Finite Group Databasebeta

← #24919490

Subgroups — #24919490

Order 10624.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128393C211yesmaximal, characteristic, complemented
2#19128335C211yesmaximal, characteristic
3#19128064C211yesmaximal, characteristic, complemented
4#19128027C211yesmaximal, characteristic, complemented
5#19128012C211yesmaximal, characteristic, complemented
6#19128000C211yesmaximal, characteristic, complemented
7#19127950C211yesmaximal, characteristic, complemented
8#19127899C211yesmaximal, characteristic, complemented
9#19127851C211yesmaximal, characteristic, complemented
10#19127850C211yesmaximal, characteristic, complemented
11#19127804C211yesmaximal, characteristic, complemented
12#19127772C211yesmaximal, characteristic, complemented
13#19127704C211yesmaximal, characteristic, complemented
14C2×#14581714C211yesmaximal, characteristic, complemented
15#465×C83C211yesmaximal, characteristic, complemented
16C22×D332C2211yescharacteristic
17C2×D4×D83—21no—
18C2×D4×D83C2211yescharacteristic
19C2×#13050577C2211yescharacteristic
20C2×#13050577C2211yescharacteristic, complemented
21C2×#13050573C2211yescharacteristic, complemented
22#14581788C2211yescharacteristic
23#14581787—21no—
24#14581783—21no—
25#14581783C2211yescharacteristic, complemented
26#14581776—21no—
27#14581768C2211yescharacteristic, complemented
28#14581767C2211yescharacteristic
29#14581765—21no—
30#14581762—21no—
31#14581760—21no—
32Q8×Dic83C2211yescharacteristic, complemented
33#14581754C2211yescharacteristic
34C4×C22⋊D83C2211yescharacteristic
35#14581750C2211yescharacteristic, complemented
36#14581748—21no—
37C2×C4⋊Dic83C2211yescharacteristic, complemented
38#14581725C2211yescharacteristic, complemented
39#14581725C2211yescharacteristic, complemented
40Dic83⋊C8C2211yescharacteristic, complemented
41#14581714C2211yescharacteristic
42#14581714C2211yescharacteristic
43#14581714C2211yescharacteristic, complemented
44#14581714C2211yescharacteristic, complemented
45#14581713C2211yescharacteristic, complemented
46#14581713C2211yescharacteristic, complemented
47#14581712C2211yescharacteristic
48#14581712C2211yescharacteristic, complemented
49C22⋊(C83⋊C8)C2211yescharacteristic, complemented
50(C2×D83)⋊C8C2211yescharacteristic, complemented
51#14581702C2211yescharacteristic, complemented
52C2×SD16×C83C2211yescharacteristic, complemented
53C22×C8×C83C2211yescharacteristic, complemented
54C22⋊Q8×C83C2211yescharacteristic, complemented
55#121×C83C2211yescharacteristic
56#106×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60C23×D83—21no—
61C2×D332—22no—
62C2×D332—21no—
63C2×D332—21no—
64C2×D332—41no—
65C2×D332C2311yescharacteristic
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C22⋊D83D411yescharacteristic
71D4×D83—22no—
72D4×D83—41no—
73D4×D83—44no—
74C22×Dic83—21no—
75C2×C4×D83—21no—
76C2×C4×D83C2311yescharacteristic
77#13050577—22no—
78#13050577—41no—
79#13050573—41no—
80C2×C83⋊C8C2311yescharacteristic
81C2×C83⋊C8C2311yescharacteristic
82C4⋊Dic83—21no—
83C4⋊Dic83—21no—
84C4⋊Dic83—21no—
85C4⋊Dic83C2311yescharacteristic
86C4⋊Dic83C2311yescharacteristic
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89Dic83⋊C4D411yescharacteristic
90C22⋊Dic83D411yescharacteristic
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4D411yescharacteristic
96C4×Dic83—21no—
97C4×Dic83—21no—
98C4×Dic83C2311yescharacteristic
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83—21no—
101C2×D4×C83C2311yescharacteristic
102C22×C4×C83C2311yescharacteristic
103SD16×C83—41no—
104C2×C8×C83—41no—
105C2×C8×C83C2311yescharacteristic
106C2×C8×C83C2311yescharacteristic
107C4⋊C4×C83—21no—
108C4⋊C4×C83C2311yescharacteristic
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C23×C83—21no—
113C23×C83C2×D411yescharacteristic
114C22×D83—21no—
115C22×D83—21no—
116C22×D83—42no—
117C22×D83—42no—
118C22×D83—42no—
119C22×D83—42no—
120C22×D83—81no—
121C22×D83—81no—
122C22×D83C2×D411yescharacteristic
123Q8×C83—21no—
124Q8×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—42no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2411yescharacteristic
135C22⋊D83—22no—
136C22⋊D83—42no—
137C22⋊D83—42no—
138C22⋊D83—81no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—42no—
142C2×Dic83C2×D411yescharacteristic
143C2×Dic83C2×D411yescharacteristic
144C2×Dic83C2×D411yescharacteristic
145D332—21no—
146D332—21no—
147D332—21no—
148D332—42no—
149D332—42no—
150D332#4112yes—
151C4×D83—21no—
152C4×D83—21no—
153C4×D83—42no—
154C8×C83—21no—
155C8×C83—21no—
156C83⋊C8—21no—
157C83⋊C8—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83AGL(1,ℤ/8)12yes—
164C22×C83C22×D411yescharacteristic
165C2×D83—21no—
166C2×D83—22no—
167C2×D83—21no—
168C2×D83—21no—
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—84no—
175C2×D83—84no—
176C4×C83—21no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83C2×#4111yescharacteristic
181C4×C83H(5,𝔽2)11yescharacteristic
182Dic83—21no—
183Dic83—21no—
184Dic83—21no—
185Dic83—41no—
186Dic83—41no—
187C2×C83—22no—
188C2×C83—81no—
189C2×C83#54611yescharacteristic
190C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C83#57711yescharacteristic
192D83—41no—
193D83—42no—
194D83—81no—
195#3310—831nomaximal, Sylow
196C83#331011yescharacteristic, complemented, Sylow
197#552—831no—
198D42—831no—
199#490—831no—
200#482—831no—
201#480—831no—
202#478—831no—
203#468—831no—
204#465D8311yescharacteristic, complemented
205#460—831no—
206#450—831no—
207#450—831no—
208#440—831no—
209#435—831no—
210#418—831no—
211C2×#102—831no—
212C22×D4—831no—
213C22×D4—831no—
214C22×D4—1661no—
215C2×SD16—831no—
216C2×SD16—831no—
217C2×SD16C2×D8311yescharacteristic, complemented
218C2×M16—831no—
219C22×C8C2×D8311yescharacteristic, complemented
220#127—831no—
221#126—1661no—
222#125—831no—
223#125—1661no—
224#123—1661no—
225C22⋊Q8—831no—
226C22⋊Q8C2×D8311yescharacteristic, complemented
227#121—831no—
228#121—1661no—
229#121C2×D8311yescharacteristic
230#120—1661no—
231#120—1661no—
232C4×Q8—831no—
233C4×D4—831no—
234C4×D4—831no—
235#117—831no—
236#117—1661no—
237C2×C4⋊C4—831no—
238#107—831no—
239#107—831no—
240#106C2×D8311yescharacteristic, complemented
241C4⋊C8—831no—
242#103C2×D8311yescharacteristic, complemented
243#102—831no—
244#102—831no—
245#102—831no—
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102—831no—
251#102C2×D8311yescharacteristic, complemented
252C22⋊C8—831no—
253C22⋊C8—831no—
254#97—831no—
255C24—1661no—
256C24—1661no—
257C2×Q8C22×D8311yescharacteristic
258C2×D4—21no—
259C2×D4—3321no—
260C2×D4—3324no—
261C2×D4—3321no—
262C2×D4—831no—
263C2×D4—831no—
264C2×D4—831no—
265C2×D4—1661no—
266C2×D4—1662no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1662no—
271C2×D4—1661no—
272C2×D4C22×D8311yescharacteristic
273C22×C4—831no—
274C22×C4—1661no—
275C22×C4—1661no—
276C22×C4C22×D8311yescharacteristic
277SD16—41no—
278SD16—3321no—
279SD16—1662no—
280M16—3321no—
281C2×C8—41no—
282C2×C8—831no—
283C2×C8—831no—
284C2×C8C22×D8311yescharacteristic
285C2×C8C22×D8311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—831no—
288C4⋊C4—831no—
289C4⋊C4—831no—
290C4⋊C4—1661no—
291C4⋊C4—1661no—
292C4⋊C4—1661no—
293C4⋊C4—1661no—
294C4⋊C4—1661no—
295C4⋊C4C22×D8311yescharacteristic
296C4⋊C4C22×D8311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—831no—
300C22⋊C4—831no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C42—831no—
306C42—1661no—
307C42—1661no—
308C23—21no—
309C23—3322no—
310C23—3322no—
311C23—3322no—
312C23—3322no—
313C23—831no—
314C23—6641no—
315C23—6641no—
316C23—1661no—
317C23—1661no—
318C23D4×D8311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—3322no—
325D4—3322no—
326D4—3322no—
327D4—3322no—
328D4—832no—
329D4—6641no—
330D4—1662no—
331D4—1661no—
332D4—1661no—
333D4—1661no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—3322no—
340C2×C4—3322no—
341C2×C4—831no—
342C2×C4—831no—
343C2×C4—831no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4D4×D8311yescharacteristic
349C2×C4C23×D8311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1661no—
353C8—1661no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—3321no—
360C22—3321no—
361C22—3321no—
362C22—3321no—
363C22—3321no—
364C22—6644no—
365C22—6644no—
366C22—1661no—
367C22—1662no—
368C22—1661no—
369C22—1661no—
370C22#1458181912yes—
371C22C2×D4×D8311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—3321no—
377C4—3321no—
378C4—1661no—
379C4—1661no—
380C4—1661no—
381C4#41×D8311yescharacteristic
382C4#1458183711yescharacteristic
383C2—22no—
384C2—81no—
385C2—3322no—
386C2—3321no—
387C2—6641no—
388C2#1912834011yescharacteristic
389C2C2×#1458181911yescharacteristic
390C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#465×C83C2
socleC22×C83C22×D4
prime radical#19128335C2
hypercenter#465D83
hypocenterC83#3310
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #19128393C2semidirect
3. #19128064C2semidirect
4. #19128027C2semidirect
5. #19128012C2semidirect
6. #19128000C2semidirect
7. #19127950C2semidirect
8. #19127899C2semidirect
9. #19127851C2semidirect
10. #19127850C2semidirect
11. #19127804C2semidirect
12. #19127772C2semidirect
13. #19127704C2semidirect
14. C2×#14581714C2semidirect
15. #465×C83C2semidirect
20. C2×#13050577C22semidirect
21. C2×#13050573C22semidirect
25. #14581783C22semidirect
27. #14581768C22semidirect
32. Q8×Dic83C22semidirect
35. #14581750C22semidirect
37. C2×C4⋊Dic83C22semidirect
38. #14581725C22semidirect
39. #14581725C22semidirect
40. Dic83⋊C8C22semidirect
43. #14581714C22semidirect
44. #14581714C22semidirect
45. #14581713C22semidirect
46. #14581713C22semidirect
48. #14581712C22semidirect
49. C22⋊(C83⋊C8)C22semidirect
50. (C2×D83)⋊C8C22semidirect
51. #14581702C22semidirect
52. C2×SD16×C83C22semidirect
53. C22×C8×C83C22semidirect
54. C22⋊Q8×C83C22semidirect
56. #106×C83C22semidirect
57. #103×C83C22semidirect
58. #102×C83C22semidirect
196. C83#3310semidirect
204. #465D83semidirect
217. C2×SD16C2×D83semidirect
219. C22×C8C2×D83semidirect
226. C22⋊Q8C2×D83semidirect
240. #106C2×D83semidirect
242. #103C2×D83semidirect
251. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283932
#191283352
#191280642
#191280272
#191280122
#191280002
#191279502
#191278992
#191278512
#191278502
#191278042
#191277722
#191277042
C2×#145817142
#465×C832
#331083