Finite Group Databasebeta

← #29836852

Subgroups — #29836852

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457506C211yesmaximal, characteristic
2#21448053C211yesmaximal, characteristic, complemented
3#21448016C211yesmaximal, characteristic, complemented
4#21448010C211yesmaximal, characteristic, complemented
5#21447973C211yesmaximal, characteristic, complemented
6#21447917C211yesmaximal, characteristic, complemented
7#21447908C211yesmaximal, characteristic, complemented
8#21447881C211yesmaximal, characteristic, complemented
9#21447757C211yesmaximal, characteristic, complemented
10#21447747C211yesmaximal, characteristic, complemented
11C4×#14519720C211yesmaximal, characteristic, complemented
12D4×#11758185C211yesmaximal, characteristic, complemented
13#21447524C211yesmaximal, characteristic, complemented
14C2×#16839562C211yesmaximal, characteristic, complemented
15C61×#3336C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×D488C2211yescharacteristic
18#16840408—21no—
19C2×#14519723C2211yescharacteristic, complemented
20C2×#14519720C2211yescharacteristic
21C2×#14519720C2211yescharacteristic
22C22×#11758185C2211yescharacteristic
23C22×#11758185C2211yescharacteristic, complemented
24#16840146C2211yescharacteristic
25#16840003C2211yescharacteristic
26#16839998C2211yescharacteristic
27#16839978—21no—
28#16839915—21no—
29#16839909—21no—
30C4×D488C2211yescharacteristic
31#16839589C2211yescharacteristic, complemented
32#16839589C2211yescharacteristic, complemented
33#16839587C2211yescharacteristic, complemented
34C4⋊#11758185C2211yescharacteristic, complemented
35#16839566C2211yescharacteristic, complemented
36#16839565C2211yescharacteristic, complemented
37#16839565C2211yescharacteristic, complemented
38#16839565C2211yescharacteristic, complemented
39#16839562C2211yescharacteristic
40#16839562C2211yescharacteristic
41#16839562C2211yescharacteristic
42#16839562C2211yescharacteristic
43#16839562C2211yescharacteristic
44#16839562C2211yescharacteristic
45C22⋊#11758185C2211yescharacteristic
46C22⋊#11758185C2211yescharacteristic, complemented
47C4×#11758185C2211yescharacteristic, complemented
48C61×#552—21no—
49C61×#499C2211yescharacteristic, complemented
50C61×#485—21no—
51C61×#483—21no—
52C61×#479—21no—
53C61×#467C2211yescharacteristic
54C61×#467C2211yescharacteristic, complemented
55C4×Q16×C61C2211yescharacteristic, complemented
56C8×D4×C61C2211yescharacteristic
57C61×#427C2211yescharacteristic, complemented
58C2×#107×C61C2211yescharacteristic
59C2×#11758222—21no—
60C22×D244—21no—
61#14519799—22no—
62#14519799—41no—
63C2×Dic244C2311yescharacteristic
64C2×#11758190—21no—
65C2×#11758190—21no—
66C2×D488—22no—
67C2×D488—21no—
68C2×D488—41no—
69C2×D488C2311yescharacteristic
70C2×D488C2311yescharacteristic
71#14519765—21no—
72#14519762—41no—
73#14519756—21no—
74#14519751—21no—
75#14519750—41no—
76#14519745—41no—
77C4×D244—21no—
78C22⋊C4×D61—41no—
79#14519723—41no—
80#14519720—22no—
81#14519720—41no—
82C2×#11758185—41no—
83C2×#11758185—41no—
84C2×#11758185C2311yescharacteristic
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519702—21no—
90#14519699—21no—
91#14519699—21no—
92C2×Q16×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2311yescharacteristic
95#126×C61—41no—
96#125×C61—21no—
97#125×C61—41no—
98#123×C61—41no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×D4×C61D411yescharacteristic
103#117×C61—21no—
104#117×C61—41no—
105C2×C4⋊C4×C61—21no—
106#107×C61—21no—
107#107×C61C2311yescharacteristic
108#107×C61C2311yescharacteristic
109#107×C61C2311yescharacteristic
110#106×C61—21no—
111#106×C61—21no—
112C4⋊C8×C61D411yescharacteristic
113#103×C61—21no—
114#103×C61—21no—
115#103×C61—21no—
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119C23×D61—41no—
120C2×Q8×C61—21no—
121C2×D4×C61D811yescharacteristic
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127#11758222—42no—
128#11758222—81no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—42no—
132C2×D244—41no—
133C2×D244—81no—
134C2×C4×D61—41no—
135C2×C4×D61—41no—
136C2×Dic122—21no—
137Q16×C61—21no—
138Q16×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2411yescharacteristic
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61—41no—
153C4⋊C4×C61—41no—
154C4⋊C4×C61—41no—
155C4⋊C4×C61—41no—
156C4⋊C4×C61D811yescharacteristic
157C22⋊C4×C61—41no—
158C22⋊C4×C61—41no—
159C22⋊C4×C61—41no—
160C22⋊C4×C61D811yescharacteristic
161C22⋊C4×C61D811yescharacteristic
162C42×C61—41no—
163C42×C61—41no—
164C42×C61C2×D411yescharacteristic
165C22⋊Dic61—41no—
166#11758198—41no—
167#11758198—41no—
168#11758198—41no—
169#11758198—41no—
170C4⋊Dic61—21no—
171#11758196—41no—
172Dic244—21no—
173Dic244—21no—
174D488—21no—
175D488—21no—
176D488—21no—
177D488—42no—
178D488#4112yes—
179#11758190—42no—
180#11758190—81no—
181#11758185—21no—
182#11758185—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61C2×D811yescharacteristic
186C23×C61C2×D811yescharacteristic
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—82no—
190C22×D61—82no—
191C22×D61—161no—
192Q8×C61—41no—
193Q8×C61—41no—
194D4×C61—22no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—82no—
204C2×C4×C61C2×D811yescharacteristic
205C2×C4×C61C2×D811yescharacteristic
206C2×C4×C61C2×D811yescharacteristic
207C2×C4×C61C2×D811yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C22⋊D61—82no—
210C22⋊D61—161no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213D244—22no—
214D244—41no—
215D244—41no—
216D244—41no—
217D244—82no—
218C4×D61—82no—
219C4×D61—161no—
220Dic122—41no—
221Dic122—41no—
222C8×C61—21no—
223C8×C61—21no—
224C8×C61—21no—
225C8×C61C2×#4111yescharacteristic
226C8×C61H(5,𝔽2)11yescharacteristic
227#33863—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61#50812yes—
234C22×C61C22×D811yescharacteristic
235C2×D61—42no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—164no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—81no—
247C4×C61—81no—
248C4×C61—81no—
249C4×C61—81no—
250C4×C61#54611yescharacteristic
251C4×C61#57711yescharacteristic
252Dic61—81no—
253Dic61—81no—
254#3336D6111yescharacteristic, complemented
255#3280—611no—
256#2242—611no—
257#2235—611no—
258#2234—611no—
259#2228—611no—
260#2217—611no—
261#2216—611no—
262#2211—611no—
263#2189—611no—
264#2187—611no—
265C4×SD32—611no—
266D4×C16—611no—
267#2138—611no—
268C2×#357—611no—
269C2×C61—22no—
270C2×C61—41no—
271C2×C61#329411yescharacteristic
272C2×C61C2×#50811yescharacteristic
273C2×C61#341611yescharacteristic
274D61—82no—
275D61—161no—
276C2×#135—611no—
277C22×D8—611no—
278#552—21no—
279#546—1221no—
280C2×Q32—611no—
281C2×SD32—611no—
282C2×SD32—611no—
283C22×C16—611no—
284C22×C16—611no—
285#499C2×D6111yescharacteristic, complemented
286#495—611no—
287#485—21no—
288#483—21no—
289#479—21no—
290#467—611no—
291#467C2×D6111yescharacteristic
292#467C2×D6111yescharacteristic, complemented
293#466—611no—
294#463—1221no—
295#450—1221no—
296#449—1221no—
297C4×Q16C2×D6111yescharacteristic, complemented
298C4×D8—611no—
299C8×D4C2×D6111yescharacteristic
300#427C2×D6111yescharacteristic, complemented
301C2×#107C2×D6111yescharacteristic
302#367—611no—
303#367—611no—
304#366—611no—
305C4⋊C16—611no—
306#358—611no—
307#358—611no—
308#358—611no—
309#358—611no—
310#357—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315#357—611no—
316C22⋊C16—611no—
317C22⋊C16—611no—
318C4×C16—611no—
319C61#3386311yescharacteristic, complemented, Sylow
320C2×#41—1221no—
321C22×D4—1221no—
322#135—1222no—
323#135—2441no—
324C2×Q16—611no—
325C2×Q16C22×D6111yescharacteristic
326C2×SD16—1221no—
327C2×SD16—1221no—
328C2×D8—611no—
329C2×D8—611no—
330C2×D8—1222no—
331C2×D8—1221no—
332C2×D8—2441no—
333C22×C8C22×D6111yescharacteristic
334C22×C8C22×D6111yescharacteristic
335#126—41no—
336#125—21no—
337#125—41no—
338#124—1221no—
339#123—41no—
340#123—2441no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—1221no—
344#121—1221no—
345#121—2441no—
346#120—2441no—
347C4×Q8—21no—
348C4×D4—1221no—
349C4×D4C22⋊D6111yescharacteristic
350#117—21no—
351#117—41no—
352C2×C4⋊C4—21no—
353C2×C22⋊C4—2441no—
354Q32—2441no—
355SD32—1222no—
356SD32—2441no—
357C2×C16—611no—
358C2×C16—611no—
359C2×C16—611no—
360C2×C16—611no—
361C2×C16—2441no—
362C2×C16—2441no—
363#107—21no—
364#107—611no—
365#107C22×D6111yescharacteristic
366#107C22×D6111yescharacteristic
367#107C22×D6111yescharacteristic
368#106—21no—
369#106—21no—
370C4⋊C8C22⋊D6111yescharacteristic
371#103—21no—
372#103—21no—
373#103—21no—
374#103—1221no—
375#102—1221no—
376#102—1221no—
377C22⋊C8C22⋊D6111yescharacteristic
378C22⋊C8C22⋊D6111yescharacteristic
379C4×C8C22×D6111yescharacteristic
380C24—2441no—
381#41—4881no—
382#41—2442no—
383C2×Q8—21no—
384C2×Q8—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—4881no—
388C2×D4—2442no—
389C2×D4—2441no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4#1175819911yescharacteristic
393C22×C4—41no—
394C22×C4—2441no—
395C22×C4—2441no—
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1221no—
401Q16—1221no—
402SD16—4881no—
403SD16—2442no—
404D8—612no—
405D8—1221no—
406D8—1221no—
407D8—1221no—
408D8—2442no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D6111yescharacteristic
414C2×C8C2×C22⋊D6111yescharacteristic
415C2×C8C2×C22⋊D6111yescharacteristic
416C2×C8C23×D6111yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—1221no—
427C4⋊C4—2441no—
428C4⋊C4#1175819911yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4#1175819911yescharacteristic
438C22⋊C4#1175819911yescharacteristic
439C42—41no—
440C42—41no—
441C42C2×C22⋊D6111yescharacteristic
442C16—1221no—
443C16—1221no—
444C16—1221no—
445C16—1221no—
446C23—9761no—
447C23—4882no—
448C23—4882no—
449C23—2441no—
450C23—2441no—
451C23C2×#1175819911yescharacteristic
452C23C2×#1175819911yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—2441no—
456Q8—2441no—
457D4—22no—
458D4—1222no—
459D4—9761no—
460D4—4882no—
461D4—4882no—
462D4—2441no—
463D4—2441no—
464D4—2441no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—9761no—
475C2×C4—4882no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4C2×#1175819911yescharacteristic
479C2×C4C2×#1175819911yescharacteristic
480C2×C4C2×#1175819911yescharacteristic
481C2×C4C2×#1175819911yescharacteristic
482C2×C4C22×C22⋊D6111yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D6111yescharacteristic
487C8#1451982311yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—9764no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22—4881no—
498C22—4881no—
499C22—2442no—
500C22#1684019212yes—
501C22C22×#1175819911yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—4881no—
511C4—4881no—
512C4#1684040011yescharacteristic
513C4#1684057011yescharacteristic
514C2—22no—
515C2—41no—
516C2—9761no—
517C2—4882no—
518C2#2145762611yescharacteristic
519C2C2×#1684019211yescharacteristic
520C2#2145896311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3336C2
socleC22×C61C22×D8
prime radical#21457506C2
hypercenter#3336D61
hypocenterC61#33863
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #21448053C2semidirect
3. #21448016C2semidirect
4. #21448010C2semidirect
5. #21447973C2semidirect
6. #21447917C2semidirect
7. #21447908C2semidirect
8. #21447881C2semidirect
9. #21447757C2semidirect
10. #21447747C2semidirect
11. C4×#14519720C2semidirect
12. D4×#11758185C2semidirect
13. #21447524C2semidirect
14. C2×#16839562C2semidirect
15. C61×#3336C2semidirect
19. C2×#14519723C22semidirect
23. C22×#11758185C22semidirect
31. #16839589C22semidirect
32. #16839589C22semidirect
33. #16839587C22semidirect
34. C4⋊#11758185C22semidirect
35. #16839566C22semidirect
36. #16839565C22semidirect
37. #16839565C22semidirect
38. #16839565C22semidirect
46. C22⋊#11758185C22semidirect
47. C4×#11758185C22semidirect
49. C61×#499C22semidirect
54. C61×#467C22semidirect
55. C4×Q16×C61C22semidirect
57. C61×#427C22semidirect
254. #3336D61semidirect
285. #499C2×D61semidirect
292. #467C2×D61semidirect
297. C4×Q16C2×D61semidirect
300. #427C2×D61semidirect
319. C61#33863semidirect

Maximal subgroups

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

subgroupindex
#214575062
#214480532
#214480162
#214480102
#214479732
#214479172
#214479082
#214478812
#214477572
#214477472
C4×#145197202
D4×#117581852
#214475242
C2×#168395622
C61×#33362
#3386361