Finite Group Databasebeta

← #29656267

Subgroups — #29656267

Order 15616.

1316 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456269C211yesmaximal, characteristic, complemented
2#21455941C211yesmaximal, characteristic, complemented
3#21455910C211yesmaximal, characteristic, complemented
4#21444946C214yesmaximal, complemented
5#21444884C214yesmaximal, complemented
6#21444533C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic, complemented
9C2×C61×#550C211yesmaximal, characteristic, complemented
10C2×#14519803—22no—
11#16840312C2211yescharacteristic
12#16840149—22no—
13#16840138C2212yescomplemented
14#16840134C2212yescomplemented
15#16840080—21no—
16#16840075—21no—
17#16840057C2212yescomplemented
18#16840052C2212yescomplemented
19#16839962C2212yescomplemented
20#16839957C2212yescomplemented
21#16839901—21no—
22#16839896—21no—
23#16839872—22no—
24#16839860—22no—
25#16839819C2211yescharacteristic
26#16839819C2211yescharacteristic
27C4×#11758194C2211yescharacteristic
28#16839506C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C22×#41×C61—21no—
32C61×#550—22no—
33C61×#550—22no—
34C61×#550—22no—
35C61×#550C2214yes—
36C2×#127×C61—21no—
37C2×#127×C61C2211yescharacteristic
38C2×#121×C61—22no—
39C2×#121×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758222—21no—
44#14519803—48no—
45C2×#11758200—24no—
46C2×#11758201—22no—
47C2×#11758199—22no—
48C2×#11758199—24no—
49C2×#11758194—21no—
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—22no—
62#14519698—22no—
63#14519697—22no—
64#14519697—22no—
65C22⋊#92418—21no—
66C22⋊#92418—21no—
67#14519688—21no—
68#14519688—21no—
69#14519688—21no—
70C4×#92418C2312yes—
71C2×#41×C61—24no—
72C2×#41×C61—24no—
73C2×#41×C61—48no—
74C22×Q8×C61D411yescharacteristic
75C22×D4×C61—22no—
76C22×D4×C61—22no—
77C22×D4×C61—21no—
78C22×D4×C61—21no—
79C22×D4×C61D411yescharacteristic
80C22×D4×C61D412yes—
81C23×C4×C61—22no—
82C23×C4×C61—21no—
83#127×C61—24no—
84#127×C61—24no—
85#127×C61—21no—
86#127×C61—21no—
87#127×C61D412yes—
88#127×C61C2312yes—
89#121×C61—24no—
90#121×C61—28no—
91#121×C61—28no—
92#121×C61—24no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97C4×D4×C61—24no—
98C4×D4×C61—24no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61D412yes—
102C4×D4×C61C2312yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C24×C61—22no—
111C24×C61—21no—
112#41×C61—416no—
113#41×C61—48no—
114#41×C61—48no—
115C2×Q8×C61—24no—
116C2×Q8×C61—21no—
117C2×Q8×C61C2×D412yes—
118C2×D4×C61—28no—
119C2×D4×C61—28no—
120C2×D4×C61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—42no—
127C2×D4×C61—44no—
128C2×D4×C61—48no—
129C2×D4×C61—44no—
130C2×D4×C61—44no—
131C2×D4×C61—84no—
132C2×D4×C61C2×D412yes—
133C2×D4×C61C2×D414yes—
134C22×C4×C61—21no—
135C22×C4×C61—28no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—48no—
141C22×C4×C61—44no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×C22⋊D61—41no—
146#11758222—44no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×C4×D61—41no—
151C2×Dic122—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61C2×D412yes—
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—28no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C42×C61—22no—
162C42×C61—22no—
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2411yescharacteristic
167#11758201—44no—
168#11758200—48no—
169#11758199—48no—
170#11758199—44no—
171#11758198—41no—
172#11758198—41no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175#11758194—24no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—22no—
180C23×C61—24no—
181C23×C61—44no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—44no—
185C23×C61—44no—
186C23×C61—42no—
187C23×C61#12011yescharacteristic
188C22×D61—41no—
189Q8×C61—24no—
190Q8×C61—22no—
191Q8×C61AGL(1,ℤ/8)14yescomplemented
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—44no—
195D4×C61—42no—
196D4×C61—44no—
197D4×C61—416no—
198D4×C61—44no—
199D4×C61—44no—
200D4×C61—44no—
201D4×C61—82no—
202D4×C61—88no—
203C2×C4×C61—24no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—416no—
211C2×C4×C61—44no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—44no—
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C22⋊D61—82no—
222C2×Dic61—41no—
223D244—42no—
224D244—42no—
225D244—42no—
226C4×D61—82no—
227Dic122—42no—
228#92418—24no—
229#92418—42no—
230#92418—42no—
231#18545—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—44no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—44no—
238C22×C61—44no—
239C22×C61—42no—
240C22×C61—88no—
241C22×C61—84no—
242C22×C61#45312yes—
243C22×C61C2×#12011yescharacteristic
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—24no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61C2×AGL(1,ℤ/8)12yes—
255C4×C61#57512yes—
256Dic61—81no—
257C2×#550D6111yescharacteristic, complemented
258#3151—611no—
259#3112—611no—
260#3109—611no—
261#1631—614no—
262#1621—614no—
263#1555—611no—
264#1491—611no—
265#1490—611no—
266C2×C61—22no—
267C2×C61—82no—
268C2×C61—84no—
269C2×C61#300311yescharacteristic
270C2×C61#300511yescharacteristic
271C2×C61C2×#45311yescharacteristic
272D61—161no—
273C22×#41—21no—
274C2×AGL(1,ℤ/8)—1222no—
275#550—22no—
276#550—22no—
277#550—22no—
278#550C2×D6114yes—
279#532—611no—
280C2×#127—21no—
281C2×#127C2×D6111yescharacteristic
282C2×#121—22no—
283C2×#121—21no—
284C2×C4×Q8C2×D6111yescharacteristic
285C2×C4×D4—21no—
286C2×C4×D4C2×D6111yescharacteristic
287#496—1222no—
288#493—612no—
289#492—612no—
290#481—1221no—
291#480—1221no—
292#477—612no—
293#476—612no—
294#460—612no—
295#459—612no—
296#448—1221no—
297#447—1221no—
298#442—1222no—
299#440—1222no—
300#432—611no—
301#432—611no—
302C4×M16—611no—
303#331—614no—
304Q8⋊C8—614no—
305D4⋊C8—614no—
306C61#1854511yescharacteristic, complemented, Sylow
307C2×#41—24no—
308C2×#41—24no—
309C2×#41—48no—
310C2×#41—1221no—
311C22×Q8C22⋊D6111yescharacteristic
312C22×D4—22no—
313C22×D4—22no—
314C22×D4—21no—
315C22×D4—21no—
316C22×D4C22⋊D6111yescharacteristic
317C22×D4C22⋊D6112yes—
318C23×C4—22no—
319C23×C4—21no—
320AGL(1,ℤ/8)—2448no—
321C2×SD16—1222no—
322C2×SD16—1224no—
323C2×D8—1224no—
324C2×D8—1222no—
325C2×M16—1221no—
326C4⋊Q8—611no—
327#127—24no—
328#127—24no—
329#127—21no—
330#127—21no—
331#127—611no—
332#127C22⋊D6112yes—
333#127C22×D6112yes—
334#124—1221no—
335#121—24no—
336#121—28no—
337#121—28no—
338#121—24no—
339#121—1221no—
340#121—1221no—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×Q8C22⋊D6112yes—
344C4×Q8C22×D6112yes—
345C4×D4—24no—
346C4×D4—24no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—1221no—
350C4×D4—1221no—
351C4×D4C22⋊D6112yes—
352C4×D4C22×D6112yes—
353C2×C4⋊C4—21no—
354C2×C4⋊C4C22⋊D6111yescharacteristic
355C2×C4⋊C4C22⋊D6111yescharacteristic
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C42—21no—
359C2×C42C22×D6111yescharacteristic
360#107—1222no—
361#106—1222no—
362C4⋊C8—612no—
363C4⋊C8—612no—
364#103—1222no—
365#102—1222no—
366#102—1222no—
367#102—1222no—
368C22⋊C8—1221no—
369C22⋊C8—1221no—
370#97—1221no—
371#97—1221no—
372#97—1221no—
373C4×C8—612no—
374C24—22no—
375C24—21no—
376#41—416no—
377#41—48no—
378#41—48no—
379#41—2444no—
380C2×Q8—24no—
381C2×Q8—21no—
382C2×Q8—1221no—
383C2×Q8C2×C22⋊D6112yes—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—48no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—84no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—2441no—
402C2×D4C2×C22⋊D6112yes—
403C2×D4C2×C22⋊D6114yes—
404C22×C4—21no—
405C22×C4—28no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—2441no—
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6111yescharacteristic
416SD16—2448no—
417SD16—2444no—
418D8—2448no—
419D8—2444no—
420M16—1224no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1222no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4C2×C22⋊D6112yes—
431C4⋊C4C2×C22⋊D6112yes—
432C22⋊C4—28no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C42—22no—
438C42—22no—
439C42C2×C22⋊D6111yescharacteristic
440C42C2×C22⋊D6111yescharacteristic
441C42C2×C22⋊D6111yescharacteristic
442C42C23×D6111yescharacteristic
443C23—24no—
444C23—22no—
445C23—44no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—2441no—
452C23#1451974611yescharacteristic
453Q8—24no—
454Q8—22no—
455Q8—2442no—
456Q8#1451980314yescomplemented
457D4—22no—
458D4—22no—
459D4—44no—
460D4—42no—
461D4—44no—
462D4—416no—
463D4—44no—
464D4—44no—
465D4—44no—
466D4—82no—
467D4—88no—
468D4—4882no—
469D4—2442no—
470D4—2442no—
471D4—2442no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—416no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—4882no—
485C2×C4—2441no—
486C2×C4#1451974611yescharacteristic
487C2×C4#1451974611yescharacteristic
488C2×C4#1451974611yescharacteristic
489C2×C4C22×C22⋊D6111yescharacteristic
490C2×C4C22×C22⋊D6111yescharacteristic
491C2×C4C22×C22⋊D6111yescharacteristic
492C8—1224no—
493C8—2442no—
494C8—2442no—
495C22—21no—
496C22—21no—
497C22—44no—
498C22—42no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22—88no—
504C22—84no—
505C22—4881no—
506C22—4881no—
507C22—4881no—
508C22#1683993012yes—
509C22C2×#1451974611yescharacteristic
510C4—21no—
511C4—21no—
512C4—24no—
513C4—21no—
514C4—21no—
515C4—42no—
516C4—42no—
517C4—4881no—
518C4C2×#1451980312yes—
519C4#1684056012yes—
520C2—22no—
521C2—84no—
522C2—82no—
523C2—9761no—
524C2#2145504611yescharacteristic
525C2#2145506111yescharacteristic
526C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#550C2
socleC22×C61C2×#120
hypercenterC2×#550D61
hypocenterC61#18545
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #21456269C2semidirect
2. #21455941C2semidirect
3. #21455910C2semidirect
4. #21444946C2semidirect
5. #21444884C2semidirect
6. #21444533C2semidirect
7. #21444266C2semidirect
8. #21444262C2semidirect
9. C2×C61×#550C2semidirect
13. #16840138C22semidirect
14. #16840134C22semidirect
17. #16840057C22semidirect
18. #16840052C22semidirect
19. #16839962C22semidirect
20. #16839957C22semidirect
28. #16839506C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
191. Q8×C61AGL(1,ℤ/8)semidirect
257. C2×#550D61semidirect
306. C61#18545semidirect
456. Q8#14519803semidirect

Maximal subgroups

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

subgroupindex
#214562692
#214559412
#214559102
#214449462
#214448842
#214445332
#214442662
#214442622
C2×C61×#5502
#1854561