Finite Group Databasebeta

← #29658722

Subgroups — #29658722

Order 15616.

1288 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456737C211yesmaximal, characteristic, complemented
2#21455850C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic, complemented
4#21445019C214yesmaximal, complemented
5#21444884C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×C61×#550C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12#16840138C2212yescomplemented
13#16840108C2212yescomplemented
14#16840080—21no—
15#16840047C2212yescomplemented
16#16839977—22no—
17#16839967C2212yescomplemented
18#16839962C2212yescomplemented
19#16839957C2212yescomplemented
20#16839954—22no—
21#16839901—21no—
22#16839896—22no—
23C4×#11758201—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
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—44no—
45C2×#11758200—22no—
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#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519700—22no—
64#14519698—24no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—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—21no—
111C24×C61—22no—
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—22no—
127C2×D4×C61—22no—
128C2×D4×C61—48no—
129C2×D4×C61—44no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61—82no—
133C2×D4×C61—84no—
134C2×D4×C61C2×D412yes—
135C2×D4×C61C2×D414yes—
136C22×C4×C61—21no—
137C22×C4×C61—28no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—48no—
143C22×C4×C61—44no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—41no—
148#11758222—42no—
149#11758222—42no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×C4×D61—41no—
154C2×Dic122—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C2×D412yes—
160C4⋊C4×C61C2×D412yes—
161C22⋊C4×C61—28no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169C42×C61C2411yescharacteristic
170#11758201—44no—
171#11758200—44no—
172#11758199—44no—
173#11758199—84no—
174#11758198—41no—
175#11758198—41no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178#11758194—42no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—22no—
183C23×C61—24no—
184C23×C61—44no—
185C23×C61—42no—
186C23×C61—44no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—44no—
190C23×C61#12011yescharacteristic
191C22×D61—41no—
192Q8×C61—24no—
193Q8×C61—22no—
194Q8×C61AGL(1,ℤ/8)14yescomplemented
195D4×C61—22no—
196D4×C61—22no—
197D4×C61—44no—
198D4×C61—416no—
199D4×C61—44no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—44no—
203D4×C61—42no—
204D4×C61—44no—
205D4×C61—82no—
206D4×C61—88no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—416no—
214C2×C4×C61—44no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—44no—
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#13512yes—
222C2×C4×C61AGL(1,ℤ/8)12yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C22⋊D61—82no—
227C2×Dic61—41no—
228D244—42no—
229D244—41no—
230D244—41no—
231D244—42no—
232C4×D61—82no—
233Dic122—41no—
234Dic122—41no—
235#92418—42no—
236#92418—42no—
237#92418—42no—
238#18770—611nomaximal, Sylow
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—42no—
242C22×C61—44no—
243C22×C61—44no—
244C22×C61—42no—
245C22×C61—44no—
246C22×C61—42no—
247C22×C61—84no—
248C22×C61—88no—
249C22×C61#44912yes—
250C22×C61C2×#12011yescharacteristic
251C2×D61—81no—
252C2×D61—81no—
253C2×D61—81no—
254C4×C61—24no—
255C4×C61—21no—
256C4×C61—21no—
257C4×C61—21no—
258C4×C61—42no—
259C4×C61—42no—
260C4×C61#44912yes—
261C4×C61C2×#13511yescharacteristic
262C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C4×C61C2×AGL(1,ℤ/8)12yes—
264Dic61—81no—
265C2×#550D6111yescharacteristic, complemented
266#3205—611no—
267#3103—611no—
268#3036—611no—
269#1642—614no—
270#1621—614no—
271#1545—611no—
272#1487—611no—
273#1478—611no—
274C2×C61—22no—
275C2×C61—82no—
276C2×C61—84no—
277C2×C61C2×#44911yescharacteristic
278C2×C61#300311yescharacteristic
279C2×C61#301911yescharacteristic
280D61—161no—
281C22×#41—21no—
282C2×AGL(1,ℤ/8)—1221no—
283#550—22no—
284#550—22no—
285#550—22no—
286#550C2×D6114yes—
287#532—611no—
288C2×#127—21no—
289C2×#127C2×D6111yescharacteristic
290C2×#121—22no—
291C2×#121—21no—
292C2×C4×Q8C2×D6111yescharacteristic
293C2×C4×D4—21no—
294C2×C4×D4C2×D6111yescharacteristic
295#493—612no—
296#486—612no—
297#481—1221no—
298#475—612no—
299#463—1222no—
300#461—612no—
301#460—612no—
302#459—612no—
303#459—1222no—
304#448—1221no—
305#447—1222no—
306C4×SD16—1222no—
307#431—611no—
308#431—611no—
309#423—611no—
310#417—1221no—
311#331—614no—
312Q8⋊C8—614no—
313D4⋊C8—614no—
314C61#1877011yescharacteristic, complemented, Sylow
315C2×#41—24no—
316C2×#41—24no—
317C2×#41—48no—
318C2×#41—1221no—
319C22×Q8C22⋊D6111yescharacteristic
320C22×D4—22no—
321C22×D4—22no—
322C22×D4—21no—
323C22×D4—21no—
324C22×D4C22⋊D6111yescharacteristic
325C22×D4C22⋊D6112yes—
326C23×C4—22no—
327C23×C4—21no—
328AGL(1,ℤ/8)—2444no—
329C2×SD16—1222no—
330C2×SD16—1222no—
331C2×D8—1224no—
332C2×D8—1222no—
333C2×M16—1221no—
334C4⋊Q8—611no—
335#127—24no—
336#127—24no—
337#127—21no—
338#127—21no—
339#127—611no—
340#127C22⋊D6112yes—
341#127C22×D6112yes—
342#124—1221no—
343#121—24no—
344#121—28no—
345#121—28no—
346#121—24no—
347#121—1221no—
348#121—1221no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8C22⋊D6112yes—
352C4×Q8C22×D6112yes—
353C4×D4—24no—
354C4×D4—24no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—1221no—
358C4×D4—1221no—
359C4×D4C22⋊D6112yes—
360C4×D4C22×D6112yes—
361C2×C4⋊C4—21no—
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22⋊D6111yescharacteristic
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C42—21no—
367C2×C42C22×D6111yescharacteristic
368#106—1222no—
369C4⋊C8—612no—
370C4⋊C8—612no—
371C4⋊C8—1221no—
372C4⋊C8—1221no—
373#103—1222no—
374#103—1222no—
375#102—1224no—
376#102—1222no—
377#102—1222no—
378C22⋊C8—1221no—
379C22⋊C8—1221no—
380C4×C8—612no—
381C4×C8—1221no—
382C24—21no—
383C24—22no—
384#41—416no—
385#41—48no—
386#41—48no—
387#41—2442no—
388#41—2442no—
389C2×Q8—24no—
390C2×Q8—21no—
391C2×Q8—1221no—
392C2×Q8C2×C22⋊D6112yes—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—82no—
408C2×D4—84no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—2441no—
413C2×D4C2×C22⋊D6112yes—
414C2×D4C2×C22⋊D6114yes—
415C22×C4—21no—
416C22×C4—28no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—2441no—
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C2×C22⋊D6111yescharacteristic
427SD16—2444no—
428SD16—2444no—
429D8—4884no—
430D8—2444no—
431M16—2442no—
432C2×C8—1222no—
433C2×C8—1222no—
434C2×C8—1222no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—1221no—
440C4⋊C4—1221no—
441C4⋊C4C2×C22⋊D6112yes—
442C4⋊C4C2×C22⋊D6112yes—
443C22⋊C4—28no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C42—22no—
449C42—22no—
450C42C2×C22⋊D6111yescharacteristic
451C42C2×C22⋊D6111yescharacteristic
452C42C2×C22⋊D6111yescharacteristic
453C42C23×D6111yescharacteristic
454C23—24no—
455C23—22no—
456C23—44no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—2441no—
463C23#1451974611yescharacteristic
464Q8—24no—
465Q8—22no—
466Q8—2441no—
467Q8—2441no—
468Q8#1451980314yescomplemented
469D4—22no—
470D4—22no—
471D4—44no—
472D4—416no—
473D4—44no—
474D4—42no—
475D4—42no—
476D4—44no—
477D4—42no—
478D4—44no—
479D4—82no—
480D4—88no—
481D4—4882no—
482D4—2442no—
483D4—2441no—
484D4—2441no—
485D4—2442no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—416no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—4882no—
498C2×C4—2441no—
499C2×C4#1451974611yescharacteristic
500C2×C4#1451974611yescharacteristic
501C2×C4#1451974611yescharacteristic
502C2×C4#1451979712yes—
503C2×C4#1451980212yes—
504C2×C4C22×C22⋊D6111yescharacteristic
505C2×C4C22×C22⋊D6111yescharacteristic
506C2×C4C22×C22⋊D6111yescharacteristic
507C8—2442no—
508C8—2442no—
509C8—2442no—
510C22—21no—
511C22—21no—
512C22—44no—
513C22—42no—
514C22—42no—
515C22—44no—
516C22—44no—
517C22—42no—
518C22—84no—
519C22—88no—
520C22—4881no—
521C22—4881no—
522C22—4881no—
523C22#1683990712yes—
524C22C2×#1451974611yescharacteristic
525C4—21no—
526C4—21no—
527C4—24no—
528C4—21no—
529C4—42no—
530C4—42no—
531C4—4881no—
532C4#1683990712yes—
533C4C2×#1451979711yescharacteristic
534C4C2×#1451980211yescharacteristic
535C4C2×#1451980312yes—
536C2—22no—
537C2—84no—
538C2—82no—
539C2—9761no—
540C2C2×#1683990711yescharacteristic
541C2#2145504611yescharacteristic
542C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#550C2
socleC22×C61C2×#120
hypercenterC2×#550D61
hypocenterC61#18770
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. #21456737C2semidirect
2. #21455850C2semidirect
3. #21455283C2semidirect
4. #21445019C2semidirect
5. #21444884C2semidirect
6. #21444487C2semidirect
7. #21444248C2semidirect
8. #21444210C2semidirect
9. C2×C61×#550C2semidirect
12. #16840138C22semidirect
13. #16840108C22semidirect
15. #16840047C22semidirect
17. #16839967C22semidirect
18. #16839962C22semidirect
19. #16839957C22semidirect
28. #16839506C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
194. Q8×C61AGL(1,ℤ/8)semidirect
265. C2×#550D61semidirect
314. C61#18770semidirect
468. Q8#14519803semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214558502
#214552832
#214450192
#214448842
#214444872
#214442482
#214442102
C2×C61×#5502
#1877061