Finite Group Databasebeta

← #29662855

Subgroups — #29662855

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456212C211yesmaximal, characteristic, complemented
2#21455815C211yesmaximal, characteristic
3C2×#16839968C211yesmaximal, characteristic, complemented
4#21454554C211yesmaximal, characteristic, complemented
5#21445379C212yesmaximal, complemented
6#21445365C212yesmaximal, complemented
7#21445255C212yesmaximal, complemented
8#21445241C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444505C211yesmaximal, characteristic, complemented
11C61×#1547C211yesmaximal, characteristic, complemented
12C22×#11758190—21no—
13C2×M16×D61—21no—
14C2×#14519777C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16#16840121—22no—
17#16840112C2212yescomplemented
18#16840108C2212yescomplemented
19#16840073—21no—
20#16840039C2211yescharacteristic
21#16840039C2211yescharacteristic
22#16839968—21no—
23#16839968—21no—
24#16839968C2212yes—
25#16839968C2212yes—
26#16839954C2211yescharacteristic
27#16839954C2211yescharacteristic
28#16839895—21no—
29C4×#11758199—21no—
30C4×#11758199—21no—
31#16839839C2212yes—
32#16839832—22no—
33#16839827C2212yes—
34#16839815C2211yescharacteristic
35#16839763—21no—
36#16839763—21no—
37C2×#14519702—21no—
38#16839725—21no—
39C4×#11758194C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839507C2212yescomplemented
43#16839507C2212yescomplemented
44C61×#532C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yescomplemented
48C61×#331C2212yescomplemented
49C22×Dic122—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52C2×#11758190—22no—
53C2×#11758190—22no—
54C2×#11758190—44no—
55C2×#11758199—21no—
56C2×#11758199—21no—
57C2×#11758194D411yescharacteristic
58C2×#11758194D411yescharacteristic
59C2×#11758189—22no—
60M16×D61—44no—
61C2×C8×D61—22no—
62#14519777—21no—
63#14519777—21no—
64#14519777—21no—
65#14519777C2312yes—
66C4×D244—21no—
67C4×D244—21no—
68C4×D244D411yescharacteristic
69C4×D244D411yescharacteristic
70C4×D244C2311yescharacteristic
71C4×D244C2311yescharacteristic
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×#11758198—21no—
76#14519710—21no—
77#14519710—21no—
78C4⋊#92418C2311yescharacteristic
79C4⋊#92418C2311yescharacteristic
80#14519702—22no—
81#14519702—22no—
82#14519701—22no—
83#14519701—22no—
84#14519698—21no—
85#14519698—21no—
86#14519698—22no—
87#14519698—22no—
88#14519697—21no—
89#14519697—21no—
90C22⋊#92418—21no—
91#14519689—22no—
92#14519689—22no—
93#14519688D412yes—
94C4×#92418—21no—
95C4×#92418C2312yes—
96C2×#41×C61—21no—
97C2×M16×C61—21no—
98C22×C8×C61—21no—
99C4⋊Q8×C61C2311yescharacteristic
100#127×C61C2311yescharacteristic
101#124×C61—21no—
102#121×C61—21no—
103#121×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61—21no—
109C4⋊C8×C61C2312yes—
110C4⋊C8×C61C2312yes—
111C23×D61—21no—
112#41×C61—42no—
113#41×C61—42no—
114C2×Q8×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—41no—
125C2×D244—22no—
126C2×D244—21no—
127C2×D244—41no—
128C2×D244C2×D411yescharacteristic
129C2×D244C2×D411yescharacteristic
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C2×Dic122—22no—
137C2×Dic122—42no—
138C2×Dic122—42no—
139C2×Dic122—41no—
140M16×C61—42no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—42no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61#4111yescharacteristic
151C42×C61C2411yescharacteristic
152#11758199—42no—
153#11758199—42no—
154#11758198—21no—
155#11758198—21no—
156#11758198—21no—
157#11758198—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—41no—
162C4⋊Dic61—41no—
163C4⋊Dic61C2×D411yescharacteristic
164C4⋊Dic61C2×D411yescharacteristic
165#11758194—22no—
166#11758194—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418C2×D412yes—
170C2×#92418C2×D412yes—
171#11758190—44no—
172#11758190—44no—
173#11758189—44no—
174C8×D61—44no—
175C23×C61—41no—
176C23×C61#12111yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61—81no—
182Q8×C61—41no—
183Q8×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—81no—
189D4×C61—82no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—82no—
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#13512yes—
199C2×C4×C61AGL(1,ℤ/8)12yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—82no—
208D244—21no—
209D244—21no—
210D244—21no—
211D244—42no—
212D244AGL(1,ℤ/8)12yes—
213C4×D61—41no—
214C4×D61—41no—
215C4×D61—41no—
216C4×D61—41no—
217C4×D61—42no—
218C4×D61—42no—
219Dic122—42no—
220Dic122—42no—
221Dic122—82no—
222C8×C61—42no—
223C8×C61—42no—
224#92418—22no—
225#92418—22no—
226#92418—41no—
227#92418—41no—
228#19169—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61#46312yes—
235C22×C61C2×#12111yescharacteristic
236C2×D61—22no—
237C2×D61—41no—
238C2×D61—41no—
239C2×D61—41no—
240C2×D61—41no—
241C2×D61—41no—
242C2×D61—84no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—81no—
247C4×C61#46312yes—
248C4×C61C2×#13511yescharacteristic
249C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C61#57511yescharacteristic
252Dic61—41no—
253Dic61—41no—
254Dic61—82no—
255#3141—611no—
256#3100—611no—
257C2×#461—611no—
258D4×M16—611no—
259#1701—612no—
260#1699—612no—
261#1681—612no—
262#1679—612no—
263#1565—611no—
264#1549—611no—
265#1547D6111yescharacteristic, complemented
266C2×C61—22no—
267C2×C61—161no—
268C2×C61C2×#46311yescharacteristic
269C2×C61#305511yescharacteristic
270C2×C61#306911yescharacteristic
271D61—42no—
272D61—81no—
273C22×SD16—1221no—
274C22×M16—1221no—
275#532C2×D6111yescharacteristic
276C2×C4⋊Q8—611no—
277C2×C4×D4—611no—
278#489—1222no—
279#487—612no—
280#486—612no—
281#480—1221no—
282#474—611no—
283#474—611no—
284#461—612no—
285#461—612no—
286#461—1221no—
287#461—1221no—
288#459—611no—
289#459—611no—
290#447—1221no—
291C4×D8—1221no—
292C4×D8—1221no—
293#436—612no—
294#435—1222no—
295C8×D4—612no—
296#431—611no—
297#423C2×D6111yescharacteristic
298C2×C4⋊C8C2×D6111yescharacteristic
299#417—1221no—
300#417—1221no—
301C2×#103—1221no—
302#407—1221no—
303C4×M16—611no—
304#335—612no—
305#334—612no—
306#332—612no—
307#332—612no—
308#332C2×D6112yescomplemented
309#331C2×D6112yescomplemented
310C61#1916911yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C22×Q8—1221no—
313C22×D4—611no—
314C23×C4—1221no—
315C2×SD16—1222no—
316C2×SD16—1222no—
317C2×SD16—2444no—
318C2×D8—1221no—
319C2×D8—1221no—
320C2×M16—21no—
321C2×M16—611no—
322C2×M16—611no—
323C2×M16—1222no—
324C2×M16—2444no—
325C22×C8—21no—
326C22×C8—1222no—
327C4⋊Q8—612no—
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4⋊Q8C22×D6111yescharacteristic
332#127C22×D6111yescharacteristic
333#124—21no—
334#121—21no—
335#121—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—611no—
339C4×D4—611no—
340C4×D4—611no—
341C4×D4—611no—
342C4×D4—1221no—
343C4×D4—1221no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C42C22×D6111yescharacteristic
349#107—1221no—
350#107—1221no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—611no—
354C4⋊C8—611no—
355C4⋊C8C22×D6112yes—
356C4⋊C8C22×D6112yes—
357#103—1222no—
358#103—1222no—
359#103—1222no—
360#103—1222no—
361#102—1221no—
362#102—1221no—
363#102—1221no—
364#102—1221no—
365#102—1222no—
366#102—1222no—
367C22⋊C8—1222no—
368C22⋊C8—1222no—
369C22⋊C8—1221no—
370#97—612no—
371C4×C8—612no—
372C4×C8—1221no—
373C24—1221no—
374#41—42no—
375#41—42no—
376C2×Q8—21no—
377C2×Q8—1222no—
378C2×Q8—2442no—
379C2×Q8—2442no—
380C2×Q8—2441no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—41no—
385C2×D4—611no—
386C2×D4—611no—
387C2×D4—1222no—
388C2×D4—1221no—
389C2×D4—2441no—
390C22×C4—41no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—1221no—
395C22×C4—1221no—
396C22×C4—2441no—
397C22×C4—2442no—
398C22×C4—2442no—
399C22×C4#1175822211yescharacteristic
400C22×C4D4×D6111yescharacteristic
401C22×C4D4×D6111yescharacteristic
402SD16—2444no—
403SD16—2444no—
404D8—2442no—
405D8—2442no—
406M16—42no—
407M16—1222no—
408M16—1222no—
409M16—2444no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—42no—
413C2×C8—612no—
414C2×C8—612no—
415C2×C8—1221no—
416C2×C8—1221no—
417C2×C8—2444no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—611no—
421C4⋊C4—611no—
422C4⋊C4—1221no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—1221no—
430C22⋊C4—1221no—
431C22⋊C4—1221no—
432C22⋊C4—1221no—
433C42#1175822211yescharacteristic
434C42D4×D6111yescharacteristic
435C42D4×D6111yescharacteristic
436C42C23×D6111yescharacteristic
437C23—41no—
438C23—1221no—
439C23—1221no—
440C23—4881no—
441C23—2442no—
442C23—2442no—
443C23#1451975011yescharacteristic
444Q8—41no—
445Q8—41no—
446Q8—4882no—
447Q8—2442no—
448Q8—2442no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—81no—
454D4—82no—
455D4—612no—
456D4—1221no—
457D4—1221no—
458D4—1221no—
459D4—2442no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—82no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—4882no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4#1451975011yescharacteristic
477C2×C4#1451975011yescharacteristic
478C2×C4#1451975011yescharacteristic
479C2×C4#1451979812yes—
480C2×C4#1451980112yes—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4C2×#1175822211yescharacteristic
484C8—42no—
485C8—42no—
486C8—1222no—
487C8—1222no—
488C8—2441no—
489C8—2441no—
490C22—21no—
491C22—21no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—1222no—
496C22—4884no—
497C22—2441no—
498C22—2441no—
499C22—2441no—
500C22—2441no—
501C22—2441no—
502C22#1683997612yes—
503C22C2×#1451975011yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—81no—
508C4—4882no—
509C4—2441no—
510C4—2441no—
511C4#1683997612yes—
512C4C2×#1451979811yescharacteristic
513C4AGL(1,ℤ/8)×D6111yescharacteristic
514C4C2×#1451980111yescharacteristic
515C4#1684056111yescharacteristic
516C2—22no—
517C2—161no—
518C2—4881no—
519C2—2442no—
520C2C2×#1683997611yescharacteristic
521C2#2145544111yescharacteristic
522C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1547C2
socleC22×C61C2×#121
prime radical#21455815C2
hypercenter#1547D61
hypocenterC61#19169
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
1. #21456212C2semidirect
3. C2×#16839968C2semidirect
4. #21454554C2semidirect
5. #21445379C2semidirect
6. #21445365C2semidirect
7. #21445255C2semidirect
8. #21445241C2semidirect
9. #21444569C2semidirect
10. #21444505C2semidirect
11. C61×#1547C2semidirect
17. #16840112C22semidirect
18. #16840108C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839507C22semidirect
43. #16839507C22semidirect
47. C61×#332C22semidirect
48. C61×#331C22semidirect
265. #1547D61semidirect
308. #332C2×D61semidirect
309. #331C2×D61semidirect
310. C61#19169semidirect

Maximal subgroups

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

subgroupindex
#214562122
#214558152
C2×#168399682
#214545542
#214453792
#214453652
#214452552
#214452412
#214445692
#214445052
C61×#15472
#1916961