Finite Group Databasebeta

← #29662792

Subgroups — #29662792

Order 15616.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456278C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3C2×#16840042C211yesmaximal, characteristic, complemented
4#21445308C212yesmaximal, complemented
5#21445284C212yesmaximal, complemented
6#21445184C212yesmaximal, complemented
7#21445160C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444497C211yesmaximal, characteristic, complemented
10#21444492C211yesmaximal, characteristic, complemented
11D4×M16×C61C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13C2×#14519810—21no—
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840156—22no—
17#16840142C2212yescomplemented
18#16840135C2212yescomplemented
19#16840075—21no—
20#16840042—21no—
21#16840042—21no—
22#16840042C2212yes—
23#16840042C2211yescharacteristic
24#16840042C2212yes—
25#16840042C2211yescharacteristic
26#16839957C2211yescharacteristic
27#16839957C2211yescharacteristic
28#16839896—21no—
29C4×#11758199—21no—
30C4×#11758199—21no—
31#16839815C2211yescharacteristic
32C2×#14519710—21no—
33C2×C4⋊#92418C2211yescharacteristic
34C2×#14519698—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839509C2212yescomplemented
38#16839509C2212yescomplemented
39#16839509C2212yescomplemented
40#16839505C2212yescomplemented
41C22×M16×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#436C2212yes—
44C61×#435—22no—
45C8×D4×C61C2212yes—
46C61×#423C2211yescharacteristic
47C61×#407—21no—
48C4×M16×C61C2211yescharacteristic
49C2×#11758222—21no—
50C22×Dic122—21no—
51#14519810—44no—
52#14519810—44no—
53C2×Dic244—22no—
54C2×Dic244—22no—
55C2×#11758190—22no—
56C2×#11758190—22no—
57C2×#11758199—21no—
58C2×#11758199—21no—
59C22×#92418—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777C2312yes—
64#14519777C2311yescharacteristic
65#14519774C2311yescharacteristic
66#14519765—21no—
67#14519751—21no—
68#14519751—21no—
69C4×D244—21no—
70C4×D244—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73#14519710—22no—
74#14519710—22no—
75#14519710—21no—
76#14519710—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2311yescharacteristic
79C4⋊#92418C2311yescharacteristic
80C4⋊#92418C2312yes—
81#14519698—22no—
82#14519698—22no—
83#14519698—21no—
84#14519698—21no—
85#14519697—21no—
86#14519697—21no—
87C22⋊#92418—21no—
88C4×#92418—21no—
89C22×D4×C61D411yescharacteristic
90C23×C4×C61—21no—
91C2×M16×C61—21no—
92C2×M16×C61—22no—
93C2×M16×C61—44no—
94C2×M16×C61D411yescharacteristic
95C2×M16×C61D411yescharacteristic
96C22×C8×C61—22no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61D411yescharacteristic
100C4×D4×C61D411yescharacteristic
101C4×D4×C61C2311yescharacteristic
102C4×D4×C61C2311yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108C22⋊C8×C61—22no—
109C22⋊C8×C61—22no—
110#97×C61D412yes—
111C4×C8×C61C2312yes—
112C24×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—41no—
116C2×D4×C61C2×D411yescharacteristic
117C2×D4×C61C2×D411yescharacteristic
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C2×C22⋊D61—41no—
129C22×Dic61—41no—
130#11758222—42no—
131#11758222—42no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×C4×D61—41no—
136C2×Dic122—22no—
137C2×Dic122—21no—
138C2×Dic122—42no—
139C2×Dic122—42no—
140C2×Dic122—41no—
141M16×C61—22no—
142M16×C61—42no—
143M16×C61—44no—
144M16×C61D814yescomplemented
145C2×C8×C61—22no—
146C2×C8×C61—44no—
147C2×C8×C61C2×D412yes—
148C2×C8×C61C2×D412yes—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D411yescharacteristic
151C4⋊C4×C61C2×D411yescharacteristic
152C22⋊C4×C61—21no—
153C22⋊C4×C61—21no—
154C22⋊C4×C61—21no—
155C22⋊C4×C61—21no—
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61#4111yescharacteristic
159C42×C61C2411yescharacteristic
160#11758199—42no—
161#11758199—42no—
162#11758198—41no—
163#11758198—41no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—41no—
169C4⋊Dic61—41no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418—42no—
174Dic244—44no—
175Dic244—44no—
176#11758190—44no—
177#11758190—44no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—81no—
183C23×C61#12111yescharacteristic
184C22×D61—41no—
185D4×C61—21no—
186D4×C61—21no—
187D4×C61—21no—
188D4×C61—42no—
189D4×C61#13712yescomplemented
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61C2×D812yes—
206C2×C4×C61AGL(1,ℤ/8)12yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C22⋊D61—82no—
211C2×Dic61—42no—
212C2×Dic61—41no—
213C2×Dic61—82no—
214D244—41no—
215D244—41no—
216D244—41no—
217D244—41no—
218D244—81no—
219C4×D61—82no—
220Dic122—42no—
221Dic122—41no—
222Dic122—42no—
223Dic122—41no—
224Dic122—82no—
225C8×C61—22no—
226C8×C61—42no—
227C8×C61C2×D814yes—
228#92418—41no—
229#92418—42no—
230#92418—41no—
231#19163—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—22no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—84no—
241C22×C61#45912yes—
242C22×C61C2×#12111yescharacteristic
243C2×D61—81no—
244C2×D61—81no—
245C2×D61—81no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61#45912yes—
252C4×C61C22×D811yescharacteristic
253C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C61C2×#13711yescharacteristic
255C4×C61#57511yescharacteristic
256Dic61—82no—
257Dic61—81no—
258#3152—611no—
259#3100—611no—
260C2×#474—611no—
261D4×M16D6111yescharacteristic, complemented
262#1689—612no—
263#1685—612no—
264#1669—612no—
265#1665—612no—
266#1565—611no—
267#1547—611no—
268#1546—611no—
269C2×C61—22no—
270C2×C61—42no—
271C2×C61—81no—
272C2×C61C2×#45911yescharacteristic
273C2×C61#305611yescharacteristic
274C2×C61#306911yescharacteristic
275D61—161no—
276C2×#137—1221no—
277C2×#137—1221no—
278C22×M16—21no—
279#532—611no—
280C2×C4⋊Q8—611no—
281C2×C4×D4C2×D6111yescharacteristic
282#497—1222no—
283#494—612no—
284#492—612no—
285#480—1221no—
286#474—611no—
287#474—611no—
288#474—612no—
289#474—612no—
290#474—1221no—
291#474—1221no—
292#459—611no—
293#459—611no—
294#447—1221no—
295C4×D8—1221no—
296C4×D8—1221no—
297#436C2×D6112yes—
298#435—22no—
299C8×D4C2×D6112yes—
300#431—611no—
301C2×#107—1221no—
302#423C2×D6111yescharacteristic
303C2×C4⋊C8—611no—
304C2×#102—1221no—
305#407—21no—
306C4×M16C2×D6111yescharacteristic
307#335—612no—
308#334—612no—
309#332—612no—
310#332—612no—
311#332—612no—
312#331—612no—
313C61#1916311yescharacteristic, complemented, Sylow
314C2×#41—1221no—
315C22×Q8—1221no—
316C22×D4D24411yescharacteristic
317C23×C4—21no—
318#137—2444no—
319#137—2444no—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C2×D8—1221no—
325C2×D8—1221no—
326C2×M16—21no—
327C2×M16—22no—
328C2×M16—44no—
329C2×M16C22⋊D6111yescharacteristic
330C2×M16C22⋊D6111yescharacteristic
331C22×C8—22no—
332C22×C8—1221no—
333C4⋊Q8—612no—
334C4⋊Q8—611no—
335C4⋊Q8—1221no—
336C4⋊Q8—1221no—
337C4⋊Q8—1221no—
338#127—611no—
339#124—1221no—
340#121—1221no—
341#121—1221no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—1221no—
345C4×D4—1221no—
346C4×D4D24411yescharacteristic
347C4×D4D24411yescharacteristic
348C4×D4C22×D6111yescharacteristic
349C4×D4C22×D6111yescharacteristic
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4D24411yescharacteristic
353C2×C22⋊C4—21no—
354C2×C42C22×D6111yescharacteristic
355#107—1222no—
356#107—1222no—
357#107—1221no—
358#107—1221no—
359C4⋊C8—21no—
360C4⋊C8—612no—
361C4⋊C8—611no—
362C4⋊C8—611no—
363C4⋊C8—1221no—
364C4⋊C8C22×D6112yes—
365#102—1222no—
366#102—1222no—
367#102—1221no—
368#102—1221no—
369#102—1221no—
370#102—1221no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—1221no—
374#97C22⋊D6112yes—
375C4×C8—1221no—
376C4×C8C22×D6112yes—
377C24—21no—
378#41—2442no—
379#41—2442no—
380C2×Q8—1221no—
381C2×Q8—1222no—
382C2×Q8—2442no—
383C2×Q8—2442no—
384C2×Q8—2441no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—2441no—
392C2×D4C2×D24411yescharacteristic
393C2×D4C2×D24411yescharacteristic
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—2441no—
402C22×C4—2441no—
403C22×C4C2×D24411yescharacteristic
404C22×C4#1175822211yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406Q16—2444no—
407Q16—2444no—
408SD16—2444no—
409SD16—2444no—
410D8—2442no—
411D8—2442no—
412M16—22no—
413M16—42no—
414M16—44no—
415M16#1175819914yescomplemented
416C2×C8—22no—
417C2×C8—44no—
418C2×C8—1221no—
419C2×C8—1221no—
420C2×C8—1222no—
421C2×C8—2442no—
422C2×C8C2×C22⋊D6112yes—
423C2×C8C2×C22⋊D6112yes—
424C4⋊C4—21no—
425C4⋊C4—1222no—
426C4⋊C4—1221no—
427C4⋊C4—1222no—
428C4⋊C4—1221no—
429C4⋊C4—2441no—
430C4⋊C4—2441no—
431C4⋊C4C2×D24411yescharacteristic
432C4⋊C4C2×D24411yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C42C2×D24411yescharacteristic
440C42#1175822211yescharacteristic
441C42C2×C22⋊D6111yescharacteristic
442C42C23×D6111yescharacteristic
443C23—21no—
444C23—21no—
445C23—42no—
446C23—42no—
447C23—81no—
448C23—2441no—
449C23#1451975111yescharacteristic
450Q8—4882no—
451Q8—2442no—
452Q8—2441no—
453Q8—2442no—
454Q8—2441no—
455D4—21no—
456D4—21no—
457D4—21no—
458D4—42no—
459D4—4881no—
460D4—4882no—
461D4—2441no—
462D4—2441no—
463D4—2441no—
464D4—2441no—
465D4#1451981012yescomplemented
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—4882no—
479C2×C4—4882no—
480C2×C4—2441no—
481C2×C4—2442no—
482C2×C4#1451975111yescharacteristic
483C2×C4#1451975111yescharacteristic
484C2×C4#1451975111yescharacteristic
485C2×C4C2×#1175819912yes—
486C2×C4#1451980212yes—
487C2×C4C22×C22⋊D6111yescharacteristic
488C2×C4C22×D24411yescharacteristic
489C2×C4C2×#1175822211yescharacteristic
490C8—22no—
491C8—42no—
492C8—2441no—
493C8—2442no—
494C8—2441no—
495C8C2×#1175819914yes—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—84no—
505C22—4881no—
506C22—4881no—
507C22—4881no—
508C22#1683995712yes—
509C22C2×#1451975111yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—4881no—
516C4—4882no—
517C4#1683995712yes—
518C4C22×#1175819911yescharacteristic
519C4C2×#1451980211yescharacteristic
520C4C2×#1451981011yescharacteristic
521C4#1684056211yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—9761no—
526C2C2×#1683995711yescharacteristic
527C2#2145544911yescharacteristic
528C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×M16×C61C2
socleC22×C61C2×#121
prime radical#21455819C2
hypercenterD4×M16D61
hypocenterC61#19163
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #21456278C2semidirect
3. C2×#16840042C2semidirect
4. #21445308C2semidirect
5. #21445284C2semidirect
6. #21445184C2semidirect
7. #21445160C2semidirect
8. #21444571C2semidirect
9. #21444497C2semidirect
10. #21444492C2semidirect
11. D4×M16×C61C2semidirect
17. #16840142C22semidirect
18. #16840135C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839509C22semidirect
38. #16839509C22semidirect
39. #16839509C22semidirect
40. #16839505C22semidirect
144. M16×C61D8semidirect
189. D4×C61#137semidirect
261. D4×M16D61semidirect
313. C61#19163semidirect
415. M16#11758199semidirect
465. D4#14519810semidirect

Maximal subgroups

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

subgroupindex
#214562782
#214558192
C2×#168400422
#214453082
#214452842
#214451842
#214451602
#214445712
#214444972
#214444922
D4×M16×C612
#1916361