Finite Group Databasebeta

← #29836981

Subgroups — #29836981

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458166C211yesmaximal, characteristic, complemented
2#21457505C211yesmaximal, characteristic, complemented
3#21448058C211yesmaximal, characteristic, complemented
4#21448018C211yesmaximal, characteristic, complemented
5#21448006C211yesmaximal, characteristic, complemented
6#21447923C211yesmaximal, characteristic, complemented
7#21447916C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447743C211yesmaximal, characteristic, complemented
10#21447732C211yesmaximal, characteristic, complemented
11#21447693C211yesmaximal, characteristic, complemented
12#21447643C211yesmaximal, characteristic, complemented
13#21447528C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C61×#2226C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18#16840415—21no—
19#16840407—21no—
20C2×#14519722C2211yescharacteristic, complemented
21C2×D976C2211yescharacteristic
22C2×#14519718C2211yescharacteristic
23C2×#14519716C2211yescharacteristic, complemented
24C2×C16×D61C2211yescharacteristic
25#16840146C2211yescharacteristic
26#16840077—21no—
27#16840077—21no—
28#16840039—21no—
29#16840004C2211yescharacteristic
30#16839999C2211yescharacteristic
31#16839999C2211yescharacteristic
32#16839999C2211yescharacteristic, complemented
33#16839977—21no—
34#16839914—21no—
35#16839908—21no—
36C4×D488C2211yescharacteristic, complemented
37C8×D244C2211yescharacteristic
38#107×D61C2211yescharacteristic
39#107×D61C2211yescharacteristic, complemented
40#16839589C2211yescharacteristic, complemented
41#16839567C2211yescharacteristic, complemented
42#16839565C2211yescharacteristic, complemented
43#16839565C2211yescharacteristic, complemented
44#16839564C2211yescharacteristic
45#16839563C2211yescharacteristic
46#16839562C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic, complemented
49#16839541C2211yescharacteristic
50#16839541C2211yescharacteristic, complemented
51#16839538C2211yescharacteristic, complemented
52C61×#500C2211yescharacteristic, complemented
53C4×D8×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic, complemented
55C61×#367C2211yescharacteristic, complemented
56C61×C4⋊C16C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×#11758226—41no—
60C2×#11758224—21no—
61C2×D4×D61—21no—
62#14519796—22no—
63#14519796—41no—
64C2×Dic244C2311yescharacteristic
65C2×#11758190—21no—
66C2×#11758200—21no—
67C2×D488C2311yescharacteristic
68C2×D488C2311yescharacteristic
69C2×#11758199—21no—
70D8×D61—22no—
71D8×D61—41no—
72C2×C8×D61C2311yescharacteristic
73C2×C8×D61C2311yescharacteristic
74#14519765—21no—
75#14519762—41no—
76#14519762—41no—
77#14519761—41no—
78#14519757—21no—
79#14519753—41no—
80#14519752—21no—
81#14519752—21no—
82#14519752—21no—
83#14519751—41no—
84#14519747—41no—
85C4×D244—21no—
86C4×D244D411yescharacteristic
87C4⋊C4×D61—21no—
88C4⋊C4×D61—21no—
89C2×#11758198—41no—
90#14519722—41no—
91D976—41no—
92#14519718—22no—
93#14519716—41no—
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96C16×D61—41no—
97#14519711C2311yescharacteristic
98#14519710—21no—
99#14519710—21no—
100C4⋊#92418D411yescharacteristic
101#14519700—21no—
102#14519699—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519689D411yescharacteristic
107#14519689D411yescharacteristic
108C2×D8×C61C2311yescharacteristic
109C4⋊Q8×C61—21no—
110C4×D4×C61—21no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113#107×C61—21no—
114#107×C61C2311yescharacteristic
115#107×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#102×C61—21no—
118C4×C8×C61C2311yescharacteristic
119C23×D61—41no—
120C2×Q8×C61—41no—
121C2×D4×C61—21no—
122C22×C4×C61—41no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C22×Dic61—41no—
126#11758226—84no—
127#11758224—42no—
128#11758224—81no—
129D4×D61—42no—
130D4×D61—81no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—41no—
134C2×D244—41no—
135C2×D244—41no—
136C2×D244D811yescharacteristic
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×Dic122—21no—
143D8×C61—21no—
144D8×C61#4112yescomplemented
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2411yescharacteristic
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C42×C61C2×D411yescharacteristic
153#11758200—42no—
154#11758199—42no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198—41no—
160#11758198—41no—
161#11758198—41no—
162#11758198D811yescharacteristic
163#11758198D811yescharacteristic
164C4⋊Dic61—21no—
165C4⋊Dic61—41no—
166C4⋊Dic61D811yescharacteristic
167#11758196—41no—
168#11758196—41no—
169C2×#92418C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171Dic244—21no—
172Dic244—21no—
173D488—21no—
174D488—21no—
175D488—21no—
176D488—21no—
177#11758190—81no—
178C8×D61—21no—
179C8×D61—21no—
180C8×D61—21no—
181C8×D61—21no—
182C16×C61—21no—
183C16×C61—21no—
184#11758185—21no—
185#11758185—21no—
186C23×C61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—82no—
190C22×D61—82no—
191C22×D61—161no—
192C22×D61C2×D811yescharacteristic
193C22×D61C2×D811yescharacteristic
194Q8×C61—82no—
195D4×C61—22no—
196D4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—161no—
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C22⋊D61—82no—
205C22⋊D61—82no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—82no—
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211D244—22no—
212D244—41no—
213D244—41no—
214D244—41no—
215D244—41no—
216D244—82no—
217D244—82no—
218D244—161no—
219C4×D61—21no—
220C4×D61—21no—
221C4×D61—21no—
222C4×D61—21no—
223C4×D61—82no—
224C4×D61—82no—
225C4×D61—82no—
226Dic122—41no—
227Dic122—41no—
228C8×C61—21no—
229C8×C61C2×#4111yescharacteristic
230C8×C61H(5,𝔽2)11yescharacteristic
231#92418—21no—
232#92418—21no—
233#33873—611nomaximal, Sylow
234C22×C61—42no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61C22×D811yescharacteristic
238C2×D61—21no—
239C2×D61—21no—
240C2×D61—21no—
241C2×D61—21no—
242C2×D61—21no—
243C2×D61—81no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—164no—
250C2×D61#50912yes—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—81no—
254C4×C61—81no—
255C4×C61—81no—
256C4×C61#54611yescharacteristic
257C4×C61#57711yescharacteristic
258Dic61—21no—
259Dic61—21no—
260Dic61—81no—
261Dic61—81no—
262#3334—611no—
263#3280—611no—
264#2243—611no—
265#2235—611no—
266#2233—611no—
267#2226D6111yescharacteristic, complemented
268#2218—611no—
269#2217—611no—
270#2210—611no—
271#2186—611no—
272#2184—611no—
273#2177—611no—
274#2168—611no—
275#2139—611no—
276C2×#357—611no—
277C2×C61—82no—
278C2×C61#329411yescharacteristic
279C2×C61C2×#50911yescharacteristic
280C2×C61#341611yescharacteristic
281D61—22no—
282D61—41no—
283D61—161no—
284D61—161no—
285C2×#135—611no—
286C22×D8—611no—
287#547—1221no—
288#546—1221no—
289C2×SD32—611no—
290C2×D16—611no—
291C2×D16—611no—
292C2×M32—611no—
293C22×C16—611no—
294#500C2×D6111yescharacteristic, complemented
295#495—611no—
296#480—1221no—
297#480—1221no—
298#474—1221no—
299#467—611no—
300#466—611no—
301#466—611no—
302#466—611no—
303#463—1221no—
304#450—1221no—
305#449—1221no—
306C4×D8—611no—
307C4×D8C2×D6111yescharacteristic
308C8×D4—611no—
309C2×#107—611no—
310C2×#107—611no—
311#367—611no—
312#367C2×D6111yescharacteristic, complemented
313#367C2×D6111yescharacteristic, complemented
314C4⋊C16C2×D6111yescharacteristic, complemented
315#358—611no—
316#358—611no—
317#358—611no—
318#357—611no—
319#357—611no—
320#357—611no—
321#357—611no—
322#357—611no—
323#357C2×D6111yescharacteristic
324#357C2×D6111yescharacteristic
325C22⋊C16—611no—
326C22⋊C16—611no—
327#346—611no—
328C61#3387311yescharacteristic, complemented, Sylow
329C2×#41—1221no—
330C2×#41—2441no—
331C22×D4—1221no—
332#135—1222no—
333#135—2441no—
334C2×Q16—611no—
335C2×SD16—1221no—
336C2×SD16—1221no—
337C2×D8—611no—
338C2×D8—611no—
339C2×D8—1221no—
340C2×D8—1222no—
341C2×D8—2441no—
342C2×D8C22×D6111yescharacteristic
343C22×C8—611no—
344C22×C8—611no—
345C4⋊Q8—21no—
346#124—1221no—
347#123—2441no—
348#123—2441no—
349#123—2441no—
350C22⋊Q8—1221no—
351C22⋊Q8—2441no—
352#121—1221no—
353#121—1221no—
354#121—1221no—
355#121—2441no—
356#120—2441no—
357C4×D4—21no—
358C4×D4—611no—
359C4×D4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C22⋊C4—2441no—
363SD32—2441no—
364D16—1222no—
365D16—2441no—
366M32—2441no—
367C2×C16—611no—
368C2×C16—611no—
369C2×C16—2441no—
370C2×C16C22×D6111yescharacteristic
371C2×C16C22×D6111yescharacteristic
372#107—21no—
373#107—611no—
374#107—1221no—
375#107—1221no—
376#107C22×D6111yescharacteristic
377#107C22×D6111yescharacteristic
378#107C22×D6111yescharacteristic
379C4⋊C8—611no—
380#103—1221no—
381#102—21no—
382#102—1221no—
383#102—1221no—
384#102—1221no—
385#102—1221no—
386C22⋊C8—611no—
387C22⋊C8—611no—
388C4×C8C22×D6111yescharacteristic
389C24—2441no—
390#41—4881no—
391#41—4884no—
392#41—2442no—
393C2×Q8—41no—
394C2×Q8—1221no—
395C2×D4—21no—
396C2×D4—611no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—4881no—
400C2×D4—2441no—
401C2×D4—2441no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4—2442no—
405C2×D4—2441no—
406C22×C4—41no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4—2441no—
412C22×C4—2441no—
413Q16—1221no—
414Q16—1221no—
415SD16—4881no—
416SD16—2442no—
417D8—21no—
418D8—1221no—
419D8—1221no—
420D8—1221no—
421D8—1221no—
422D8—2442no—
423D8#1175822612yescomplemented
424C2×C8—611no—
425C2×C8—611no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8D4×D6111yescharacteristic
431C2×C8C23×D6111yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—611no—
437C4⋊C4—1221no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C22⋊C4—41no—
442C22⋊C4—611no—
443C22⋊C4—611no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C22⋊C4—2441no—
451C42D4×D6111yescharacteristic
452C16—21no—
453C16—21no—
454C16—1221no—
455C16—1221no—
456C23—41no—
457C23—611no—
458C23—611no—
459C23—9761no—
460C23—4882no—
461C23—4882no—
462C23—2441no—
463C23—2441no—
464Q8—82no—
465Q8—2441no—
466Q8—2441no—
467D4—22no—
468D4—41no—
469D4—1222no—
470D4—9761no—
471D4—4882no—
472D4—4882no—
473D4—4882no—
474D4—4882no—
475D4—2441no—
476D4—2441no—
477D4—2441no—
478D4—2441no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—161no—
483C2×C4—611no—
484C2×C4—611no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—4882no—
490C2×C4—4882no—
491C2×C4—4882no—
492C2×C4—4882no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4D8×D6111yescharacteristic
496C2×C4D8×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C8—21no—
499C8—1221no—
500C8—1221no—
501C8C2×#1175822611yescharacteristic
502C8#1451982311yescharacteristic
503C22—42no—
504C22—81no—
505C22—81no—
506C22—612no—
507C22—1221no—
508C22—1221no—
509C22—1221no—
510C22—1221no—
511C22—1221no—
512C22—9764no—
513C22—4881no—
514C22—4881no—
515C22—4881no—
516C22—4881no—
517C22—4881no—
518C22—4881no—
519C22C2×D8×D6111yescharacteristic
520C4—21no—
521C4—21no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—1221no—
526C4—1221no—
527C4—4881no—
528C4—4881no—
529C4#1684040311yescharacteristic
530C4#1684057111yescharacteristic
531C2—82no—
532C2—1222no—
533C2—9761no—
534C2—9761no—
535C2—2441no—
536C2#2145762911yescharacteristic
537C2#509×D6111yescharacteristic
538C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2226C2
socleC22×C61C22×D8
hypercenter#2226D61
hypocenterC61#33873
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #21458166C2semidirect
2. #21457505C2semidirect
3. #21448058C2semidirect
4. #21448018C2semidirect
5. #21448006C2semidirect
6. #21447923C2semidirect
7. #21447916C2semidirect
8. #21447875C2semidirect
9. #21447743C2semidirect
10. #21447732C2semidirect
11. #21447693C2semidirect
12. #21447643C2semidirect
13. #21447528C2semidirect
14. #357×D61C2semidirect
15. C61×#2226C2semidirect
20. C2×#14519722C22semidirect
23. C2×#14519716C22semidirect
32. #16839999C22semidirect
36. C4×D488C22semidirect
39. #107×D61C22semidirect
40. #16839589C22semidirect
41. #16839567C22semidirect
42. #16839565C22semidirect
43. #16839565C22semidirect
48. #16839562C22semidirect
50. #16839541C22semidirect
51. #16839538C22semidirect
52. C61×#500C22semidirect
54. C61×#367C22semidirect
55. C61×#367C22semidirect
56. C61×C4⋊C16C22semidirect
144. D8×C61#41semidirect
267. #2226D61semidirect
294. #500C2×D61semidirect
312. #367C2×D61semidirect
313. #367C2×D61semidirect
314. C4⋊C16C2×D61semidirect
328. C61#33873semidirect
423. D8#11758226semidirect

Maximal subgroups

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

subgroupindex
#214581662
#214575052
#214480582
#214480182
#214480062
#214479232
#214479162
#214478752
#214477432
#214477322
#214476932
#214476432
#214475282
#357×D612
C61×#22262
#3387361