Finite Group Databasebeta

← #29833792

Subgroups — #29833792

Order 15616.

1150 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
1C22×#14519796C211yesmaximal, characteristic, complemented
2C2×#16840193C212yesmaximal, complemented
3#21455402C211yesmaximal, characteristic, complemented
4#21447756C212yesmaximal, complemented
5#21447752C211yesmaximal, characteristic, complemented
6#21447743C212yesmaximal, complemented
7#21447738C211yesmaximal, characteristic, complemented
8#21447731C211yesmaximal, characteristic, complemented
9#21447523C212yesmaximal, complemented
10C22⋊C16×D61C211yesmaximal, characteristic, complemented
11C61×#2187C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C2×#14519800C2211yescharacteristic, complemented
14C2×#14519796—28no—
15C2×#14519796—21no—
16C2×#14519796—21no—
17C2×#14519796C2212yes—
18C2×#14519796C2212yes—
19C22×Dic244C2211yescharacteristic
20C22×#11758200—21no—
21C22×C8×D61C2211yescharacteristic
22#16840275—21no—
23#16840193—28no—
24C2×#14519723C2212yescomplemented
25C2×#14519722C2212yescomplemented
26C2×#14519718C2212yescomplemented
27C2×C16×D61C2212yescomplemented
28#16840000C2211yescharacteristic, complemented
29#16839999C2211yescharacteristic, complemented
30#16839998C2211yescharacteristic
31#16839994—21no—
32#16839993—21no—
33#16839799C2211yescharacteristic, complemented
34#16839757—21no—
35#16839567C2212yescomplemented
36#16839565C2212yescomplemented
37#16839563C2212yescomplemented
38C22⋊#11758185C2211yescharacteristic
39#16839541C2212yescomplemented
40C22×D8×C61C2211yescharacteristic
41C2×SD32×C61C2212yescomplemented
42C61×#467C2211yescharacteristic
43C61×#357C2212yescomplemented
44C61×C22⋊C16C2211yescharacteristic
45C2×#11758226—21no—
46C2×#11758224—28no—
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—44no—
50C2×#11758224—41no—
51C2×#11758224—41no—
52C22×Dic122—21no—
53C22×C22⋊D61—41no—
54C23×Dic61—41no—
55C22×C4×D61D411yescharacteristic
56#14519800—24no—
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×#11758201—21no—
70C2×D488C2311yescharacteristic
71C22×#92418D411yescharacteristic
72C2×C8×D61—24no—
73C2×C8×D61D411yescharacteristic
74C2×C8×D61D411yescharacteristic
75C2×C8×D61C2311yescharacteristic
76C2×C8×D61C2311yescharacteristic
77#14519762—41no—
78#14519753—21no—
79#14519752—21no—
80#14519751—21no—
81#14519747—41no—
82#14519739—41no—
83#14519737—41no—
84#14519735—21no—
85#14519723—24no—
86#14519722—24no—
87#14519718—24no—
88C2×#11758185C2312yes—
89C16×D61—24no—
90#14519711C2311yescharacteristic
91#14519708—21no—
92#14519701—21no—
93#14519699—21no—
94#14519697—21no—
95C22×D4×C61—21no—
96C2×Q16×C61C2311yescharacteristic
97C2×D8×C61—24no—
98C2×D8×C61—21no—
99C2×D8×C61C2312yes—
100C22×C8×C61C2311yescharacteristic
101C22⋊Q8×C61—21no—
102SD32×C61—24no—
103C2×C16×C61C2312yes—
104#107×C61C2311yescharacteristic
105#103×C61—21no—
106C24×C61—41no—
107C23×D61D811yescharacteristic
108C2×Q8×C61—21no—
109C2×D4×C61—24no—
110C2×D4×C61—22no—
111C2×D4×C61—42no—
112C2×D4×C61—41no—
113C22×C4×C61C2×D411yescharacteristic
114C2×C22⋊D61—48no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61—41no—
118C22×Dic61—48no—
119C22×Dic61—41no—
120C22×Dic61—42no—
121C22×Dic61—42no—
122C22×Dic61D811yescharacteristic
123#11758226—42no—
124#11758226—42no—
125#11758224—28no—
126#11758224—28no—
127#11758224—48no—
128#11758224—48no—
129#11758224—44no—
130#11758224—44no—
131C2×D244—21no—
132C2×D244—41no—
133C2×D244—41no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—41no—
137C2×C4×D61D811yescharacteristic
138C2×C4×D61D811yescharacteristic
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61C2×D411yescharacteristic
141C2×Dic122—24no—
142C2×Dic122—22no—
143C2×Dic122—42no—
144C2×Dic122—41no—
145Q16×C61—22no—
146D8×C61—24no—
147D8×C61—22no—
148D8×C61C2×D414yes—
149C2×C8×C61—22no—
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61C2411yescharacteristic
152C4⋊C4×C61—21no—
153C4⋊C4×C61—41no—
154C22⋊C4×C61—41no—
155#11758201—42no—
156#11758200—28no—
157#11758200—28no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198—41no—
162C4⋊Dic61—21no—
163#11758196—41no—
164C4×Dic61—41no—
165C2×#92418—22no—
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168Dic244—24no—
169Dic244—22no—
170Dic244C2×D414yes—
171D488—22no—
172C8×D61—22no—
173C8×D61—22no—
174C8×D61—22no—
175C8×D61C2×D414yes—
176C16×C61—22no—
177#11758185—22no—
178C23×C61—44no—
179C23×C61—42no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61C2×D811yescharacteristic
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61C2×D811yescharacteristic
188C22×D61C2×D811yescharacteristic
189Q8×C61—41no—
190Q8×C61—41no—
191D4×C61—24no—
192D4×C61—44no—
193D4×C61—42no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61C2×D811yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C22⋊D61—48no—
201C22⋊D61—48no—
202C22⋊D61—82no—
203C2×Dic61—22no—
204C2×Dic61—48no—
205C2×Dic61—44no—
206C2×Dic61—42no—
207C2×Dic61—48no—
208C2×Dic61—41no—
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211D244—41no—
212D244—41no—
213D244—82no—
214D244—82no—
215C4×D61—22no—
216C4×D61—21no—
217C4×D61—21no—
218C4×D61—22no—
219C4×D61—82no—
220C4×D61#12012yes—
221C4×D61AGL(1,ℤ/8)12yes—
222Dic122—24no—
223Dic122—44no—
224Dic122—42no—
225C8×C61—21no—
226C8×C61C22×D412yes—
227#92418—21no—
228#92418#12012yes—
229#33516—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—44no—
233C22×C61—88no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61#50812yescomplemented
237C22×C61C22×D811yescharacteristic
238C2×D61—21no—
239C2×D61—21no—
240C2×D61—21no—
241C2×D61—21no—
242C2×D61—21no—
243C2×D61—44no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61#44712yes—
248C4×C61—21no—
249C4×C61—81no—
250C4×C61—81no—
251C4×C61C2×#12011yescharacteristic
252C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
253Dic61—21no—
254Dic61—44no—
255Dic61—81no—
256Dic61#44712yes—
257C22×#135—611no—
258C2×#508—612no—
259#3050—611no—
260#2189—612no—
261#2188—611no—
262#2187D6111yescharacteristic, complemented
263#2186—612no—
264#2185—611no—
265#2184—611no—
266#2138—612no—
267C2×C22⋊C16—611no—
268C2×C61—22no—
269C2×C61—84no—
270C2×C61C2×#44711yescharacteristic
271C2×C61C2×#50811yescharacteristic
272C2×C61#341411yescharacteristic
273D61—22no—
274D61—41no—
275D61—161no—
276C22×#41—1221no—
277C2×#135—611no—
278C2×#135—612no—
279C2×#135—612no—
280C2×#135—1221no—
281C2×#135—1221no—
282C2×#135—1228no—
283C22×Q16—611no—
284C22×SD16—1221no—
285C22×D8C2×D6111yescharacteristic
286C23×C8—611no—
287#525—1221no—
288#508—1228no—
289C2×Q32—612no—
290C2×SD32—612no—
291C2×SD32C2×D6112yescomplemented
292C2×D16—612no—
293C22×C16—612no—
294#467—611no—
295#467C2×D6111yescharacteristic
296#466—611no—
297#466—611no—
298#465—1221no—
299#465—1221no—
300#427—611no—
301#416—1221no—
302#358—612no—
303#358—612no—
304#357—612no—
305#357C2×D6112yescomplemented
306C22⋊C16—611no—
307C22⋊C16—612no—
308C22⋊C16C2×D6111yescharacteristic
309C61#3351611yescharacteristic, complemented, Sylow
310C2×#41—1222no—
311C2×#41—1222no—
312C2×#41—1221no—
313C2×#41—1228no—
314C2×#41—2441no—
315C2×#41—2441no—
316C2×#41—2444no—
317C22×Q8—1221no—
318C22×D4—21no—
319C22×D4—2441no—
320C23×C4—611no—
321C23×C4—2441no—
322#135—618no—
323#135—1224no—
324#135—1224no—
325#135—1224no—
326#135—1228no—
327#135—1228no—
328#135—1224no—
329C2×Q16—612no—
330C2×Q16—1224no—
331C2×Q16—1221no—
332C2×Q16C22×D6111yescharacteristic
333C2×SD16—1222no—
334C2×SD16—1222no—
335C2×SD16—1221no—
336C2×SD16—1228no—
337C2×D8—24no—
338C2×D8—21no—
339C2×D8—611no—
340C2×D8C22×D6112yes—
341C22×C8—611no—
342C22×C8—611no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8—611no—
346C22×C8—1224no—
347C22×C8C22×D6111yescharacteristic
348#123—2441no—
349C22⋊Q8—21no—
350C22⋊Q8—1221no—
351#121—1221no—
352#121—1221no—
353#120—2441no—
354C4×D4—2441no—
355C4×D4—2441no—
356#117—1221no—
357Q32—1224no—
358SD32—24no—
359SD32—1224no—
360D16—1224no—
361C2×C16—612no—
362C2×C16—1224no—
363C2×C16C22×D6112yes—
364#107—611no—
365#107C22×D6111yescharacteristic
366#106—1221no—
367#103—21no—
368#103—1221no—
369#102—1221no—
370#102—1221no—
371C24—41no—
372C24—611no—
373#41—1228no—
374#41—1228no—
375#41—2442no—
376#41—2442no—
377#41—2444no—
378#41—2448no—
379#41—2448no—
380#41—2444no—
381C2×Q8—21no—
382C2×Q8—1224no—
383C2×Q8—1222no—
384C2×Q8—2442no—
385C2×Q8—2441no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—41no—
390C2×D4—1221no—
391C2×D4—2441no—
392C2×D4—2441no—
393C2×D4—2441no—
394C2×D4—2442no—
395C2×D4—2442no—
396C2×D4—2448no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—2441no—
405C22×C4—2442no—
406C22×C4—2442no—
407C22×C4—2448no—
408C22×C4—2441no—
409C22×C4D4×D6111yescharacteristic
410Q16—22no—
411Q16—614no—
412Q16—1224no—
413Q16—1222no—
414SD16—1228no—
415SD16—1228no—
416SD16—2442no—
417D8—24no—
418D8—22no—
419D8—1222no—
420D8D4×D6114yes—
421C2×C8—22no—
422C2×C8—611no—
423C2×C8—614no—
424C2×C8—611no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—1222no—
429C2×C8D4×D6111yescharacteristic
430C2×C8C23×D6111yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—1221no—
434C4⋊C4—2441no—
435C22⋊C4—41no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C42—2441no—
441C16—22no—
442C16—1222no—
443C23—44no—
444C23—42no—
445C23—82no—
446C23—82no—
447C23—611no—
448C23—611no—
449C23—1222no—
450C23—1222no—
451C23—2441no—
452C23—2441no—
453C23D8×D6111yescharacteristic
454Q8—41no—
455Q8—41no—
456Q8—1224no—
457Q8—2444no—
458Q8—2442no—
459D4—24no—
460D4—44no—
461D4—42no—
462D4—4882no—
463D4—4882no—
464D4—4882no—
465D4—2448no—
466D4—2448no—
467D4—2441no—
468D4—2441no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—611no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—611no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—1221no—
481C2×C4—1221no—
482C2×C4—4882no—
483C2×C4—2448no—
484C2×C4—2448no—
485C2×C4—2441no—
486C2×C4—2444no—
487C2×C4—2442no—
488C2×C4D8×D6111yescharacteristic
489C2×C4C2×D4×D6111yescharacteristic
490C8—21no—
491C8—612no—
492C8—1221no—
493C8C2×D4×D6112yes—
494C22—21no—
495C22—21no—
496C22—44no—
497C22—88no—
498C22—82no—
499C22—82no—
500C22—612no—
501C22—1221no—
502C22—1221no—
503C22—1221no—
504C22—1221no—
505C22—1221no—
506C22—4881no—
507C22—4881no—
508C22—4881no—
509C22—2444no—
510C22#1684019312yescomplemented
511C22C2×D8×D6111yescharacteristic
512C4—21no—
513C4—81no—
514C4—81no—
515C4—612no—
516C4—1221no—
517C4—4881no—
518C4—2444no—
519C4#120×D6111yescharacteristic
520C4AGL(1,ℤ/8)×D6111yescharacteristic
521C2—22no—
522C2—84no—
523C2—1222no—
524C2—9761no—
525C2—2441no—
526C2#447×D6111yescharacteristic
527C2C2×#1684019311yescharacteristic
528C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2187C2
socleC22×C61C22×D8
hypercenter#2187D61
hypocenterC61#33516
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. C22×#14519796C2semidirect
2. C2×#16840193C2semidirect
3. #21455402C2semidirect
4. #21447756C2semidirect
5. #21447752C2semidirect
6. #21447743C2semidirect
7. #21447738C2semidirect
8. #21447731C2semidirect
9. #21447523C2semidirect
10. C22⋊C16×D61C2semidirect
11. C61×#2187C2semidirect
13. C2×#14519800C22semidirect
24. C2×#14519723C22semidirect
25. C2×#14519722C22semidirect
26. C2×#14519718C22semidirect
27. C2×C16×D61C22semidirect
28. #16840000C22semidirect
29. #16839999C22semidirect
33. #16839799C22semidirect
35. #16839567C22semidirect
36. #16839565C22semidirect
37. #16839563C22semidirect
39. #16839541C22semidirect
41. C2×SD32×C61C22semidirect
43. C61×#357C22semidirect
62. #14519796D4semidirect
236. C22×C61#508semidirect
262. #2187D61semidirect
291. C2×SD32C2×D61semidirect
305. #357C2×D61semidirect
309. C61#33516semidirect
510. C22#16840193semidirect

Maximal subgroups

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

subgroupindex
C22×#145197962
C2×#168401932
#214554022
#214477562
#214477522
#214477432
#214477382
#214477312
#214475232
C22⋊C16×D612
C61×#21872
#3351661