Finite Group Databasebeta

← #29834751

Subgroups — #29834751

Order 15616.

524 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456328C211yesmaximal, characteristic, complemented
2#443×D61C211yesmaximal, characteristic
3#21447994C212yesmaximal, complemented
4#21447993C212yesmaximal, complemented
5#21447991C211yesmaximal, characteristic, complemented
6#21447990C211yesmaximal, characteristic, complemented
7#21447987C211yesmaximal, characteristic, complemented
8#21447590C212yesmaximal, complemented
9#364×D61C211yesmaximal, characteristic, complemented
10#21447553C212yesmaximal
11C61×#2230C211yesmaximal, characteristic, complemented
12#135×D61C2211yescharacteristic
13#131×D61—21no—
14#16840317—21no—
15M32×D61C2212yes—
16#16840165—21no—
17#16840162—21no—
18#16840162C2212yescomplemented
19#16840160C2211yescharacteristic, complemented
20#16839880C2211yescharacteristic
21#16839879C2211yescharacteristic
22#16839878C2211yescharacteristic
23#16839877—21no—
24#16839876—21no—
25#108×D61C2211yescharacteristic
26#16839797—21no—
27#16839797C2212yes—
28#104×D61—21no—
29C4×C8×D61C2211yescharacteristic
30#16839586C2212yescomplemented
31#16839585C2212yescomplemented
32#16839584C2212yescomplemented
33#16839583C2211yescharacteristic, complemented
34#16839582C2212yescomplemented
35#16839573C2212yes—
36#16839572C2212yes—
37#16839571C2212yes—
38C61×#498C2211yescharacteristic, complemented
39C61×#443C2211yescharacteristic
40C61×#365C2212yescomplemented
41C61×#364C2211yescharacteristic, complemented
42C61×#360C2212yes—
43#41×D61—21no—
44#14519800Q811yescharacteristic, complemented
45#14519799C2311yescharacteristic
46#14519798—21no—
47#14519797—21no—
48#14519796Q811yescharacteristic, complemented
49Q16×D61Q811yescharacteristic, complemented
50SD16×D61—21no—
51D8×D61Q811yescharacteristic, complemented
52#14519785—21no—
53#14519785—41no—
54#14519784—21no—
55#14519783—21no—
56#14519783—41no—
57M16×D61—21no—
58M16×D61—41no—
59C2×C8×D61—41no—
60C2×C8×D61D411yescharacteristic
61C2×C8×D61C2311yescharacteristic
62#14519777—21no—
63#14519768—22no—
64#14519757—41no—
65#14519753—41no—
66Q8×Dic61—41no—
67#14519742—41no—
68#14519735—22no—
69C42×D61D411yescharacteristic
70#14519717C2312yes—
71#14519716—22no—
72C16×D61—22no—
73#14519713C2311yescharacteristic
74#14519712—21no—
75#14519710—22no—
76#14519710—22no—
77#14519709—21no—
78#14519709C2312yes—
79#14519705—21no—
80#14519704—21no—
81#14519703—21no—
82C4×#92418D411yescharacteristic
83C8×Dic61D411yescharacteristic
84C8×Dic61C2311yescharacteristic
85#135×C61C2311yescharacteristic
86#131×C61—21no—
87C4⋊Q8×C61—21no—
88M32×C61C2312yes—
89#108×C61C2311yescharacteristic
90#106×C61—21no—
91#106×C61C2312yes—
92#104×C61—21no—
93C4×C8×C61C2311yescharacteristic
94#41×C61—21no—
95C2×Q8×C61—41no—
96#11758226—21no—
97Q8×D61—21no—
98#11758224—21no—
99#11758224—41no—
100D4×D61—21no—
101D4×D61—41no—
102#11758222—21no—
103#11758222—41no—
104C2×C4×D61—21no—
105C2×C4×D61—41no—
106C2×C4×D61C2×D411yescharacteristic
107C2×Dic122—41no—
108Q16×C61C2×Q811yescharacteristic
109SD16×C61—21no—
110D8×C61C2×Q811yescharacteristic
111M16×C61—21no—
112M16×C61—41no—
113C2×C8×C61—41no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2411yescharacteristic
116C4⋊C4×C61—22no—
117C4⋊C4×C61—41no—
118C42×C61C2×D411yescharacteristic
119#11758202—21no—
120#11758201—21no—
121#11758200—21no—
122#11758199—21no—
123#11758198—42no—
124C4⋊Dic61—22no—
125C4⋊Dic61—42no—
126C4⋊Dic61—41no—
127#11758196—42no—
128C4×Dic61—21no—
129C4×Dic61—42no—
130C4×Dic61C2×D411yescharacteristic
131#11758194—21no—
132#11758194—41no—
133C2×#92418—41no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136Dic244C2×Q811yescharacteristic
137D488C2×Q811yescharacteristic
138#11758190—21no—
139#11758189—41no—
140#11758189—41no—
141#11758189—41no—
142C8×D61—41no—
143C8×D61—41no—
144C8×D61—41no—
145C8×D61SD1611yescharacteristic
146C8×D61SD1611yescharacteristic
147C8×D61SD1611yescharacteristic
148C8×D61SD1611yescharacteristic
149C8×D61C2×Q811yescharacteristic
150C8×D61C2×Q811yescharacteristic
151C8×D61#4111yescharacteristic
152C8×D61#4111yescharacteristic
153C16×C61—22no—
154#11758185—22no—
155C22×D61—41no—
156C22×D61AGL(1,ℤ/8)11yescharacteristic
157Q8×C61—21no—
158Q8×C61—82no—
159D4×C61—21no—
160D4×C61—41no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—41no—
166C22⋊D61—41no—
167C22⋊D61—41no—
168C2×Dic61—21no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61AGL(1,ℤ/8)11yescharacteristic
172D244—21no—
173D244—41no—
174C4×D61—42no—
175C4×D61—41no—
176C4×D61—41no—
177C4×D61—41no—
178C4×D61C22⋊Q811yescharacteristic
179C4×D61C22⋊Q811yescharacteristic
180C4×D61C2×SD1611yescharacteristic
181C4×D61C2×SD1611yescharacteristic
182Dic122—21no—
183Dic122—41no—
184Dic122—82no—
185C8×C61—41no—
186C8×C61C2×SD1611yescharacteristic
187C8×C61C2×SD1611yescharacteristic
188C8×C61C22×Q811yescharacteristic
189C8×C61C2×#4111yescharacteristic
190#92418—41no—
191#92418C22⋊Q811yescharacteristic
192#92418C22⋊Q811yescharacteristic
193#92418C2×SD1611yescharacteristic
194#92418C2×SD1611yescharacteristic
195#33637—611nomaximal, Sylow
196C22×C61—41no—
197C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
198C2×D61—41no—
199C2×D61—41no—
200C2×D61—82no—
201C2×D61#47611yescharacteristic
202C2×D61#47611yescharacteristic
203C4×C61—22no—
204C4×C61—41no—
205C4×C61—82no—
206C4×C61C2×C22⋊Q811yescharacteristic
207C4×C61C22×SD1611yescharacteristic
208Dic61—41no—
209Dic61—41no—
210Dic61—82no—
211Dic61#47611yescharacteristic
212Dic61#47611yescharacteristic
213#3160—611no—
214C2×#443—611no—
215#2231—612no—
216#2231—612no—
217#2230—611no—
218#2230—611no—
219#2230—611no—
220#2230D6111yescharacteristic, complemented
221#2155—612no—
222C2×#364—611no—
223#2145—612no—
224C2×C61—21no—
225C2×C61—81no—
226C2×C61C2×#47611yescharacteristic
227D61—21no—
228D61—21no—
229D61—81no—
230C2×#135—611no—
231C2×#131—1221no—
232#533—1221no—
233C2×M32—612no—
234#500—1221no—
235#499—612no—
236#499—1221no—
237#498—611no—
238#498C2×D6111yescharacteristic, complemented
239#443—611no—
240#443—611no—
241#443—611no—
242#443—1221no—
243#443—1221no—
244#443C2×D6111yescharacteristic
245C2×#108—611no—
246#427—612no—
247#427—1221no—
248C2×#104—1221no—
249C2×C4×C8—611no—
250#365—612no—
251#365—612no—
252#365—612no—
253#365C2×D6112yescomplemented
254#364—612no—
255#364—611no—
256#364C2×D6111yescharacteristic, complemented
257#360—612no—
258#360—612no—
259#360—612no—
260#360C2×D6112yes—
261C61#3363711yescharacteristic, complemented, Sylow
262C2×#41—1221no—
263#135—611no—
264#135—611no—
265#135—611no—
266#135—1221no—
267#135—1221no—
268#135C22×D6111yescharacteristic
269C2×Q16—611no—
270C2×SD16—1221no—
271C2×D8—611no—
272#131—21no—
273#131—1221no—
274#131—1221no—
275#131—1221no—
276#131—2441no—
277#131—2441no—
278C2×M16—1221no—
279C2×M16—2441no—
280C22×C8—611no—
281C22×C8—611no—
282C22×C8—2441no—
283C4⋊Q8—21no—
284C4⋊Q8—1221no—
285#125—1222no—
286C22⋊Q8—2441no—
287C22⋊Q8—2441no—
288C4×Q8—2441no—
289C4×Q8—2441no—
290#117—1222no—
291C2×C42—611no—
292M32—612no—
293M32—1222no—
294M32C22×D6112yes—
295C2×C16—1222no—
296#108—611no—
297#108—1221no—
298#108C22×D6111yescharacteristic
299#107—1222no—
300#107—1222no—
301#106—21no—
302#106—612no—
303#106—1221no—
304#106C22×D6112yes—
305#104—21no—
306#104—1221no—
307#104—1221no—
308#104—1221no—
309C4×C8—611no—
310C4×C8—611no—
311C4×C8—611no—
312C4×C8C22×D6111yescharacteristic
313#41—21no—
314#41—1221no—
315#41—1221no—
316#41—1221no—
317#41—2441no—
318#41—2441no—
319C2×Q8—41no—
320C2×Q8—1221no—
321C2×Q8—2441no—
322C2×D4—1221no—
323C2×D4—2441no—
324C22×C4—611no—
325C22×C4—1221no—
326C22×C4—2441no—
327Q16—611no—
328Q16—1221no—
329Q16Q8×D6111yescharacteristic
330SD16—21no—
331SD16—1221no—
332SD16—1221no—
333SD16—1221no—
334D8—611no—
335D8—1221no—
336D8Q8×D6111yescharacteristic
337M16—21no—
338M16—41no—
339M16—1221no—
340M16—2441no—
341M16—2441no—
342M16—2441no—
343M16—2441no—
344C2×C8—41no—
345C2×C8—611no—
346C2×C8—611no—
347C2×C8—611no—
348C2×C8—611no—
349C2×C8—611no—
350C2×C8—611no—
351C2×C8—611no—
352C2×C8—611no—
353C2×C8—611no—
354C2×C8—611no—
355C2×C8—2441no—
356C2×C8—2441no—
357C2×C8—2441no—
358C2×C8—2441no—
359C2×C8D4×D6111yescharacteristic
360C2×C8C23×D6111yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—1222no—
364C4⋊C4—2441no—
365C4⋊C4—2442no—
366C4⋊C4—2442no—
367C22⋊C4—2442no—
368C42—611no—
369C42—1221no—
370C42—2442no—
371C42D4×D6111yescharacteristic
372C16—22no—
373C16—1222no—
374C23—611no—
375C23—2441no—
376Q8—21no—
377Q8—82no—
378Q8—1221no—
379Q8—4882no—
380Q8—2441no—
381D4—21no—
382D4—41no—
383D4—1221no—
384D4—2441no—
385D4—2441no—
386D4—2441no—
387D4—2441no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—611no—
392C2×C4—611no—
393C2×C4—611no—
394C2×C4—611no—
395C2×C4—611no—
396C2×C4—1221no—
397C2×C4—2442no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2442no—
402C2×C4—2441no—
403C2×C4C2×D4×D6111yescharacteristic
404C8—41no—
405C8—611no—
406C8—611no—
407C8—611no—
408C8—611no—
409C8—2441no—
410C8SD16×D6111yescharacteristic
411C8SD16×D6111yescharacteristic
412C8C2×Q8×D6111yescharacteristic
413C8#41×D6111yescharacteristic
414C22—41no—
415C22—611no—
416C22—611no—
417C22—4882no—
418C22—2441no—
419C22—2441no—
420C22AGL(1,ℤ/8)×D6111yescharacteristic
421C4—22no—
422C4—41no—
423C4—82no—
424C4—611no—
425C4—611no—
426C4—4882no—
427C4—2441no—
428C4—2441no—
429C4C22⋊Q8×D6111yescharacteristic
430C4C2×SD16×D6111yescharacteristic
431C2—21no—
432C2—81no—
433C2—1221no—
434C2—1221no—
435C2—4881no—
436C2#476×D6111yescharacteristic
subgroupG/H
centerC2#476×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2230C2
socleC2×C61C2×#476
prime radical#135×D61C22
hypercenter#2230D61
hypocenterC61#33637
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2230

Complements

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

NH ≅ G/Ntype
1. #21456328C2semidirect
3. #21447994C2semidirect
4. #21447993C2semidirect
5. #21447991C2semidirect
6. #21447990C2semidirect
7. #21447987C2semidirect
8. #21447590C2semidirect
9. #364×D61C2semidirect
11. C61×#2230C2semidirect
18. #16840162C22semidirect
19. #16840160C22semidirect
30. #16839586C22semidirect
31. #16839585C22semidirect
32. #16839584C22semidirect
33. #16839583C22semidirect
34. #16839582C22semidirect
38. C61×#498C22semidirect
40. C61×#365C22semidirect
41. C61×#364C22semidirect
44. #14519800Q8semidirect
48. #14519796Q8semidirect
49. Q16×D61Q8semidirect
51. D8×D61Q8semidirect
220. #2230D61semidirect
238. #498C2×D61semidirect
253. #365C2×D61semidirect
256. #364C2×D61semidirect
261. C61#33637semidirect

Maximal subgroups

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

subgroupindex
#214563282
#443×D612
#214479942
#214479932
#214479912
#214479902
#214479872
#214475902
#364×D612
#214475532
C61×#22302
#3363761