Finite Group Databasebeta

← #29834408

Subgroups — #29834408

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458954C212yesmaximal, complemented
2#21456248C211yesmaximal, characteristic, complemented
3#443×D61C211yesmaximal, characteristic, complemented
4#21447897C211yesmaximal, characteristic, complemented
5#21447896C211yesmaximal, characteristic, complemented
6#21447895C211yesmaximal, characteristic, complemented
7#21447891C212yesmaximal, complemented
8#21447890C212yesmaximal, complemented
9#364×D61C211yesmaximal, characteristic, complemented
10#21447567C212yesmaximal, complemented
11C61×#2214C211yesmaximal, characteristic, complemented
12C2×#14519800—21no—
13C2×#14519800C2212yes—
14#135×D61C2211yescharacteristic
15#131×D61—21no—
16#16840313—21no—
17#16840204—22no—
18#16840203C2212yes—
19#16840200C2212yes—
20#16840199—22no—
21#16840197C2212yes—
22#16840195—22no—
23#16840193—22no—
24M32×D61C2212yes—
25#16840147—21no—
26#16840147C2212yescomplemented
27#16840139C2211yescharacteristic, complemented
28#16840136—21no—
29#16839880C2211yescharacteristic
30#16839879C2211yescharacteristic
31#16839878C2211yescharacteristic
32#16839877—21no—
33#16839876—21no—
34#108×D61C2211yescharacteristic
35#104×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839583C2211yescharacteristic, complemented
38#16839582C2212yescomplemented
39#16839578C2212yescomplemented
40#16839576C2212yescomplemented
41#16839574C2212yescomplemented
42C61×#510C2212yes—
43C61×#494C2211yescharacteristic, complemented
44C61×#443C2211yescharacteristic
45C61×#364C2211yescharacteristic, complemented
46C61×#362C2212yescomplemented
47C2×#11758226—22no—
48#41×D61—21no—
49#14519800—22no—
50#14519800—22no—
51#14519800—22no—
52#14519800—22no—
53#14519800—41no—
54#14519800—41no—
55#14519800—41no—
56#14519800—41no—
57#14519800D411yescharacteristic, complemented
58#14519799C2311yescharacteristic
59#14519798—21no—
60#14519797—21no—
61#14519796D411yescharacteristic, complemented
62Q16×D61D411yescharacteristic, complemented
63C2×#11758201—22no—
64C2×#11758201—22no—
65SD16×D61—21no—
66C2×D488—21no—
67C2×D488C2312yes—
68D8×D61D411yescharacteristic, complemented
69#14519785—21no—
70#14519785—41no—
71#14519784—21no—
72#14519783—21no—
73#14519783—41no—
74M16×D61—21no—
75M16×D61—41no—
76C2×C8×D61—41no—
77C2×C8×D61D411yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519774—21no—
80#14519766—22no—
81#14519752—42no—
82#14519739—42no—
83C42×D61D411yescharacteristic
84#14519723—22no—
85#14519722—22no—
86#14519720—22no—
87#14519720—22no—
88D976—22no—
89#14519718—22no—
90#14519717C2312yes—
91#14519716—22no—
92C16×D61—22no—
93#14519713C2311yescharacteristic
94#14519712—21no—
95#14519705—21no—
96#14519704—21no—
97#14519703—21no—
98C4×#92418D411yescharacteristic
99C8×Dic61D411yescharacteristic
100C8×Dic61C2311yescharacteristic
101#135×C61C2311yescharacteristic
102C2×Q16×C61—21no—
103C2×Q16×C61C2312yes—
104#131×C61—21no—
105C4⋊Q8×C61—21no—
106Q32×C61—22no—
107SD32×C61—22no—
108M32×C61C2312yes—
109#108×C61C2311yescharacteristic
110#104×C61—21no—
111C4×C8×C61C2311yescharacteristic
112#41×C61—21no—
113C2×Q8×C61—22no—
114#11758226—21no—
115#11758226—42no—
116#11758226—42no—
117#11758226—42no—
118#11758226—42no—
119Q8×D61—21no—
120#11758224—21no—
121#11758224—41no—
122D4×D61—21no—
123D4×D61—41no—
124#11758222—21no—
125#11758222—41no—
126C2×D244—22no—
127C2×D244—42no—
128C2×D244—42no—
129C2×C4×D61—21no—
130C2×C4×D61—41no—
131C2×C4×D61—42no—
132C2×C4×D61C2×D411yescharacteristic
133Q16×C61—22no—
134Q16×C61—22no—
135Q16×C61—41no—
136Q16×C61—41no—
137Q16×C61C2×D411yescharacteristic
138SD16×C61—21no—
139D8×C61C2×D411yescharacteristic
140M16×C61—21no—
141M16×C61—41no—
142C2×C8×C61—41no—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2411yescharacteristic
145C4⋊C4×C61—42no—
146C42×C61C2×D411yescharacteristic
147#11758202—21no—
148#11758201—21no—
149#11758201—42no—
150#11758201—42no—
151#11758201—42no—
152#11758201—42no—
153#11758200—21no—
154#11758199—21no—
155#11758198—44no—
156C4×Dic61—21no—
157C4×Dic61C2×D411yescharacteristic
158#11758194—21no—
159#11758194—41no—
160C2×#92418—41no—
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163Dic244C2×D411yescharacteristic
164D488—22no—
165D488—22no—
166D488—41no—
167D488—41no—
168D488C2×D411yescharacteristic
169#11758190—21no—
170#11758189—41no—
171#11758189—41no—
172#11758189—41no—
173C8×D61—41no—
174C8×D61—41no—
175C8×D61—41no—
176C8×D61D811yescharacteristic
177C8×D61D811yescharacteristic
178C8×D61D811yescharacteristic
179C8×D61D811yescharacteristic
180C8×D61C2×D411yescharacteristic
181C8×D61C2×D411yescharacteristic
182C8×D61#4111yescharacteristic
183C8×D61#4111yescharacteristic
184C16×C61—22no—
185#11758185—22no—
186C22×D61—41no—
187C22×D61—42no—
188C22×D61AGL(1,ℤ/8)11yescharacteristic
189Q8×C61—21no—
190Q8×C61—42no—
191Q8×C61—42no—
192D4×C61—21no—
193D4×C61—41no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—41no—
199C22⋊D61—41no—
200C22⋊D61—41no—
201C2×Dic61—21no—
202C2×Dic61—41no—
203C2×Dic61AGL(1,ℤ/8)11yescharacteristic
204D244—21no—
205D244—41no—
206D244—42no—
207D244—42no—
208D244—84no—
209D244—82no—
210D244—82no—
211C4×D61—42no—
212C4×D61—41no—
213C4×D61—41no—
214C4×D61—41no—
215C4×D61—82no—
216C4×D61—82no—
217C4×D61#12111yescharacteristic
218C4×D61#12111yescharacteristic
219C4×D61C2×D811yescharacteristic
220C4×D61C2×D811yescharacteristic
221Dic122—21no—
222Dic122—41no—
223C8×C61—41no—
224C8×C61C2×D811yescharacteristic
225C8×C61C2×D811yescharacteristic
226C8×C61C22×D411yescharacteristic
227C8×C61C2×#4111yescharacteristic
228#92418—41no—
229#92418#12111yescharacteristic
230#92418#12111yescharacteristic
231#92418C2×D811yescharacteristic
232#92418C2×D811yescharacteristic
233#33594—611nomaximal, Sylow
234C22×C61—41no—
235C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
236C2×D61—41no—
237C2×D61—41no—
238C2×D61—82no—
239C2×D61—84no—
240C2×D61—82no—
241C2×D61#45911yescharacteristic
242C2×D61#45911yescharacteristic
243C4×C61—22no—
244C4×C61—41no—
245C4×C61—82no—
246C4×C61C2×#12111yescharacteristic
247C4×C61C22×D811yescharacteristic
248Dic61—41no—
249Dic61—41no—
250Dic61#45911yescharacteristic
251Dic61#45911yescharacteristic
252#3414—612no—
253#3147—611no—
254C2×#443—611no—
255#2214—611no—
256#2214D6111yescharacteristic, complemented
257#2213—611no—
258#2213—611no—
259#2212—612no—
260#2212—612no—
261C2×#364—611no—
262#2148—612no—
263C2×C61—21no—
264C2×C61—81no—
265C2×C61C2×#45911yescharacteristic
266D61—21no—
267D61—21no—
268D61—81no—
269D61—162no—
270C2×#135—611no—
271C2×#135—612no—
272C2×#135—1221no—
273C2×#131—1221no—
274#532—1221no—
275#510—612no—
276#510—1222no—
277#510C2×D6112yes—
278#509—612no—
279#509—612no—
280#509—1222no—
281#508—1222no—
282#508—1222no—
283C2×M32—612no—
284#495—612no—
285#495—1221no—
286#494C2×D6111yescharacteristic, complemented
287#493—611no—
288#492—1221no—
289#443—611no—
290#443—611no—
291#443—611no—
292#443—1221no—
293#443—1221no—
294#443C2×D6111yescharacteristic
295C2×#108—611no—
296C2×#104—1221no—
297C2×C4×C8—611no—
298#364—612no—
299#364—611no—
300#364C2×D6111yescharacteristic, complemented
301#362—612no—
302#362C2×D6112yescomplemented
303#361—612no—
304#361—612no—
305C61#3359411yescharacteristic, complemented, Sylow
306C2×#41—1222no—
307C2×#41—1221no—
308#135—611no—
309#135—611no—
310#135—611no—
311#135—1221no—
312#135—1221no—
313#135—1222no—
314#135—1222no—
315#135—1222no—
316#135—1222no—
317#135—2441no—
318#135—2441no—
319#135—2441no—
320#135—2441no—
321#135C22×D6111yescharacteristic
322C2×Q16—21no—
323C2×Q16—611no—
324C2×Q16C22×D6112yes—
325C2×SD16—1221no—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×D8—611no—
329C2×D8—612no—
330C2×D8—1221no—
331#131—21no—
332#131—1221no—
333#131—1221no—
334#131—1221no—
335#131—2441no—
336#131—2441no—
337C2×M16—1221no—
338C2×M16—2441no—
339C22×C8—611no—
340C22×C8—611no—
341C22×C8—2441no—
342C4⋊Q8—21no—
343#127—1221no—
344#124—1222no—
345#121—2442no—
346C4×D4—2442no—
347C2×C42—611no—
348Q32—22no—
349Q32—1222no—
350SD32—22no—
351SD32—1222no—
352SD32—1222no—
353SD32—1222no—
354D16—1222no—
355D16—1222no—
356M32—612no—
357M32—1222no—
358M32C22×D6112yes—
359C2×C16—1222no—
360#108—611no—
361#108—1221no—
362#108C22×D6111yescharacteristic
363#104—21no—
364#104—1221no—
365#104—1221no—
366#104—1221no—
367C4×C8—611no—
368C4×C8—611no—
369C4×C8—611no—
370C4×C8C22×D6111yescharacteristic
371#41—21no—
372#41—1221no—
373#41—1221no—
374#41—1221no—
375#41—2441no—
376#41—2442no—
377#41—2442no—
378#41—2442no—
379#41—2442no—
380#41—2441no—
381C2×Q8—22no—
382C2×Q8—1221no—
383C2×D4—1221no—
384C2×D4—1222no—
385C2×D4—2442no—
386C2×D4—2442no—
387C2×D4—2441no—
388C22×C4—611no—
389C22×C4—1221no—
390C22×C4—2442no—
391C22×C4—2441no—
392Q16—22no—
393Q16—22no—
394Q16—41no—
395Q16—41no—
396Q16—611no—
397Q16—1221no—
398Q16D4×D6111yescharacteristic
399SD16—21no—
400SD16—1221no—
401SD16—1221no—
402SD16—1221no—
403SD16—2442no—
404SD16—2442no—
405SD16—2442no—
406SD16—2442no—
407D8—611no—
408D8—1221no—
409D8—1222no—
410D8—1222no—
411D8—2441no—
412D8—2441no—
413D8D4×D6111yescharacteristic
414M16—21no—
415M16—41no—
416M16—1221no—
417M16—2441no—
418M16—2441no—
419M16—2441no—
420M16—2441no—
421C2×C8—41no—
422C2×C8—611no—
423C2×C8—611no—
424C2×C8—611no—
425C2×C8—611no—
426C2×C8—611no—
427C2×C8—611no—
428C2×C8—611no—
429C2×C8—611no—
430C2×C8—611no—
431C2×C8—611no—
432C2×C8—2441no—
433C2×C8—2441no—
434C2×C8—2441no—
435C2×C8—2441no—
436C2×C8D4×D6111yescharacteristic
437C2×C8C23×D6111yescharacteristic
438C4⋊C4—42no—
439C22⋊C4—2444no—
440C42—611no—
441C42—1221no—
442C42D4×D6111yescharacteristic
443C16—22no—
444C16—1222no—
445C23—611no—
446C23—2441no—
447C23—2442no—
448Q8—21no—
449Q8—42no—
450Q8—42no—
451Q8—1221no—
452Q8—2441no—
453D4—21no—
454D4—41no—
455D4—1221no—
456D4—4884no—
457D4—4882no—
458D4—4882no—
459D4—2441no—
460D4—2441no—
461D4—2441no—
462D4—2441no—
463D4—2442no—
464D4—2442no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—611no—
469C2×C4—611no—
470C2×C4—611no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—1221no—
474C2×C4—4882no—
475C2×C4—4882no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4C2×D4×D6111yescharacteristic
482C8—41no—
483C8—611no—
484C8—611no—
485C8—611no—
486C8—611no—
487C8—2441no—
488C8D8×D6111yescharacteristic
489C8D8×D6111yescharacteristic
490C8C2×D4×D6111yescharacteristic
491C8#41×D6111yescharacteristic
492C22—41no—
493C22—611no—
494C22—611no—
495C22—4884no—
496C22—4882no—
497C22—4882no—
498C22—2441no—
499C22—2441no—
500C22AGL(1,ℤ/8)×D6111yescharacteristic
501C4—22no—
502C4—41no—
503C4—82no—
504C4—611no—
505C4—611no—
506C4—2441no—
507C4—2441no—
508C4#121×D6111yescharacteristic
509C4C2×D8×D6111yescharacteristic
510C2—21no—
511C2—81no—
512C2—1221no—
513C2—1221no—
514C2—9762no—
515C2—4881no—
516C2#459×D6111yescharacteristic
subgroupG/H
centerC2#459×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2214C2
socleC2×C61C2×#459
hypercenter#2214D61
hypocenterC61#33594
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2214

Complements

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

NH ≅ G/Ntype
1. #21458954C2semidirect
2. #21456248C2semidirect
3. #443×D61C2semidirect
4. #21447897C2semidirect
5. #21447896C2semidirect
6. #21447895C2semidirect
7. #21447891C2semidirect
8. #21447890C2semidirect
9. #364×D61C2semidirect
10. #21447567C2semidirect
11. C61×#2214C2semidirect
26. #16840147C22semidirect
27. #16840139C22semidirect
37. #16839583C22semidirect
38. #16839582C22semidirect
39. #16839578C22semidirect
40. #16839576C22semidirect
41. #16839574C22semidirect
43. C61×#494C22semidirect
45. C61×#364C22semidirect
46. C61×#362C22semidirect
57. #14519800D4semidirect
61. #14519796D4semidirect
62. Q16×D61D4semidirect
68. D8×D61D4semidirect
256. #2214D61semidirect
286. #494C2×D61semidirect
300. #364C2×D61semidirect
302. #362C2×D61semidirect
305. C61#33594semidirect

Maximal subgroups

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

subgroupindex
#214589542
#214562482
#443×D612
#214478972
#214478962
#214478952
#214478912
#214478902
#364×D612
#214475672
C61×#22142
#3359461