—
92C2×#11758198C2314yes—
93C2×#11758198C2314yes—
94C2×#11758198C2314yes—
95C2×#11758198C2314yes—
96C2×#11758198C2314yes—
97C2×#11758198C2314yes—
98C2×#11758198C2314yes—
99C2×#11758198C2314yes—
100C2×#11758198C2314yes—
101C2×C4×Dic61C2314yes—
102C2×C4×Dic61C2314yes—
103C2×C4×Dic61C2311yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C42×D61C2×C418yescomplemented
107#14519684C2314yes—
108#14519684C2314yes—
109#14519684C2314yes—
110#14519684C2314yes—
111#14519684C2312yes—
112#14519684C2312yes—
113#14519684C2312yes—
114#14519684C2314yes—
115#14519684C2312yes—
116#14519684C2312yes—
117#14519684C2314yes—
118#14519684C2312yes—
119C22×Q8×C61C2312yes—
120C4⋊Q8×C61C2×C418yescomplemented
121C2×C4⋊C4×C61C2314yes—
122C2×C4⋊C4×C61C2312yes—
123C2×C4⋊C4×C61C2312yes—
124C2×C42×C61C2314yes—
125C2×C42×C61C2311yescharacteristic
126#95×C61C2312yes—
127#95×C61C2312yes—
128#95×C61C2312yes—
129#95×C61C2312yes—
130#95×C61C2312yes—
131#95×C61C2312yes—
132#95×C61C2314yes—
133#95×C61C2314yes—
134C23×D61C2414yes—
135C23×D61C2411yescharacteristic
136C2×Q8×C61—42no—
137C2×Q8×C61—42no—
138C2×Q8×C61C22×C418yes—
139C22×C4×C61C2414yes—
140C22×C4×C61C2414yes—
141C22×C4×C61C2411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×C4×C61C2414yes—
145C22×Dic61C2414yes—
146C22×Dic61C2414yes—
147C22×Dic61C2411yescharacteristic
148C22×Dic61C2411yescharacteristic
149C22×Dic61C2411yescharacteristic
150#11758226—432no—
151C2×D244—28no—
152C2×D244—28no—
153C2×D244—42no—
154C2×D244—42no—
155C2×D244—42no—
156C2×D244—42no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244C22×C4116yes—
160C2×D244C22×C418yes—
161C2×D244C22×C418yes—
162C2×D244C22×C418yes—
163C2×C4×D61—24no—
164C2×C4×D61—28no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—44no—
168C2×C4×D61—44no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61—44no—
173C2×C4×D61—44no—
174C2×C4×D61—44no—
175C2×C4×D61—44no—
176C2×C4×D61—44no—
177C2×C4×D61C22×C414yes—
178C2×C4×D61C22×C4116yes—
179C2×C4×D61C22×C414yes—
180C2×C4×D61C22×C414yes—
181C4⋊C4×C61—44no—
182C4⋊C4×C61C22×C418yes—
183C4⋊C4×C61C22×C418yes—
184C42×C61—44no—
185C42×C61C22×C414yes—
186#11758198—28no—
187#11758198—44no—
188#11758198—44no—
189#11758198—44no—
190#11758198—44no—
191#11758198—44no—
192#11758198—44no—
193#11758198C22×C4116yes—
194#11758198C22×C4116yes—
195C4⋊Dic61—44no—
196#11758196—44no—
197C4×Dic61—44no—
198C4×Dic61—44no—
199C4×Dic61C22×C414yes—
200C4×Dic61C22×C414yes—
201C4×Dic61C22×C414yes—
202C23×C61C2511yescharacteristic
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—24no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61C23×C418yes—
212C22×D61C23×C412yes—
213C22×D61H(5,𝔽2)14yes—
214Q8×C61—48no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61C23×C412yes—
227C2×C4×C61C23×C412yes—
228C2×C4×C61C23×C412yes—
229C2×C4×C61C23×C418yes—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61C23×C412yes—
240C2×Dic61C23×C412yes—
241C2×Dic61C23×C412yes—
242D244—232no—
243D244—48no—
244D244—48no—
245D244—48no—
246C4×D61—28no—
247C4×D61—44no—
248C4×D61—44no—
249C4×D61—416no—
250C4×D61—88no—
251C4×D61—88no—
252C4×D61—88no—
253#34388—611nomaximal, Sylow
254C22×C61C24×C411yescharacteristic
255C22×C61C2×H(5,𝔽2)12yes—
256C22×C61#58512yes—
257C22×C61#58512yes—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—28no—
261C2×D61—28no—
262C2×D61—28no—
263C2×D61—28no—
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—44no—
268C2×D61#51814yes—
269C4×C61—22no—
270C4×C61—28no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—44no—
274C4×C61—44no—
275Dic61—22no—
276Dic61—22no—
277Dic61—22no—
278Dic61—44no—
279Dic61—44no—
280C2×#532—611no—
281#2381—611no—
282#2381D6111yescharacteristic, complemented
283#2380—611no—
284#2380—611no—
285#2364—614no—
286#2353—614no—
287#2339—614no—
288#2337—614no—
289#2325—614no—
290#2318—614no—
291#2283—611no—
292C2×#383—611no—
293C2×C61C2×#51811yescharacteristic
294C2×C61#342811yescharacteristic
295C2×C61#342811yescharacteristic
296C2×C61#356014yes—
297D61—24no—
298D61—48no—
299C22×#41—612no—
300#532—6116no—
301C2×C4⋊Q8C2×D6111yescharacteristic
302C2×#127—611no—
303C2×#124—612no—
304C2×#121—612no—
305C2×#121—612no—
306C2×C4×D4—612no—
307C2×C4×D4—612no—
308C2×C4×D4—614no—
309C2×C4×D4—614no—
310C22×C42—611no—
311#391—614no—
312#391C2×D6114yes—
313#388—614no—
314#388—614no—
315#388—614no—
316#388—614no—
317#387—614no—
318#387—614no—
319#387C2×D6114yes—
320#387C2×D6114yes—
321#386—614no—
322#386—614no—
323#386—614no—
324#386—614no—
325#386—614no—
326#386—614no—
327#386—614no—
328#386—614no—
329#385—614no—
330#385—614no—
331#383—611no—
332#383—611no—
333#383—612no—
334#383—612no—
335#383—612no—
336#383C2×D6111yescharacteristic
337#383C2×D6111yescharacteristic
338#382—612no—
339#381—614no—
340#381—612no—
341#380—614no—
342#380—612no—
343C4×C4⋊C4—612no—
344C4×C22⋊C4—614no—
345C4×C22⋊C4—614no—
346C4×C22⋊C4—614no—
347C4×C22⋊C4—612no—
348#376—612no—
349C2×#95—614no—
350C2×#95—612no—
351C2×#95—612no—
352C2×#95—612no—
353C61#3438811yescharacteristic, complemented, Sylow
354C2×#41—6116no—
355C2×#41—2444no—
356C2×#41—2444no—
357C22×Q8C22×D6112yes—
358C22×D4—614no—
359C22×D4—612no—
360C22×D4—612no—
361C22×D4—612no—
362C23×C4—611no—
363C23×C4—614no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—614no—
367C23×C4—614no—
368C23×C4—614no—
369C4⋊Q8C4×D6118yescomplemented
370#127—618no—
371#124—6116no—
372#121—6116no—
373#121—6116no—
374C4×D4—6116no—
375C4×D4—6116no—
376C4×D4—2448no—
377C4×D4—2448no—
378C2×C4⋊C4—614no—
379C2×C4⋊C4—614no—
380C2×C4⋊C4C22×D6114yes—
381C2×C4⋊C4C22×D6112yes—
382C2×C4⋊C4C22×D6112yes—
383C2×C22⋊C4—614no—
384C2×C22⋊C4—614no—
385C2×C22⋊C4—614no—
386C2×C22⋊C4—614no—
387C2×C22⋊C4—614no—
388C2×C22⋊C4—614no—
389C2×C22⋊C4—614no—
390C2×C22⋊C4—614no—
391C2×C22⋊C4—614no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—618no—
396C2×C42—614no—
397C2×C42—614no—
398C2×C42C22×D6114yes—
399C2×C42C22×D6111yescharacteristic
400#95—614no—
401#95—614no—
402#95—614no—
403#95—614no—
404#95—612no—
405#95—612no—
406#95—612no—
407#95—614no—
408#95—612no—
409#95—614no—
410#95—612no—
411#95—612no—
412#95C22×D6112yes—
413#95C22×D6112yes—
414#95C22×D6112yes—
415#95C22×D6114yes—
416#95C22×D6112yes—
417#95C22×D6114yes—
418#95C22×D6112yes—
419#95C22×D6112yes—
420C24—611no—
421C24—614no—
422#41—24432no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8C2×C4×D6118yes—
426C2×D4—6116no—
427C2×D4—618no—
428C2×D4—618no—
429C2×D4—618no—
430C2×D4—1228no—
431C2×D4—1228no—
432C2×D4—2442no—
433C2×D4—2442no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4—2442no—
437C2×D4—2442no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—614no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—6116no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—1224no—
448C22×C4—1228no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4—2444no—
452C22×C4—2444no—
453C22×C4—2444no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4—2444no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4—2444no—
460C22×C4—2444no—
461C22×C4C23×D6114yes—
462C22×C4C23×D6114yes—
463C22×C4C23×D6111yescharacteristic
464C22×C4C23×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466C22×C4C23×D6114yes—
467C4⋊C4—44no—
468C4⋊C4—2444no—
469C4⋊C4—2444no—
470C4⋊C4C2×C4×D6118yes—
471C4⋊C4C2×C4×D6118yes—
472C22⋊C4—6116no—
473C22⋊C4—6116no—
474C22⋊C4—1228no—
475C22⋊C4—2444no—
476C22⋊C4—2444no—
477C22⋊C4—2444no—
478C22⋊C4—2444no—
479C22⋊C4—2444no—
480C22⋊C4—2444no—
481C42—44no—
482C42—614no—
483C42—614no—
484C42—614no—
485C42—2444no—
486C42—2444no—
487C42C2×C4×D6114yes—
488C23—612no—
489C23—614no—
490C23—618no—
491C23—1222no—
492C23—1222no—
493C23—1224no—
494C23—1224no—
495C23—1224no—
496C23—1224no—
497C23—1224no—
498C23—1224no—
499C23C24×D6111yescharacteristic
500Q8—48no—
501D4—12232no—
502D4—2448no—
503D4—2448no—
504D4—2448no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—1228no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—1224no—
529C2×C4—4888no—
530C2×C4—4888no—
531C2×C4—4888no—
532C2×C4—2444no—
533C2×C4—2444no—
534C2×C4—24416no—
535C2×C4C22×C4×D6112yes—
536C2×C4C22×C4×D6112yes—
537C2×C4C22×C4×D6112yes—
538C2×C4C22×C4×D6118yes—
539C22—614no—
540C22—1222no—
541C22—1222no—
542C22—1228no—
543C22—1228no—
544C22—1228no—
545C22—1228no—
546C22—2444no—
547C22—2444no—
548C22—2444no—
549C22—2444no—
550C22C23×C4×D6111yescharacteristic
551C22H(5,𝔽2)×D6112yes—
552C22#1684059912yes—
553C22#1684059912yes—
554C4—22no—
555C4—28no—
556C4—22no—
557C4—22no—
558C4—44no—
559C4—44no—
560C4—1222no—
561C4—1222no—
562C4—1222no—
563C4—2444no—
564C4—2444no—
565C2—1224no—
566C2—2448no—
567C2#518×D6111yescharacteristic
568C2#2145902111yescharacteristic
569C2#2145902111yescharacteristic
570C2#2146036214yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2381C2
socle, cosocleC23×C61C25
prime radicalC2×#16840313C2
hypercenter#2381D61
hypocenterC61#34388
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. #21449000C2semidirect
3. #21448995C2semidirect
4. #21448995C2semidirect
5. #21448865C2semidirect
6. #21448758C2semidirect
7. #21448648C2semidirect
8. #21448631C2semidirect
9. #21448518C2semidirect
10. #21448451C2semidirect
11. #21448216C2semidirect
12. #383×D61C2semidirect
13. C61×#2381C2semidirect
15. #16840313C4semidirect
70. C2×#11758226C2×C4semidirect
82. #14519774C2×C4semidirect
83. #14519766C2×C4semidirect
84. #14519752C2×C4semidirect
85. #14519752C2×C4semidirect
88. #14519739C2×C4semidirect
89. #14519739C2×C4semidirect
106. C42×D61C2×C4semidirect
120. C4⋊Q8×C61C2×C4semidirect
282. #2381D61semidirect
353. C61#34388semidirect
369. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403132
#214490002
#214489952
#214488652
#214487582
#214486482
#214486312
#214485182
#214484512
#214482162
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C61×#23812
#3438861