Finite Group Databasebeta

← #29834754

Subgroups — #29834754

Order 15616.

524 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456332C211yesmaximal, characteristic, complemented
2#443×D61C211yesmaximal, characteristic
3#21447996C211yesmaximal, characteristic, complemented
4#21447995C211yesmaximal, characteristic, complemented
5#21447992C211yesmaximal, characteristic, complemented
6#21447989C212yesmaximal, complemented
7#21447988C212yesmaximal, complemented
8#21447590C212yesmaximal, complemented
9#364×D61C211yesmaximal, characteristic, complemented
10#21447553C212yesmaximal
11C61×#2231C211yesmaximal, characteristic, complemented
12#135×D61C2211yescharacteristic
13#131×D61—21no—
14#16840318—21no—
15M32×D61C2212yes—
16#16840166—21no—
17#16840164C2211yescharacteristic, complemented
18#16840161—21no—
19#16840159C2212yescomplemented
20#16839880C2211yescharacteristic
21#16839879C2211yescharacteristic
22#16839878C2211yescharacteristic
23#16839877—21no—
24#16839876—21no—
25#108×D61C2211yescharacteristic
26#16839799—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×#499C2211yescharacteristic, 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#14519776—22no—
63#14519769—21no—
64#14519755—42no—
65#14519742—42no—
66#14519735—22no—
67C42×D61D411yescharacteristic
68#14519717C2312yes—
69#14519716—22no—
70C16×D61—22no—
71#14519713C2311yescharacteristic
72#14519712—21no—
73#14519711—21no—
74#14519710—22no—
75#14519709C2312yes—
76#14519708—22no—
77#14519705—21no—
78#14519704—21no—
79#14519703—21no—
80C4×#92418D411yescharacteristic
81C8×Dic61D411yescharacteristic
82C8×Dic61C2311yescharacteristic
83#135×C61C2311yescharacteristic
84#131×C61—21no—
85#125×C61—21no—
86M32×C61C2312yes—
87#108×C61C2311yescharacteristic
88#107×C61—21no—
89#106×C61C2312yes—
90#104×C61—21no—
91C4×C8×C61C2311yescharacteristic
92#41×C61—21no—
93#11758226—21no—
94Q8×D61—21no—
95#11758224—21no—
96#11758224—41no—
97D4×D61—21no—
98D4×D61—41no—
99#11758222—21no—
100#11758222—41no—
101C2×C4×D61—21no—
102C2×C4×D61—41no—
103C2×C4×D61C2×D411yescharacteristic
104C2×Dic122—42no—
105Q16×C61C2×Q811yescharacteristic
106SD16×C61—21no—
107D8×C61C2×Q811yescharacteristic
108M16×C61—21no—
109M16×C61—41no—
110C2×C8×C61—41no—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2411yescharacteristic
113C4⋊C4×C61—22no—
114C4⋊C4×C61—42no—
115C42×C61C2×D411yescharacteristic
116#11758202—21no—
117#11758201—21no—
118#11758200—21no—
119#11758199—21no—
120#11758198—42no—
121C4⋊Dic61—22no—
122#11758196—42no—
123#11758196—42no—
124C4×Dic61—21no—
125C4×Dic61—42no—
126C4×Dic61C2×D411yescharacteristic
127#11758194—21no—
128#11758194—41no—
129C2×#92418—41no—
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132Dic244C2×Q811yescharacteristic
133D488C2×Q811yescharacteristic
134#11758190—21no—
135#11758189—41no—
136#11758189—41no—
137#11758189—41no—
138C8×D61—41no—
139C8×D61—41no—
140C8×D61—41no—
141C8×D61SD1611yescharacteristic
142C8×D61SD1611yescharacteristic
143C8×D61SD1611yescharacteristic
144C8×D61SD1611yescharacteristic
145C8×D61C2×Q811yescharacteristic
146C8×D61C2×Q811yescharacteristic
147C8×D61#4111yescharacteristic
148C8×D61#4111yescharacteristic
149C16×C61—22no—
150#11758185—22no—
151C22×D61—41no—
152C22×D61AGL(1,ℤ/8)11yescharacteristic
153Q8×C61—21no—
154D4×C61—21no—
155D4×C61—41no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61C22×D411yescharacteristic
160C22⋊D61—41no—
161C22⋊D61—41no—
162C22⋊D61—41no—
163C2×Dic61—21no—
164C2×Dic61—41no—
165C2×Dic61—42no—
166C2×Dic61AGL(1,ℤ/8)11yescharacteristic
167D244—21no—
168D244—41no—
169C4×D61—42no—
170C4×D61—41no—
171C4×D61—41no—
172C4×D61—41no—
173C4×D61C22⋊Q811yescharacteristic
174C4×D61C22⋊Q811yescharacteristic
175C4×D61C2×SD1611yescharacteristic
176C4×D61C2×SD1611yescharacteristic
177Dic122—21no—
178Dic122—41no—
179Dic122—84no—
180C8×C61—41no—
181C8×C61C2×SD1611yescharacteristic
182C8×C61C2×SD1611yescharacteristic
183C8×C61C22×Q811yescharacteristic
184C8×C61C2×#4111yescharacteristic
185#92418—41no—
186#92418C22⋊Q811yescharacteristic
187#92418C22⋊Q811yescharacteristic
188#92418C2×SD1611yescharacteristic
189#92418C2×SD1611yescharacteristic
190#33637—611nomaximal, Sylow
191C22×C61—41no—
192C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
193C2×D61—41no—
194C2×D61—41no—
195C2×D61—82no—
196C2×D61#47611yescharacteristic
197C2×D61#47611yescharacteristic
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—82no—
201C4×C61C2×C22⋊Q811yescharacteristic
202C4×C61C22×SD1611yescharacteristic
203Dic61—22no—
204Dic61—41no—
205Dic61—82no—
206Dic61#47611yescharacteristic
207Dic61#47611yescharacteristic
208#3160—611no—
209C2×#443—611no—
210#2231—611no—
211#2231—611no—
212#2231—611no—
213#2231D6111yescharacteristic, complemented
214#2230—612no—
215#2230—612no—
216#2155—612no—
217C2×#364—611no—
218#2145—612no—
219C2×C61—21no—
220C2×C61—81no—
221C2×C61C2×#47611yescharacteristic
222D61—21no—
223D61—21no—
224D61—81no—
225C2×#135—611no—
226C2×#131—1221no—
227#533—1221no—
228C2×M32—612no—
229#500—1221no—
230#499—611no—
231#499—1221no—
232#499C2×D6111yescharacteristic, complemented
233#498—612no—
234#443—611no—
235#443—611no—
236#443—611no—
237#443—1221no—
238#443—1221no—
239#443C2×D6111yescharacteristic
240C2×#108—611no—
241#427—612no—
242#427—1221no—
243C2×#104—1221no—
244C2×C4×C8—611no—
245#365—612no—
246#365—612no—
247#365—612no—
248#365C2×D6112yescomplemented
249#364—612no—
250#364—611no—
251#364C2×D6111yescharacteristic, complemented
252#360—612no—
253#360—612no—
254#360—612no—
255#360C2×D6112yes—
256C61#3363711yescharacteristic, complemented, Sylow
257C2×#41—1221no—
258#135—611no—
259#135—611no—
260#135—611no—
261#135—1221no—
262#135—1221no—
263#135C22×D6111yescharacteristic
264C2×Q16—611no—
265C2×SD16—1221no—
266C2×D8—611no—
267#131—21no—
268#131—1221no—
269#131—1221no—
270#131—1221no—
271#131—2441no—
272#131—2441no—
273C2×M16—1221no—
274C2×M16—2441no—
275C22×C8—611no—
276C22×C8—611no—
277C22×C8—2441no—
278C4⋊Q8—1222no—
279#125—21no—
280#125—1221no—
281C22⋊Q8—2442no—
282C4×Q8—2442no—
283#117—1222no—
284C2×C42—611no—
285M32—612no—
286M32—1222no—
287M32C22×D6112yes—
288C2×C16—1222no—
289#108—611no—
290#108—1221no—
291#108C22×D6111yescharacteristic
292#107—21no—
293#107—1222no—
294#107—1221no—
295#106—612no—
296#106—1222no—
297#106C22×D6112yes—
298#104—21no—
299#104—1221no—
300#104—1221no—
301#104—1221no—
302C4×C8—611no—
303C4×C8—611no—
304C4×C8—611no—
305C4×C8C22×D6111yescharacteristic
306#41—21no—
307#41—1221no—
308#41—1221no—
309#41—1221no—
310#41—2441no—
311#41—2441no—
312C2×Q8—1221no—
313C2×Q8—2442no—
314C2×D4—1221no—
315C2×D4—2441no—
316C22×C4—611no—
317C22×C4—1221no—
318C22×C4—2441no—
319Q16—611no—
320Q16—1221no—
321Q16Q8×D6111yescharacteristic
322SD16—21no—
323SD16—1221no—
324SD16—1221no—
325SD16—1221no—
326D8—611no—
327D8—1221no—
328D8Q8×D6111yescharacteristic
329M16—21no—
330M16—41no—
331M16—1221no—
332M16—2441no—
333M16—2441no—
334M16—2441no—
335M16—2441no—
336C2×C8—41no—
337C2×C8—611no—
338C2×C8—611no—
339C2×C8—611no—
340C2×C8—611no—
341C2×C8—611no—
342C2×C8—611no—
343C2×C8—611no—
344C2×C8—611no—
345C2×C8—611no—
346C2×C8—611no—
347C2×C8—2441no—
348C2×C8—2441no—
349C2×C8—2441no—
350C2×C8—2441no—
351C2×C8D4×D6111yescharacteristic
352C2×C8C23×D6111yescharacteristic
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—1222no—
356C4⋊C4—2442no—
357C4⋊C4—2442no—
358C22⋊C4—2442no—
359C42—611no—
360C42—1221no—
361C42—2442no—
362C42D4×D6111yescharacteristic
363C16—22no—
364C16—1222no—
365C23—611no—
366C23—2441no—
367Q8—21no—
368Q8—1221no—
369Q8—4884no—
370Q8—2441no—
371D4—21no—
372D4—41no—
373D4—1221no—
374D4—2441no—
375D4—2441no—
376D4—2441no—
377D4—2441no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—611no—
382C2×C4—611no—
383C2×C4—611no—
384C2×C4—611no—
385C2×C4—611no—
386C2×C4—1221no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4—2441no—
393C2×C4C2×D4×D6111yescharacteristic
394C8—41no—
395C8—611no—
396C8—611no—
397C8—611no—
398C8—611no—
399C8—2441no—
400C8SD16×D6111yescharacteristic
401C8SD16×D6111yescharacteristic
402C8C2×Q8×D6111yescharacteristic
403C8#41×D6111yescharacteristic
404C22—41no—
405C22—611no—
406C22—611no—
407C22—4882no—
408C22—2441no—
409C22—2441no—
410C22AGL(1,ℤ/8)×D6111yescharacteristic
411C4—41no—
412C4—41no—
413C4—82no—
414C4—611no—
415C4—611no—
416C4—1222no—
417C4—4882no—
418C4—2441no—
419C4C22⋊Q8×D6111yescharacteristic
420C4C2×SD16×D6111yescharacteristic
421C2—21no—
422C2—81no—
423C2—1221no—
424C2—1221no—
425C2—4881no—
426C2#476×D6111yescharacteristic
subgroupG/H
centerC2#476×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2231C2
socleC2×C61C2×#476
prime radical#135×D61C22
hypercenter#2231D61
hypocenterC61#33637
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2231

Complements

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

NH ≅ G/Ntype
1. #21456332C2semidirect
3. #21447996C2semidirect
4. #21447995C2semidirect
5. #21447992C2semidirect
6. #21447989C2semidirect
7. #21447988C2semidirect
8. #21447590C2semidirect
9. #364×D61C2semidirect
11. C61×#2231C2semidirect
17. #16840164C22semidirect
19. #16840159C22semidirect
30. #16839586C22semidirect
31. #16839585C22semidirect
32. #16839584C22semidirect
33. #16839583C22semidirect
34. #16839582C22semidirect
38. C61×#499C22semidirect
40. C61×#365C22semidirect
41. C61×#364C22semidirect
44. #14519800Q8semidirect
48. #14519796Q8semidirect
49. Q16×D61Q8semidirect
51. D8×D61Q8semidirect
213. #2231D61semidirect
232. #499C2×D61semidirect
248. #365C2×D61semidirect
251. #364C2×D61semidirect
256. C61#33637semidirect

Maximal subgroups

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

subgroupindex
#214563322
#443×D612
#214479962
#214479952
#214479922
#214479892
#214479882
#214475902
#364×D612
#214475532
C61×#22312
#3363761