Finite Group Databasebeta

← #44562184

Subgroups — #44562184

Order 32128.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632395C211yesmaximal, characteristic, complemented
2#30632321C211yesmaximal, characteristic
3#30632032C211yesmaximal, characteristic, complemented
4#30632007C211yesmaximal, characteristic, complemented
5#30632003C211yesmaximal, characteristic, complemented
6#30631976C211yesmaximal, characteristic, complemented
7#30631919C211yesmaximal, characteristic, complemented
8#30631885C211yesmaximal, characteristic, complemented
9#30631837C211yesmaximal, characteristic, complemented
10#30631836C211yesmaximal, characteristic, complemented
11#30631772C211yesmaximal, characteristic, complemented
12C8×C22⋊D251C211yesmaximal, characteristic, complemented
13C2×#22548637C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#465×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C2×D4×D251—21no—
18C2×D4×D251C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20C2×#16860797C2211yescharacteristic, complemented
21C2×C8×D251C2211yescharacteristic, complemented
22#22548714C2211yescharacteristic, complemented
23#22548711C2211yescharacteristic
24#22548705—21no—
25#22548695—21no—
26#22548692—21no—
27#22548691C2211yescharacteristic, complemented
28#22548690C2211yescharacteristic
29#22548688—21no—
30#22548685—21no—
31#22548683—21no—
32Q8×Dic251C2211yescharacteristic, complemented
33#22548677C2211yescharacteristic
34C4×C22⋊D251C2211yescharacteristic
35(C22⋊D251)⋊C4—21no—
36C2×C4⋊Dic251C2211yescharacteristic, complemented
37C4⋊C4×D251C2211yescharacteristic, complemented
38#22548646C2211yescharacteristic, complemented
39#22548646C2211yescharacteristic, complemented
40Dic251⋊C8C2211yescharacteristic, complemented
41#22548637C2211yescharacteristic
42#22548637C2211yescharacteristic
43#22548637C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic, complemented
45#22548636C2211yescharacteristic, complemented
46#22548636C2211yescharacteristic, complemented
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic, complemented
49C22⋊(C251⋊C8)C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C2×SD16×C251C2211yescharacteristic, complemented
53C22×C8×C251C2211yescharacteristic, complemented
54C22⋊Q8×C251C2211yescharacteristic, complemented
55#121×C251C2211yescharacteristic
56#106×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58#102×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C23×D251—21no—
61C2×Dic502—21no—
62C2×D1004—22no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004—41no—
66C2×D1004C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C2×C22⋊D251D411yescharacteristic
72D4×D251—22no—
73D4×D251—41no—
74D4×D251—44no—
75C22×Dic251—21no—
76C2×C4×D251—21no—
77C2×C4×D251—21no—
78C2×C4×D251C2311yescharacteristic
79#16860797—22no—
80#16860797—41no—
81C2×C251⋊C8C2311yescharacteristic
82C2×C251⋊C8C2311yescharacteristic
83C8×D251—41no—
84C4⋊Dic251—21no—
85C4⋊Dic251—21no—
86C4⋊Dic251C2311yescharacteristic
87C4⋊Dic251C2311yescharacteristic
88Dic251⋊C4—21no—
89Dic251⋊C4—21no—
90Dic251⋊C4—21no—
91Dic251⋊C4D411yescharacteristic
92C22⋊Dic251D411yescharacteristic
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4D411yescharacteristic
97C4×Dic251—21no—
98C4×Dic251C2311yescharacteristic
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251C2311yescharacteristic
103SD16×C251—41no—
104C2×C8×C251—41no—
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251—21no—
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22⋊C4×C251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—42no—
115C22×D251—42no—
116C22×D251—42no—
117C22×D251—42no—
118C22×D251—81no—
119C22×D251—81no—
120C22×D251C2×D411yescharacteristic
121Dic502—81no—
122D1004—21no—
123D1004—21no—
124D1004—21no—
125D1004—42no—
126D1004—42no—
127D1004#4112yes—
128C22⋊D251—42no—
129C22⋊D251—42no—
130C22⋊D251—81no—
131C22⋊D251SD1614yescomplemented
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—81no—
135C2×Dic251C2×D411yescharacteristic
136C2×Dic251C2×D411yescharacteristic
137C2×Dic251C2×D411yescharacteristic
138C4×D251—21no—
139C4×D251—21no—
140C4×D251—42no—
141C4×D251—81no—
142C251⋊C8—21no—
143C251⋊C8—21no—
144C23×C251—21no—
145C23×C251C2×D411yescharacteristic
146Q8×C251—21no—
147Q8×C251—21no—
148D4×C251—21no—
149D4×C251—21no—
150D4×C251—42no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251—21no—
154C2×C4×C251—21no—
155C2×C4×C251—21no—
156C2×C4×C251C2×D411yescharacteristic
157C2×C4×C251C2411yescharacteristic
158C8×C251—21no—
159C8×C251—21no—
160C22×C251—21no—
161C22×C251—21no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251C2×SD1612yes—
166C22×C251C22×D411yescharacteristic
167C2×D251—21no—
168C2×D251—21no—
169C2×D251—22no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—41no—
175C2×D251—84no—
176C2×D251—84no—
177C2×D251C2×SD1612yes—
178C4×C251—21no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251—41no—
182C4×C251C2×#4111yescharacteristic
183C4×C251H(5,𝔽2)11yescharacteristic
184Dic251—21no—
185Dic251—21no—
186Dic251—41no—
187Dic251—41no—
188Dic251C2×SD1612yes—
189C2×C251—22no—
190C2×C251—81no—
191C2×C251#54611yescharacteristic
192C2×C251C22×SD1611yescharacteristic
193C2×C251#57611yescharacteristic
194D251—22no—
195D251—42no—
196D251—81no—
197C251#329511yescharacteristic, complemented, Sylow
198#3295—2511nomaximal, Sylow
199#554—2511no—
200D42—2511no—
201#486—2511no—
202#481—2511no—
203#481—2511no—
204#476—2511no—
205#465—2511no—
206#465D25111yescharacteristic, complemented
207#460—2511no—
208#450—2511no—
209#450—2511no—
210C4×SD16—2511no—
211C8×D4—2511no—
212C2×#102—2511no—
213C2×#102—2511no—
214C22×D4—5021no—
215C22×D4—2511no—
216C22×D4—2511no—
217C2×SD16—2511no—
218C2×SD16—2511no—
219C2×SD16C2×D25111yescharacteristic, complemented
220C22×C8—2511no—
221C22×C8C2×D25111yescharacteristic, complemented
222C4⋊Q8—2511no—
223#127—2511no—
224#125—5021no—
225C22⋊Q8—5021no—
226C22⋊Q8—5021no—
227C22⋊Q8—2511no—
228C22⋊Q8C2×D25111yescharacteristic, complemented
229#121—5021no—
230#121—2511no—
231#121C2×D25111yescharacteristic
232#120—5021no—
233#120—5021no—
234C4×Q8—2511no—
235C4×D4—5021no—
236C4×D4—2511no—
237C4×D4—2511no—
238C2×C4⋊C4—2511no—
239C2×C4⋊C4—2511no—
240#106—2511no—
241#106—2511no—
242#106C2×D25111yescharacteristic, complemented
243C4⋊C8—2511no—
244#103C2×D25111yescharacteristic, complemented
245#102—2511no—
246#102—2511no—
247#102—2511no—
248#102—2511no—
249#102—2511no—
250#102—2511no—
251#102—2511no—
252#102—2511no—
253#102C2×D25111yescharacteristic, complemented
254C22⋊C8—2511no—
255C22⋊C8—2511no—
256C4×C8—2511no—
257C24—5021no—
258C24—5021no—
259C2×Q8—5021no—
260C2×Q8C22×D25111yescharacteristic
261C2×D4—21no—
262C2×D4—10041no—
263C2×D4—10044no—
264C2×D4—10041no—
265C2×D4—5021no—
266C2×D4—5022no—
267C2×D4—5022no—
268C2×D4—5021no—
269C2×D4—5021no—
270C2×D4—5021no—
271C2×D4—5021no—
272C2×D4—2511no—
273C2×D4—2511no—
274C2×D4—2511no—
275C2×D4C22×D25111yescharacteristic
276C22×C4—5021no—
277C22×C4—5021no—
278C22×C4—5021no—
279C22×C4—2511no—
280C22×C4C22×D25111yescharacteristic
281SD16—41no—
282SD16—10041no—
283SD16—5022no—
284C2×C8—41no—
285C2×C8—10041no—
286C2×C8—2511no—
287C2×C8—2511no—
288C2×C8C22×D25111yescharacteristic
289C2×C8C22×D25111yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—5021no—
292C4⋊C4—5021no—
293C4⋊C4—5021no—
294C4⋊C4—5021no—
295C4⋊C4—5021no—
296C4⋊C4—2511no—
297C4⋊C4—2511no—
298C4⋊C4—2511no—
299C4⋊C4C22×D25111yescharacteristic
300C4⋊C4C22×D25111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—5021no—
304C22⋊C4—5021no—
305C22⋊C4—5021no—
306C22⋊C4—2511no—
307C22⋊C4—2511no—
308C42—5021no—
309C42—2511no—
310C23—21no—
311C23—20081no—
312C23—20081no—
313C23—10042no—
314C23—10042no—
315C23—10042no—
316C23—10042no—
317C23—5021no—
318C23—5021no—
319C23—2511no—
320C23D4×D25111yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—20081no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—20081no—
328D4—10042no—
329D4—10042no—
330D4—10042no—
331D4—10042no—
332D4—5021no—
333D4—5021no—
334D4—5021no—
335D4—2514no—
336D4—2512no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—20081no—
343C2×C4—20081no—
344C2×C4—10042no—
345C2×C4—5021no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—2511no—
350C2×C4—2511no—
351C2×C4—2511no—
352C2×C4D4×D25111yescharacteristic
353C2×C4C23×D25111yescharacteristic
354C8—21no—
355C8—21no—
356C8—5021no—
357C8—5021no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—20084no—
364C22—20084no—
365C22—10041no—
366C22—10041no—
367C22—10041no—
368C22—10041no—
369C22—10041no—
370C22—5022no—
371C22—5021no—
372C22—5021no—
373C22—2512no—
374C22SD16×D25112yes—
375C22C2×D4×D25111yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—10041no—
381C4—10041no—
382C4—5021no—
383C4—5021no—
384C4—2512no—
385C4#41×D25111yescharacteristic
386C4#2254876011yescharacteristic
387C2—22no—
388C2—81no—
389C2—20081no—
390C2—10042no—
391C2—5022no—
392C2#3063232611yescharacteristic
393C2C2×SD16×D25111yescharacteristic
394C2#3063248911yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#465×C251C2
socleC22×C251C22×D4
prime radical#30632321C2
hypercenter#465D251
hypocenterC251#3295
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #30632395C2semidirect
3. #30632032C2semidirect
4. #30632007C2semidirect
5. #30632003C2semidirect
6. #30631976C2semidirect
7. #30631919C2semidirect
8. #30631885C2semidirect
9. #30631837C2semidirect
10. #30631836C2semidirect
11. #30631772C2semidirect
12. C8×C22⋊D251C2semidirect
13. C2×#22548637C2semidirect
14. #102×D251C2semidirect
15. #465×C251C2semidirect
20. C2×#16860797C22semidirect
21. C2×C8×D251C22semidirect
22. #22548714C22semidirect
27. #22548691C22semidirect
32. Q8×Dic251C22semidirect
36. C2×C4⋊Dic251C22semidirect
37. C4⋊C4×D251C22semidirect
38. #22548646C22semidirect
39. #22548646C22semidirect
40. Dic251⋊C8C22semidirect
43. #22548637C22semidirect
44. #22548637C22semidirect
45. #22548636C22semidirect
46. #22548636C22semidirect
48. #22548635C22semidirect
49. C22⋊(C251⋊C8)C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. C8×Dic251C22semidirect
52. C2×SD16×C251C22semidirect
53. C22×C8×C251C22semidirect
54. C22⋊Q8×C251C22semidirect
56. #106×C251C22semidirect
57. #103×C251C22semidirect
58. #102×C251C22semidirect
131. C22⋊D251SD16semidirect
197. C251#3295semidirect
206. #465D251semidirect
219. C2×SD16C2×D251semidirect
221. C22×C8C2×D251semidirect
228. C22⋊Q8C2×D251semidirect
242. #106C2×D251semidirect
244. #103C2×D251semidirect
253. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323952
#306323212
#306320322
#306320072
#306320032
#306319762
#306319192
#306318852
#306318372
#306318362
#306317722
C8×C22⋊D2512
C2×#225486372
#102×D2512
#465×C2512
#3295251