Finite Group Databasebeta

← #29658491

Subgroups — #29658491

Order 15616.

1634 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic, complemented
4#21444998C214yesmaximal, complemented
5#21444935C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×D42×C61C211yesmaximal, characteristic
10#16840319C2211yescharacteristic, complemented
11#16840134C2212yescomplemented
12#16840108C2212yescomplemented
13#16840092—21no—
14#16840062—22no—
15#16840052C2212yescomplemented
16#16840052C2212yescomplemented
17#16840042C2212yescomplemented
18#16839967C2212yescomplemented
19#16839959—22no—
20#16839913—22no—
21#16839913—21no—
22C4×#11758200—22no—
23#16839815C2211yescharacteristic, complemented
24#16839815C2211yescharacteristic, complemented
25#16839802—21no—
26#16839786C2211yescharacteristic, complemented
27#16839767—21no—
28#16839506C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C23×D4×C61—21no—
32C23×D4×C61—21no—
33D42×C61—22no—
34D42×C61—22no—
35D42×C61—22no—
36D42×C61C2214yes—
37C2×#127×C61C2211yescharacteristic
38C2×#121×C61—21no—
39C2×#121×C61—21no—
40C2×#120×C61—22no—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758200—22no—
44C2×#11758200—24no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic, complemented
47#14519777C2311yescharacteristic, complemented
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—22no—
55#14519708—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yescomplemented
60C4⋊#92418C2312yescomplemented
61#14519700—22no—
62#14519698—24no—
63#14519698—22no—
64#14519698—22no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yescomplemented
70C25×C61—22no—
71C22×D4×C61—24no—
72C22×D4×C61—24no—
73C22×D4×C61—22no—
74C22×D4×C61—22no—
75C22×D4×C61—24no—
76C22×D4×C61—24no—
77C22×D4×C61—21no—
78C22×D4×C61—21no—
79C22×D4×C61—48no—
80C22×D4×C61—44no—
81C22×D4×C61—44no—
82C22×D4×C61D411yescharacteristic
83C22×D4×C61D411yescharacteristic
84C22×D4×C61D412yes—
85C23×C4×C61—21no—
86C23×C4×C61—21no—
87#127×C61—21no—
88#127×C61—21no—
89#127×C61D412yes—
90#127×C61C2312yes—
91#121×C61—24no—
92#121×C61—24no—
93#121×C61—24no—
94#121×C61—24no—
95#120×C61—28no—
96#120×C61—28no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61D412yes—
102C4×D4×C61D412yes—
103C4×D4×C61C2312yes—
104C4×D4×C61C2312yes—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61—22no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61C2311yescharacteristic
111C24×C61—21no—
112C24×C61—21no—
113C24×C61—28no—
114C24×C61—22no—
115C24×C61—48no—
116C24×C61—48no—
117C24×C61—84no—
118C2×D4×C61—24no—
119C2×D4×C61—28no—
120C2×D4×C61—28no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—44no—
130C2×D4×C61—416no—
131C2×D4×C61—48no—
132C2×D4×C61—48no—
133C2×D4×C61—416no—
134C2×D4×C61—48no—
135C2×D4×C61—48no—
136C2×D4×C61—48no—
137C2×D4×C61—48no—
138C2×D4×C61—42no—
139C2×D4×C61—42no—
140C2×D4×C61—82no—
141C2×D4×C61SD1614yescomplemented
142C2×D4×C61C2×D412yes—
143C2×D4×C61C2×D412yes—
144C2×D4×C61C2×D414yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—44no—
152C22×C4×C61—42no—
153C22×C4×C61—42no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C22⋊C4×C61—28no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C42×C61C2×D411yescharacteristic
169C42×C61C2×D411yescharacteristic
170C42×C61C2×D411yescharacteristic
171C42×C61C2411yescharacteristic
172C22⋊Dic61—41no—
173#11758200—44no—
174#11758200—84no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179#11758196—41no—
180#11758196—41no—
181C4×Dic61—41no—
182#11758194—42no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61—24no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—44no—
195C23×C61—416no—
196C23×C61—416no—
197C23×C61—44no—
198C23×C61—42no—
199C23×C61—44no—
200C23×C61—88no—
201C23×C61—832no—
202C23×C61#12011yescharacteristic
203D4×C61—24no—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—24no—
207D4×C61—44no—
208D4×C61—44no—
209D4×C61—416no—
210D4×C61—416no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—44no—
214D4×C61—44no—
215D4×C61—82no—
216D4×C61C2×SD1614yes—
217D4×C61AGL(1,ℤ/8)14yes—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—44no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—44no—
228C2×C4×C61—44no—
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61C2×SD1612yes—
233C2×C4×C61AGL(1,ℤ/8)12yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×Dic61—41no—
238C2×Dic61—41no—
239Dic122—42no—
240Dic122—42no—
241#92418—42no—
242#92418—42no—
243#92418—42no—
244#18747—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—24no—
248C22×C61—24no—
249C22×C61—48no—
250C22×C61—48no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—88no—
259C22×C61—832no—
260C22×C61#45012yes—
261C22×C61C2×#12011yescharacteristic
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61#45012yes—
268C4×C61C22×SD1611yescharacteristic
269C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C4×C61C2×AGL(1,ℤ/8)12yes—
271Dic61—81no—
272Dic61—81no—
273C2×D42D6111yescharacteristic
274#3208—611no—
275#3101—611no—
276#3077—611no—
277#1639—614no—
278#1629—614no—
279#1545—611no—
280#1486—611no—
281#1478—611no—
282C2×C61—22no—
283C2×C61—44no—
284C2×C61—44no—
285C2×C61—84no—
286C2×C61C2×#45011yescharacteristic
287C2×C61#300311yescharacteristic
288C2×C61#301911yescharacteristic
289C23×D4—21no—
290C23×D4—21no—
291D42—22no—
292D42—22no—
293D42—22no—
294D42C2×D6114yes—
295#533—611no—
296C2×#127C2×D6111yescharacteristic
297C2×#121—21no—
298C2×#121—21no—
299C2×#120—22no—
300C2×C4×D4C2×D6111yescharacteristic
301C2×C4×D4C2×D6111yescharacteristic
302#492—612no—
303#486—612no—
304#483—1221no—
305#478—1222no—
306#476—612no—
307#476—612no—
308#474—612no—
309#461—612no—
310#460—1222no—
311#450—1221no—
312#450—1222no—
313C4×SD16—1222no—
314#431—611no—
315#431—611no—
316#428—1221no—
317#423—611no—
318#418—1221no—
319#331—614no—
320D4⋊C8—614no—
321D4⋊C8—614no—
322C61#1874711yescharacteristic, complemented, Sylow
323C25—22no—
324C22×D4—22no—
325C22×D4—22no—
326C22×D4—24no—
327C22×D4—24no—
328C22×D4—24no—
329C22×D4—24no—
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—48no—
333C22×D4—44no—
334C22×D4—44no—
335C22×D4C22⋊D6111yescharacteristic
336C22×D4C22⋊D6111yescharacteristic
337C22×D4C22⋊D6112yes—
338C23×C4—21no—
339C23×C4—21no—
340C2×SD16—1224no—
341C2×SD16—1222no—
342C2×M16—1221no—
343C4⋊Q8—611no—
344C4⋊Q8—611no—
345#127—21no—
346#127—21no—
347#127C22⋊D6112yes—
348#127C22×D6112yes—
349#125—1221no—
350C22⋊Q8—1221no—
351C22⋊Q8—1221no—
352#121—24no—
353#121—24no—
354#121—24no—
355#121—24no—
356#120—28no—
357#120—28no—
358C4×Q8—1221no—
359C4×Q8—1221no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4C22⋊D6112yes—
365C4×D4C22⋊D6112yes—
366C4×D4C22×D6112yes—
367C4×D4C22×D6112yes—
368#117—1221no—
369C2×C4⋊C4C22⋊D6111yescharacteristic
370C2×C4⋊C4C22⋊D6111yescharacteristic
371C2×C22⋊C4—22no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C42C22×D6111yescharacteristic
375#107—1222no—
376#106—1222no—
377#106—1222no—
378C4⋊C8—612no—
379C4⋊C8—612no—
380C4⋊C8—1221no—
381C4⋊C8—1221no—
382#103—1222no—
383#102—1222no—
384#102—1222no—
385#102—1222no—
386#102—1224no—
387C22⋊C8—1221no—
388C22⋊C8—1221no—
389C4×C8—612no—
390C4×C8—1221no—
391C24—21no—
392C24—21no—
393C24—28no—
394C24—22no—
395C24—48no—
396C24—48no—
397C24—84no—
398C2×Q8—1221no—
399C2×Q8—1221no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—44no—
412C2×D4—416no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—416no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—82no—
423C2×D4#1175820014yescomplemented
424C2×D4C2×C22⋊D6112yes—
425C2×D4C2×C22⋊D6112yes—
426C2×D4C2×C22⋊D6114yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439SD16—4884no—
440SD16—2444no—
441M16—2442no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—2441no—
452C4⋊C4—2441no—
453C4⋊C4C2×C22⋊D6112yes—
454C4⋊C4C2×C22⋊D6112yes—
455C22⋊C4—28no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—2441no—
461C42—2441no—
462C42C2×C22⋊D6111yescharacteristic
463C42C2×C22⋊D6111yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C23×D6111yescharacteristic
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—416no—
476C23—416no—
477C23—44no—
478C23—42no—
479C23—44no—
480C23—88no—
481C23—832no—
482C23#1451974611yescharacteristic
483Q8—2442no—
484Q8—2442no—
485D4—24no—
486D4—22no—
487D4—22no—
488D4—24no—
489D4—44no—
490D4—44no—
491D4—416no—
492D4—416no—
493D4—42no—
494D4—42no—
495D4—44no—
496D4—44no—
497D4—82no—
498D4C2×#1175820014yes—
499D4#1451980314yes—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4#1451974611yescharacteristic
514C2×C4#1451974611yescharacteristic
515C2×C4#1451974611yescharacteristic
516C2×C4C2×#1175820012yes—
517C2×C4#1451980312yes—
518C2×C4C22×C22⋊D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C8—2442no—
522C8—2442no—
523C8—2442no—
524C22—21no—
525C22—21no—
526C22—24no—
527C22—24no—
528C22—48no—
529C22—48no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—88no—
538C22—832no—
539C22#1683991312yes—
540C22C2×#1451974611yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—42no—
545C4—42no—
546C4—4881no—
547C4—4881no—
548C4#1683991312yes—
549C4C22×#1175820011yescharacteristic
550C4C2×#1451980311yescharacteristic
551C4C2×#1451980312yes—
552C2—22no—
553C2—44no—
554C2—44no—
555C2—84no—
556C2C2×#1683991311yescharacteristic
557C2#2145504611yescharacteristic
558C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×D42×C61C2
socleC22×C61C2×#120
hypercenterC2×D42D61
hypocenterC61#18747
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21455827C2semidirect
3. #21455616C2semidirect
4. #21444998C2semidirect
5. #21444935C2semidirect
6. #21444487C2semidirect
7. #21444244C2semidirect
8. #21444210C2semidirect
10. #16840319C22semidirect
11. #16840134C22semidirect
12. #16840108C22semidirect
15. #16840052C22semidirect
16. #16840052C22semidirect
17. #16840042C22semidirect
18. #16839967C22semidirect
23. #16839815C22semidirect
24. #16839815C22semidirect
26. #16839786C22semidirect
28. #16839506C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
46. #14519777C23semidirect
47. #14519777C23semidirect
59. C4⋊#92418C23semidirect
60. C4⋊#92418C23semidirect
69. C4×#92418C23semidirect
141. C2×D4×C61SD16semidirect
322. C61#18747semidirect
423. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214558272
#214556162
#214449982
#214449352
#214444872
#214442442
#214442102
C2×D42×C612
#1874761