Finite Group Databasebeta

← #29833791

Subgroups — #29833791

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
1C2×#16840192C212yesmaximal, complemented
2#21455400C211yesmaximal, characteristic, complemented
3#21447759C212yesmaximal, complemented
4#21447753C211yesmaximal, characteristic, complemented
5#21447748C211yesmaximal, characteristic, complemented
6#21447742C212yesmaximal, complemented
7#21447736C211yesmaximal, characteristic, complemented
8#21447731C211yesmaximal, characteristic, complemented
9#21447526C212yesmaximal, complemented
10C2×C22⋊#11758185C211yesmaximal, characteristic, complemented
11C22×#135×C61C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic, complemented
13#16840273—21no—
14#16840192—28no—
15C2×#14519723C2212yescomplemented
16C2×#14519721C2212yescomplemented
17C2×#14519720C2212yescomplemented
18C2×#14519718C2212yescomplemented
19C22×#11758185C2212yescomplemented
20#16840003C2211yescharacteristic
21#16840003C2211yescharacteristic, complemented
22#16839998C2211yescharacteristic
23#16839998C2211yescharacteristic, complemented
24#16839991—21no—
25#16839991—21no—
26#16839800C2211yescharacteristic, complemented
27#16839759—21no—
28#16839566C2212yescomplemented
29#16839566C2212yescomplemented
30#16839563C2212yescomplemented
31#16839563C2212yescomplemented
32C22⋊#11758185C2211yescharacteristic
33C22⋊#11758185C2211yescharacteristic
34C22⋊#11758185C2212yescomplemented
35C22×#41×C61—21no—
36C2×#135×C61—28no—
37C2×#135×C61—21no—
38C2×#135×C61—21no—
39C2×#135×C61C2212yes—
40C2×#135×C61C2212yes—
41C22×Q16×C61C2211yescharacteristic
42C22×SD16×C61—21no—
43C22×D8×C61C2211yescharacteristic
44C23×C8×C61C2211yescharacteristic
45C2×#11758222—21no—
46#14519799—24no—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488C2311yescharacteristic
50#14519760—41no—
51#14519756—21no—
52#14519756—21no—
53#14519751—21no—
54#14519751—21no—
55#14519746—41no—
56C4×C22⋊D61—41no—
57C4×C22⋊D61—41no—
58#14519736—21no—
59#14519723—24no—
60#14519721—24no—
61#14519720—24no—
62#14519718—24no—
63C2×#11758185—24no—
64C2×#11758185C2312yes—
65C2×#11758185C2312yes—
66#14519711C2311yescharacteristic
67#14519711C2311yescharacteristic
68#14519709—21no—
69#14519702—21no—
70#14519702—21no—
71#14519699—21no—
72#14519699—21no—
73C2×#41×C61—28no—
74C2×#41×C61—22no—
75C2×#41×C61—22no—
76C2×#41×C61—44no—
77C2×#41×C61—41no—
78C2×#41×C61—41no—
79C22×Q8×C61—21no—
80C22×D4×C61—21no—
81C22×D4×C61—41no—
82C23×C4×C61—41no—
83C23×C4×C61D411yescharacteristic
84#135×C61—28no—
85#135×C61—28no—
86#135×C61—24no—
87#135×C61—24no—
88#135×C61—24no—
89#135×C61D418yescomplemented
90C2×Q16×C61—24no—
91C2×Q16×C61—21no—
92C2×Q16×C61C2312yes—
93C2×SD16×C61—28no—
94C2×SD16×C61—22no—
95C2×SD16×C61—22no—
96C2×D8×C61—24no—
97C2×D8×C61—21no—
98C2×D8×C61C2312yes—
99C22×C8×C61—24no—
100C22×C8×C61D411yescharacteristic
101C22×C8×C61D411yescharacteristic
102C22×C8×C61D411yescharacteristic
103C22×C8×C61C2311yescharacteristic
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C24×C61—41no—
107C24×C61D811yescharacteristic
108#41×C61—28no—
109#41×C61—28no—
110#41×C61—48no—
111#41×C61—44no—
112#41×C61—48no—
113#41×C61—44no—
114C2×Q8×C61—24no—
115C2×Q8×C61—22no—
116C2×Q8×C61—42no—
117C2×Q8×C61—41no—
118C2×D4×C61—24no—
119C2×D4×C61—22no—
120C2×D4×C61—42no—
121C2×D4×C61—48no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—41no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—48no—
128C22×C4×C61—41no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61D811yescharacteristic
132C22×C4×C61D811yescharacteristic
133C22×C4×C61D811yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140#11758222—42no—
141#11758222—42no—
142C2×D244—21no—
143C2×C4×D61—41no—
144C2×Dic122—21no—
145Q16×C61—24no—
146Q16×C61—22no—
147Q16×C61C2×D414yes—
148SD16×C61—28no—
149SD16×C61—28no—
150D8×C61—24no—
151D8×C61—22no—
152D8×C61C2×D414yes—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D414yes—
162C2×C8×C61C2411yescharacteristic
163C22⋊Dic61—41no—
164C22⋊Dic61—41no—
165C22⋊Dic61—41no—
166#11758198—41no—
167#11758198—41no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170#11758196—41no—
171#11758196—41no—
172C4×Dic61—41no—
173Dic244—22no—
174D488—22no—
175#11758190—42no—
176#11758185—22no—
177#11758185—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—41no—
181C23×C61—44no—
182C23×C61—42no—
183C23×C61—82no—
184C23×C61—82no—
185C23×C61C2×D811yescharacteristic
186C23×C61C2×D811yescharacteristic
187C23×C61C2×D811yescharacteristic
188C22×D61—41no—
189Q8×C61—24no—
190Q8×C61—44no—
191Q8×C61—42no—
192D4×C61—24no—
193D4×C61—44no—
194D4×C61—48no—
195D4×C61—48no—
196D4×C61—42no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—48no—
205C2×C4×C61—44no—
206C2×C4×C61—42no—
207C2×C4×C61—48no—
208C2×C4×C61#12012yes—
209C2×C4×C61C2×D811yescharacteristic
210C2×C4×C61C2×D811yescharacteristic
211C2×C4×C61C2×D811yescharacteristic
212C2×C4×C61AGL(1,ℤ/8)12yes—
213C2×C4×C61C22×D411yescharacteristic
214C22⋊D61—82no—
215C22⋊D61—82no—
216C22⋊D61—82no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220D244—41no—
221D244—41no—
222C4×D61—82no—
223Dic122—41no—
224Dic122—41no—
225C8×C61—21no—
226C8×C61—21no—
227C8×C61#12012yes—
228C8×C61C22×D412yes—
229#33516—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—44no—
238C22×C61—44no—
239C22×C61—88no—
240C22×C61—82no—
241C22×C61—82no—
242C22×C61#44712yes—
243C22×C61#50812yescomplemented
244C22×C61C22×D811yescharacteristic
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—44no—
251C4×C61#44712yes—
252C4×C61C2×#12011yescharacteristic
253C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
254Dic61—81no—
255Dic61—81no—
256Dic61—81no—
257C22×#135D6111yescharacteristic, complemented
258C2×#508—612no—
259#3050—611no—
260#2189—612no—
261#2188—611no—
262#2187—611no—
263#2186—612no—
264#2185—611no—
265#2184—611no—
266#2138—612no—
267C2×C22⋊C16—611no—
268C2×C61—22no—
269C2×C61—22no—
270C2×C61—41no—
271C2×C61—84no—
272C2×C61C2×#44711yescharacteristic
273C2×C61C2×#50811yescharacteristic
274C2×C61#341411yescharacteristic
275D61—161no—
276C22×#41—21no—
277C2×#135—28no—
278C2×#135—21no—
279C2×#135—21no—
280C2×#135—611no—
281C2×#135C2×D6112yes—
282C2×#135C2×D6112yes—
283C22×Q16C2×D6111yescharacteristic
284C22×SD16—21no—
285C22×D8C2×D6111yescharacteristic
286C23×C8C2×D6111yescharacteristic
287#525—1221no—
288#508—1228no—
289C2×Q32—612no—
290C2×SD32—612no—
291C2×SD32—612no—
292C2×D16—612no—
293C22×C16—612no—
294#467—611no—
295#467—611no—
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#357—612no—
306C22⋊C16—611no—
307C22⋊C16—611no—
308C22⋊C16—612no—
309C61#3351611yescharacteristic, complemented, Sylow
310C2×#41—28no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—44no—
314C2×#41—41no—
315C2×#41—41no—
316C2×#41—1221no—
317C22×Q8—21no—
318C22×D4—21no—
319C22×D4—41no—
320C23×C4—41no—
321C23×C4C22⋊D6111yescharacteristic
322#135—28no—
323#135—28no—
324#135—24no—
325#135—24no—
326#135—24no—
327#135—1224no—
328#135C22⋊D6118yescomplemented
329C2×Q16—24no—
330C2×Q16—21no—
331C2×Q16—611no—
332C2×Q16C22×D6112yes—
333C2×SD16—28no—
334C2×SD16—22no—
335C2×SD16—22no—
336C2×SD16—1221no—
337C2×D8—24no—
338C2×D8—21no—
339C2×D8—611no—
340C2×D8C22×D6112yes—
341C22×C8—24no—
342C22×C8C22⋊D6111yescharacteristic
343C22×C8C22⋊D6111yescharacteristic
344C22×C8C22⋊D6111yescharacteristic
345C22×C8C22×D6111yescharacteristic
346C22×C8C22×D6111yescharacteristic
347C22×C8C22×D6111yescharacteristic
348#123—2441no—
349C22⋊Q8—1221no—
350C22⋊Q8—1221no—
351#121—1221no—
352#121—1221no—
353#120—2441no—
354C4×D4—2441no—
355C4×D4—2441no—
356#117—1221no—
357Q32—1224no—
358SD32—1224no—
359SD32—1224no—
360D16—1224no—
361C2×C16—612no—
362C2×C16—612no—
363C2×C16—1224no—
364#107—611no—
365#107—611no—
366#106—1221no—
367#103—1221no—
368#103—1221no—
369#102—1221no—
370#102—1221no—
371C24—41no—
372C24#1175819911yescharacteristic
373#41—28no—
374#41—28no—
375#41—48no—
376#41—44no—
377#41—48no—
378#41—44no—
379#41—2442no—
380#41—2442no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—42no—
384C2×Q8—41no—
385C2×Q8—1221no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—41no—
393C2×D4—1221no—
394C2×D4—2441no—
395C2×D4—2441no—
396C2×D4—2441no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—48no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—2441no—
404C22×C4#1175819911yescharacteristic
405C22×C4#1175819911yescharacteristic
406C22×C4#1175819911yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410Q16—24no—
411Q16—22no—
412Q16—1222no—
413Q16C2×C22⋊D6114yes—
414SD16—28no—
415SD16—28no—
416SD16—2442no—
417D8—24no—
418D8—22no—
419D8—1222no—
420D8C2×C22⋊D6114yes—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8C2×C22⋊D6111yescharacteristic
427C2×C8C2×C22⋊D6111yescharacteristic
428C2×C8C2×C22⋊D6114yes—
429C2×C8C2×C22⋊D6111yescharacteristic
430C2×C8C23×D6111yescharacteristic
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C22⋊C4—2441no—
440C42—2441no—
441C16—1222no—
442C16—1222no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—44no—
447C23—42no—
448C23—82no—
449C23—82no—
450C23—2441no—
451C23C2×#1175819911yescharacteristic
452C23C2×#1175819911yescharacteristic
453C23C2×#1175819911yescharacteristic
454Q8—24no—
455Q8—44no—
456Q8—42no—
457Q8—2441no—
458Q8—2441no—
459D4—24no—
460D4—44no—
461D4—48no—
462D4—48no—
463D4—42no—
464D4—4882no—
465D4—4882no—
466D4—4882no—
467D4—2441no—
468D4—2441no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—48no—
480C2×C4—4882no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4#1451974612yes—
485C2×C4C2×#1175819911yescharacteristic
486C2×C4C2×#1175819911yescharacteristic
487C2×C4C2×#1175819911yescharacteristic
488C2×C4#1451980312yes—
489C2×C4C22×C22⋊D6111yescharacteristic
490C8—21no—
491C8—21no—
492C8#1451974612yes—
493C8C22×C22⋊D6112yes—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—44no—
503C22—88no—
504C22—82no—
505C22—82no—
506C22—4881no—
507C22—4881no—
508C22—4881no—
509C22#1683989612yes—
510C22#1684019212yescomplemented
511C22C22×#1175819911yescharacteristic
512C4—21no—
513C4—21no—
514C4—44no—
515C4—4881no—
516C4—4881no—
517C4—4881no—
518C4#1683989612yes—
519C4C2×#1451974611yescharacteristic
520C4C2×#1451980311yescharacteristic
521C2—22no—
522C2—22no—
523C2—41no—
524C2—84no—
525C2—9761no—
526C2C2×#1683989611yescharacteristic
527C2C2×#1684019211yescharacteristic
528C2#2145895311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC22×#135×C61C2
socleC22×C61C22×D8
hypercenterC22×#135D61
hypocenterC61#33516
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16840192C2semidirect
2. #21455400C2semidirect
3. #21447759C2semidirect
4. #21447753C2semidirect
5. #21447748C2semidirect
6. #21447742C2semidirect
7. #21447736C2semidirect
8. #21447731C2semidirect
9. #21447526C2semidirect
10. C2×C22⋊#11758185C2semidirect
11. C22×#135×C61C2semidirect
12. C2×#14519799C22semidirect
15. C2×#14519723C22semidirect
16. C2×#14519721C22semidirect
17. C2×#14519720C22semidirect
18. C2×#14519718C22semidirect
19. C22×#11758185C22semidirect
21. #16840003C22semidirect
23. #16839998C22semidirect
26. #16839800C22semidirect
28. #16839566C22semidirect
29. #16839566C22semidirect
30. #16839563C22semidirect
31. #16839563C22semidirect
34. C22⋊#11758185C22semidirect
89. #135×C61D4semidirect
243. C22×C61#508semidirect
257. C22×#135D61semidirect
309. C61#33516semidirect
328. #135C22⋊D61semidirect
510. C22#16840192semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
#214554002
#214477592
#214477532
#214477482
#214477422
#214477362
#214477312
#214475262
C2×C22⋊#117581852
C22×#135×C612
#3351661