Finite Group Databasebeta

← #29833820

Subgroups — #29833820

Order 15616.

1192 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519796C211yesmaximal, characteristic, complemented
2C2×#16840196C212yesmaximal, complemented
3#466×D61C211yesmaximal, characteristic, complemented
4#21447743C212yesmaximal, complemented
5#21447739C212yesmaximal, complemented
6#21447738C211yesmaximal, characteristic, complemented
7#21447735C211yesmaximal, characteristic, complemented
8#21447731C211yesmaximal, characteristic, complemented
9#21447528C212yesmaximal, complemented
10#21447428C211yesmaximal, characteristic, complemented
11C61×#2184C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C2×#14519796—28no—
14C2×#14519796—21no—
15C2×#14519796—21no—
16C2×#14519796C2212yes—
17C2×#14519796C2212yescomplemented
18C22×Dic244C2211yescharacteristic, complemented
19C22×#11758200—21no—
20C2×D8×D61C2211yescharacteristic, complemented
21C22×C8×D61C2211yescharacteristic
22#121×D61—21no—
23#16840196—28no—
24C2×#14519722C2212yescomplemented
25C2×#14519721C2212yescomplemented
26C2×#14519718C2212yescomplemented
27C2×#14519716C2212yescomplemented
28#16839999C2211yescharacteristic, complemented
29#16839998C2211yescharacteristic
30#16839997—21no—
31#16839996—21no—
32#16839995C2211yescharacteristic, complemented
33#107×D61C2211yescharacteristic, complemented
34#102×D61—21no—
35#16839567C2212yescomplemented
36#16839565C2212yescomplemented
37#16839563C2212yescomplemented
38C22⋊#11758185C2211yescharacteristic, complemented
39#16839541C2212yescomplemented
40C22×D8×C61C2211yescharacteristic, complemented
41C2×D16×C61C2212yescomplemented
42C61×#466C2211yescharacteristic
43C61×#357C2212yescomplemented
44C61×C22⋊C16C2211yescharacteristic, complemented
45C2×#11758224—28no—
46C2×#11758224—22no—
47C2×#11758224—22no—
48C2×#11758224—44no—
49C2×#11758224—41no—
50C2×#11758224—41no—
51C22×Dic122—21no—
52C22×C22⋊D61—41no—
53C2×D4×D61—21no—
54C2×D4×D61—41no—
55C23×Dic61—41no—
56C22×C4×D61D411yescharacteristic
57#14519796—28no—
58#14519796—28no—
59#14519796—24no—
60#14519796—24no—
61#14519796—24no—
62#14519796D418yescomplemented
63C2×Dic244—21no—
64C2×Dic244—24no—
65C2×Dic244C2312yes—
66C2×#11758200—28no—
67C2×#11758200—22no—
68C2×#11758200—22no—
69C2×D488C2311yescharacteristic
70C2×#11758199—21no—
71D8×D61—24no—
72C22×#92418D411yescharacteristic
73C2×C8×D61—24no—
74C2×C8×D61D411yescharacteristic
75C2×C8×D61D411yescharacteristic
76C2×C8×D61C2311yescharacteristic
77C2×C8×D61C2311yescharacteristic
78#14519752—21no—
79#14519751—21no—
80#14519750—41no—
81#14519749—41no—
82#14519748—21no—
83C4⋊C4×D61—21no—
84C22⋊C4×D61—41no—
85#14519722—24no—
86#14519721—24no—
87#14519718—24no—
88#14519716—24no—
89C2×#11758185C2312yes—
90#14519711C2311yescharacteristic
91#14519710—21no—
92#14519699—21no—
93#14519698—21no—
94#14519697—21no—
95C22×D4×C61—21no—
96C2×D8×C61—24no—
97C2×D8×C61—21no—
98C2×D8×C61C2311yescharacteristic
99C2×D8×C61C2312yes—
100C22×C8×C61C2311yescharacteristic
101#121×C61—21no—
102D16×C61—24no—
103C2×C16×C61C2312yes—
104#107×C61C2311yescharacteristic
105#102×C61—21no—
106C24×C61—41no—
107C23×D61—41no—
108C23×D61D811yescharacteristic
109C2×D4×C61—24no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—42no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C22×C4×C61C2×D411yescharacteristic
116C2×C22⋊D61—48no—
117C2×C22⋊D61—42no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—48no—
123C22×Dic61—41no—
124C22×Dic61—42no—
125C22×Dic61—42no—
126C22×Dic61D811yescharacteristic
127#11758224—28no—
128#11758224—28no—
129#11758224—48no—
130#11758224—48no—
131#11758224—44no—
132#11758224—44no—
133D4×D61—42no—
134D4×D61—42no—
135D4×D61—84no—
136C2×D244—21no—
137C2×D244—41no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—41no—
141C2×C4×D61D811yescharacteristic
142C2×C4×D61D811yescharacteristic
143C2×C4×D61C2×D411yescharacteristic
144C2×C4×D61C2×D411yescharacteristic
145C2×Dic122—24no—
146C2×Dic122—22no—
147C2×Dic122—42no—
148C2×Dic122—41no—
149D8×C61—24no—
150D8×C61—22no—
151D8×C61—22no—
152D8×C61C2×D414yescomplemented
153C2×C8×C61—22no—
154C2×C8×C61C2×D411yescharacteristic
155C2×C8×C61C2411yescharacteristic
156C4⋊C4×C61—21no—
157C22⋊C4×C61—41no—
158C22⋊Dic61—41no—
159#11758200—28no—
160#11758200—28no—
161#11758199—42no—
162#11758198—41no—
163#11758198—41no—
164C4⋊Dic61—21no—
165#11758196—41no—
166C2×#92418—22no—
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169Dic244—24no—
170Dic244—22no—
171Dic244C2×D414yescomplemented
172D488—22no—
173C8×D61—22no—
174C8×D61—22no—
175C8×D61—22no—
176C8×D61C2×D414yes—
177C16×C61—22no—
178#11758185—22no—
179C23×C61—44no—
180C23×C61—42no—
181C23×C61—41no—
182C23×C61—82no—
183C23×C61—82no—
184C23×C61C2×D811yescharacteristic
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—82no—
190C22×D61—82no—
191C22×D61—82no—
192C22×D61C2×D811yescharacteristic
193C22×D61C2×D811yescharacteristic
194D4×C61—24no—
195D4×C61—44no—
196D4×C61—41no—
197D4×C61—42no—
198D4×C61—41no—
199D4×C61—82no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—41no—
203C2×C4×C61C2×D811yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C22⋊D61—48no—
206C22⋊D61—48no—
207C22⋊D61—82no—
208C22⋊D61—82no—
209C22⋊D61—82no—
210C2×Dic61—22no—
211C2×Dic61—48no—
212C2×Dic61—44no—
213C2×Dic61—42no—
214C2×Dic61—48no—
215C2×Dic61—41no—
216C2×Dic61C2×D811yescharacteristic
217C2×Dic61C2×D811yescharacteristic
218D244—41no—
219D244—41no—
220D244—82no—
221C4×D61—22no—
222C4×D61—21no—
223C4×D61—21no—
224C4×D61—22no—
225C4×D61—82no—
226C4×D61#12012yes—
227C4×D61AGL(1,ℤ/8)12yes—
228Dic122—24no—
229Dic122—44no—
230Dic122—42no—
231C8×C61—21no—
232C8×C61C22×D412yes—
233#92418—21no—
234#92418#12012yes—
235#33520—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—44no—
239C22×C61—88no—
240C22×C61—82no—
241C22×C61—82no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61#50912yescomplemented
246C22×C61C22×D811yescharacteristic
247C2×D61—21no—
248C2×D61—21no—
249C2×D61—21no—
250C2×D61—21no—
251C2×D61—21no—
252C2×D61—44no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—164no—
257C2×D61#44712yes—
258C4×C61—21no—
259C4×C61—81no—
260C4×C61C2×#12011yescharacteristic
261C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
262Dic61—21no—
263Dic61—44no—
264Dic61—81no—
265Dic61#44712yes—
266C22×#135—611no—
267C2×#509—612no—
268C2×#466—611no—
269#2186—612no—
270#2186—612no—
271#2185—611no—
272#2185—611no—
273#2184—611no—
274#2184D6111yescharacteristic, complemented
275#2139—612no—
276#2112—611no—
277C2×C61—22no—
278C2×C61—84no—
279C2×C61—161no—
280C2×C61C2×#44711yescharacteristic
281C2×C61C2×#50911yescharacteristic
282C2×C61#341411yescharacteristic
283D61—22no—
284D61—41no—
285D61—161no—
286C22×#41—1221no—
287C2×#135—612no—
288C2×#135—612no—
289C2×#135—1221no—
290C2×#135—1221no—
291C2×#135—1228no—
292C22×Q16—611no—
293C22×SD16—1221no—
294C22×D8—611no—
295C22×D8C2×D6111yescharacteristic, complemented
296C23×C8—611no—
297C2×#121—1221no—
298#509—1228no—
299C2×SD32—612no—
300C2×SD32—612no—
301C2×D16—612no—
302C2×D16C2×D6112yescomplemented
303C2×M32—612no—
304#466—611no—
305#466—611no—
306#466—611no—
307#466—1221no—
308#466—1221no—
309#466C2×D6111yescharacteristic
310C2×#107—611no—
311C2×#102—1221no—
312#358—612no—
313#358—612no—
314#357—612no—
315#357C2×D6112yescomplemented
316C22⋊C16—611no—
317C22⋊C16—612no—
318C22⋊C16C2×D6111yescharacteristic, complemented
319C61#3352011yescharacteristic, complemented, Sylow
320C2×#41—1222no—
321C2×#41—1222no—
322C2×#41—1228no—
323C2×#41—2441no—
324C2×#41—2441no—
325C2×#41—2444no—
326C22×Q8—1221no—
327C22×D4—21no—
328C22×D4—1221no—
329C22×D4—2441no—
330C22×D4—2441no—
331C23×C4—611no—
332C23×C4—2441no—
333#135—618no—
334#135—1224no—
335#135—1224no—
336#135—1228no—
337#135—1228no—
338#135—1224no—
339C2×Q16—612no—
340C2×Q16—1224no—
341C2×Q16—1221no—
342C2×SD16—1222no—
343C2×SD16—1222no—
344C2×SD16—1228no—
345C2×D8—24no—
346C2×D8—21no—
347C2×D8—611no—
348C2×D8—1224no—
349C2×D8—1221no—
350C2×D8C22×D6111yescharacteristic
351C2×D8C22×D6112yes—
352C22×C8—611no—
353C22×C8—611no—
354C22×C8—611no—
355C22×C8—611no—
356C22×C8—611no—
357C22×C8—1224no—
358C22×C8C22×D6111yescharacteristic
359#121—21no—
360#121—1221no—
361#121—1221no—
362#121—1221no—
363#121—2441no—
364#121—2441no—
365C2×C4⋊C4—1221no—
366C2×C22⋊C4—2441no—
367SD32—1224no—
368SD32—1224no—
369D16—24no—
370D16—1224no—
371M32—1224no—
372C2×C16—612no—
373C2×C16C22×D6112yes—
374#107—611no—
375#107—1221no—
376#107C22×D6111yescharacteristic
377#102—21no—
378#102—1221no—
379#102—1221no—
380#102—1221no—
381C24—41no—
382C24—611no—
383C24—2441no—
384#41—1228no—
385#41—1228no—
386#41—2444no—
387#41—2448no—
388#41—2448no—
389#41—2444no—
390C2×Q8—1224no—
391C2×Q8—1222no—
392C2×Q8—2442no—
393C2×Q8—2441no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—1221no—
401C2×D4—4884no—
402C2×D4—2441no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4—2441no—
406C2×D4—2441no—
407C2×D4—2441no—
408C2×D4—2442no—
409C2×D4—2442no—
410C2×D4—2448no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—2441no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4—2448no—
422C22×C4—2441no—
423C22×C4D4×D6111yescharacteristic
424Q16—614no—
425Q16—1224no—
426Q16—1222no—
427SD16—1228no—
428SD16—1228no—
429D8—24no—
430D8—22no—
431D8—22no—
432D8—1222no—
433D8—2442no—
434D8D4×D6114yescomplemented
435C2×C8—22no—
436C2×C8—611no—
437C2×C8—614no—
438C2×C8—611no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8D4×D6111yescharacteristic
444C2×C8C23×D6111yescharacteristic
445C4⋊C4—21no—
446C4⋊C4—1221no—
447C4⋊C4—2441no—
448C22⋊C4—41no—
449C22⋊C4—2441no—
450C22⋊C4—2441no—
451C22⋊C4—2441no—
452C16—22no—
453C16—1222no—
454C23—44no—
455C23—42no—
456C23—41no—
457C23—82no—
458C23—82no—
459C23—611no—
460C23—611no—
461C23—1222no—
462C23—1222no—
463C23—4882no—
464C23—4882no—
465C23—4882no—
466C23—2441no—
467C23—2441no—
468C23D8×D6111yescharacteristic
469Q8—1224no—
470Q8—2444no—
471Q8—2442no—
472D4—24no—
473D4—44no—
474D4—41no—
475D4—42no—
476D4—41no—
477D4—82no—
478D4—4882no—
479D4—4882no—
480D4—4882no—
481D4—4882no—
482D4—2448no—
483D4—2448no—
484D4—2441no—
485D4—2441no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—611no—
490C2×C4—612no—
491C2×C4—612no—
492C2×C4—611no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—4882no—
499C2×C4—2448no—
500C2×C4—2448no—
501C2×C4—2441no—
502C2×C4—2444no—
503C2×C4—2442no—
504C2×C4D8×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C8—21no—
507C8—612no—
508C8—1221no—
509C8C2×D4×D6112yes—
510C22—21no—
511C22—21no—
512C22—44no—
513C22—88no—
514C22—82no—
515C22—82no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—612no—
520C22—1221no—
521C22—1221no—
522C22—1221no—
523C22—1221no—
524C22—1221no—
525C22—9764no—
526C22—4881no—
527C22—4881no—
528C22—4881no—
529C22—2444no—
530C22#1684019612yescomplemented
531C22C2×D8×D6111yescharacteristic
532C4—21no—
533C4—81no—
534C4—612no—
535C4—1221no—
536C4—4881no—
537C4—2444no—
538C4#120×D6111yescharacteristic
539C4AGL(1,ℤ/8)×D6111yescharacteristic
540C2—22no—
541C2—84no—
542C2—161no—
543C2—1222no—
544C2—9761no—
545C2—2441no—
546C2#447×D6111yescharacteristic
547C2C2×#1684019611yescharacteristic
548C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2184C2
socleC22×C61C22×D8
hypercenter#2184D61
hypocenterC61#33520
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

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

NH ≅ G/Ntype
1. C22×#14519796C2semidirect
2. C2×#16840196C2semidirect
3. #466×D61C2semidirect
4. #21447743C2semidirect
5. #21447739C2semidirect
6. #21447738C2semidirect
7. #21447735C2semidirect
8. #21447731C2semidirect
9. #21447528C2semidirect
10. #21447428C2semidirect
11. C61×#2184C2semidirect
17. C2×#14519796C22semidirect
18. C22×Dic244C22semidirect
20. C2×D8×D61C22semidirect
24. C2×#14519722C22semidirect
25. C2×#14519721C22semidirect
26. C2×#14519718C22semidirect
27. C2×#14519716C22semidirect
28. #16839999C22semidirect
32. #16839995C22semidirect
33. #107×D61C22semidirect
35. #16839567C22semidirect
36. #16839565C22semidirect
37. #16839563C22semidirect
38. C22⋊#11758185C22semidirect
39. #16839541C22semidirect
40. C22×D8×C61C22semidirect
41. C2×D16×C61C22semidirect
43. C61×#357C22semidirect
44. C61×C22⋊C16C22semidirect
62. #14519796D4semidirect
152. D8×C61C2×D4semidirect
171. Dic244C2×D4semidirect
245. C22×C61#509semidirect
274. #2184D61semidirect
295. C22×D8C2×D61semidirect
302. C2×D16C2×D61semidirect
315. #357C2×D61semidirect
318. C22⋊C16C2×D61semidirect
319. C61#33520semidirect
434. D8D4×D61semidirect
530. C22#16840196semidirect

Maximal subgroups

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

subgroupindex
C22×#145197962
C2×#168401962
#466×D612
#214477432
#214477392
#214477382
#214477352
#214477312
#214475282
#214474282
C61×#21842
#3352061