Finite Group Databasebeta

← #44562379

Subgroups — #44562379

Order 32128.

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#30632371C211yesmaximal, characteristic, complemented
2D4×C22⋊D251C211yesmaximal, characteristic
3#30632050C211yesmaximal, characteristic, complemented
4#30632012C211yesmaximal, characteristic, complemented
5#30631999C211yesmaximal, characteristic, complemented
6#30631989C211yesmaximal, characteristic, complemented
7#30631935C211yesmaximal, characteristic, complemented
8#30631913C211yesmaximal, characteristic, complemented
9#30631885C211yesmaximal, characteristic, complemented
10#30631836C211yesmaximal, characteristic, complemented
11#30631788C211yesmaximal, characteristic, complemented
12#30631761C211yesmaximal, characteristic, complemented
13#30631692C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#450×C251C211yesmaximal, characteristic, complemented
16C22×C22⋊D251—21no—
17C2×D4×D251C2211yescharacteristic
18C2×#16860797C2211yescharacteristic, complemented
19C2×#16860794C2211yescharacteristic, complemented
20C2×C8×D251C2211yescharacteristic, complemented
21#22548711C2211yescharacteristic
22#22548708—21no—
23#22548706C2211yescharacteristic, complemented
24#22548705—21no—
25#22548697—21no—
26#22548691C2211yescharacteristic, complemented
27#22548689C2211yescharacteristic
28#22548687—21no—
29#22548686C2211yescharacteristic
30#22548684—21no—
31#22548683—21no—
32Q8×Dic251C2211yescharacteristic, complemented
33D4×Dic251C2211yescharacteristic
34C4×C22⋊D251C2211yescharacteristic
35#22548674C2211yescharacteristic, complemented
36#22548673—21no—
37C4⋊C4×D251C2211yescharacteristic, complemented
38#22548649C2211yescharacteristic, complemented
39#22548648C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic, complemented
45#22548636C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic, complemented
48#22548636C2211yescharacteristic, complemented
49#22548635C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51#22548625C2211yescharacteristic, complemented
52C22×D4×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic, complemented
55C22⋊Q8×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic, complemented
58C22⋊C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C2×D1004C2311yescharacteristic
61C2×C22⋊D251—21no—
62C2×C22⋊D251—21no—
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C2×C22⋊D251—21no—
66C2×C22⋊D251—44no—
67C2×C22⋊D251D411yescharacteristic
68D4×D251—22no—
69D4×D251—41no—
70C22×Dic251—21no—
71C2×C4×D251—21no—
72C2×C4×D251C2311yescharacteristic
73#16860797—41no—
74#16860794—41no—
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C8×D251—41no—
78C4⋊Dic251—21no—
79C4⋊Dic251—21no—
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4D411yescharacteristic
87C22⋊Dic251—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251—21no—
90C22⋊Dic251D411yescharacteristic
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4D411yescharacteristic
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97C24×C251—21no—
98C2×Q8×C251C2311yescharacteristic
99C2×D4×C251—22no—
100C2×D4×C251—21no—
101C2×D4×C251—41no—
102C2×D4×C251C2311yescharacteristic
103C2×D4×C251C2311yescharacteristic
104C22×C4×C251C2311yescharacteristic
105SD16×C251—22no—
106SD16×C251—41no—
107C2×C8×C251C2311yescharacteristic
108C2×C8×C251C2311yescharacteristic
109C4⋊C4×C251—21no—
110C4⋊C4×C251C2311yescharacteristic
111C22⋊C4×C251—21no—
112C22×D251—21no—
113C22×D251—42no—
114C22×D251—42no—
115C22×D251—81no—
116C22×D251C2×D411yescharacteristic
117D1004—21no—
118D1004—21no—
119C22⋊D251—22no—
120C22⋊D251—42no—
121C22⋊D251—42no—
122C22⋊D251—42no—
123C22⋊D251—42no—
124C22⋊D251—81no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—42no—
129C2×Dic251C2×D411yescharacteristic
130C2×Dic251C2×D411yescharacteristic
131C2×Dic251C2×D411yescharacteristic
132C4×D251—21no—
133C4×D251—21no—
134C4×D251—42no—
135C251⋊C8—21no—
136C251⋊C8—21no—
137C23×C251—21no—
138C23×C251—21no—
139C23×C251—42no—
140C23×C251—42no—
141C23×C251—81no—
142C23×C251C2×D411yescharacteristic
143Q8×C251—21no—
144Q8×C251—21no—
145D4×C251—21no—
146D4×C251—21no—
147D4×C251—21no—
148D4×C251—42no—
149D4×C251#4112yes—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251—21no—
154C2×C4×C251C2×D411yescharacteristic
155C2×C4×C251C2411yescharacteristic
156C8×C251—21no—
157C8×C251—21no—
158C22×C251—21no—
159C22×C251—22no—
160C22×C251—21no—
161C22×C251—21no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251—84no—
168C22×C251C22×D411yescharacteristic
169C2×D251—21no—
170C2×D251—21no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—84no—
175C2×D251AGL(1,ℤ/8)12yes—
176C4×C251—21no—
177C4×C251—41no—
178C4×C251—41no—
179C4×C251C2×#4111yescharacteristic
180C4×C251H(5,𝔽2)11yescharacteristic
181Dic251—21no—
182Dic251—21no—
183Dic251—21no—
184Dic251—41no—
185Dic251—41no—
186Dic251—41no—
187C2×C251—42no—
188C2×C251—41no—
189C2×C251—81no—
190C2×C251#54611yescharacteristic
191C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C251#57711yescharacteristic
193D251—22no—
194D251—81no—
195C251#331011yescharacteristic, complemented, Sylow
196#3310—2511nomaximal, Sylow
197#552—2511no—
198D42—2511no—
199#490—2511no—
200#482—2511no—
201#480—2511no—
202#478—2511no—
203#468—2511no—
204#465—2511no—
205#460—2511no—
206#450—2511no—
207#450D25111yescharacteristic, complemented
208#440—2511no—
209#435—2511no—
210#418—2511no—
211C2×#102—2511no—
212C22×D4—5021no—
213C22×D4—2511no—
214C22×D4C2×D25111yescharacteristic
215C2×SD16—2511no—
216C2×SD16C2×D25111yescharacteristic
217C2×SD16C2×D25111yescharacteristic, complemented
218C2×M16—2511no—
219C22×C8—2511no—
220#127—2511no—
221#126—5021no—
222#125—5021no—
223#125—2511no—
224#123—5021no—
225C22⋊Q8—2511no—
226C22⋊Q8C2×D25111yescharacteristic, complemented
227#121—5021no—
228#121—2511no—
229#121—2511no—
230#120—5021no—
231#120—5021no—
232C4×Q8—2511no—
233C4×D4—2511no—
234C4×D4—2511no—
235#117—5021no—
236#117—2511no—
237C2×C4⋊C4—2511no—
238#107—2511no—
239#107—2511no—
240#106—2511no—
241C4⋊C8—2511no—
242#103—2511no—
243#102—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102—2511no—
248#102—2511no—
249#102—2511no—
250#102C2×D25111yescharacteristic
251#102C2×D25111yescharacteristic, complemented
252C22⋊C8—2511no—
253C22⋊C8C2×D25111yescharacteristic, complemented
254#97—2511no—
255C24—21no—
256C24—5021no—
257C2×Q8C22×D25111yescharacteristic
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—10044no—
262C2×D4—10041no—
263C2×D4—5021no—
264C2×D4—5021no—
265C2×D4—5022no—
266C2×D4—5021no—
267C2×D4—5021no—
268C2×D4—5021no—
269C2×D4—2511no—
270C2×D4—2511no—
271C2×D4C22×D25111yescharacteristic
272C2×D4C22×D25111yescharacteristic
273C22×C4—5021no—
274C22×C4—5021no—
275C22×C4—2511no—
276C22×C4C22×D25111yescharacteristic
277SD16—22no—
278SD16—41no—
279SD16—10041no—
280M16—10041no—
281C2×C8—10041no—
282C2×C8—2511no—
283C2×C8—2511no—
284C2×C8C22×D25111yescharacteristic
285C2×C8C22×D25111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—5021no—
288C4⋊C4—5021no—
289C4⋊C4—5021no—
290C4⋊C4—5021no—
291C4⋊C4—5021no—
292C4⋊C4—2511no—
293C4⋊C4—2511no—
294C4⋊C4—2511no—
295C4⋊C4—2511no—
296C4⋊C4C22×D25111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—5021no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4—2511no—
304C22⋊C4—2511no—
305C42—5021no—
306C42—5021no—
307C42—2511no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—20081no—
314C23—10042no—
315C23—10042no—
316C23—5021no—
317C23—2511no—
318C23D4×D25111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—20081no—
326D4—10042no—
327D4—10042no—
328D4—10042no—
329D4—10042no—
330D4—5022no—
331D4—5021no—
332D4—5021no—
333D4#1686080912yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—10042no—
339C2×C4—10042no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4—2511no—
346C2×C4—2511no—
347C2×C4—2511no—
348C2×C4D4×D25111yescharacteristic
349C2×C4C23×D25111yescharacteristic
350C8—21no—
351C8—21no—
352C8—5021no—
353C8—5021no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—20084no—
365C22—10041no—
366C22—10041no—
367C22—10041no—
368C22—5021no—
369C22—5021no—
370C22—2512no—
371C22C2×D4×D25111yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—10041no—
376C4—10041no—
377C4—10041no—
378C4—5021no—
379C4—5021no—
380C4—5021no—
381C4C2×#1686080911yescharacteristic
382C4#2254876011yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—20081no—
387C2—5022no—
388C2#3063232411yescharacteristic
389C2AGL(1,ℤ/8)×D25111yescharacteristic
390C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#450×C251C2
socleC22×C251C22×D4
prime radicalD4×C22⋊D251C2
hypercenter#450D251
hypocenterC251#3310
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #30632371C2semidirect
3. #30632050C2semidirect
4. #30632012C2semidirect
5. #30631999C2semidirect
6. #30631989C2semidirect
7. #30631935C2semidirect
8. #30631913C2semidirect
9. #30631885C2semidirect
10. #30631836C2semidirect
11. #30631788C2semidirect
12. #30631761C2semidirect
13. #30631692C2semidirect
14. #102×D251C2semidirect
15. #450×C251C2semidirect
18. C2×#16860797C22semidirect
19. C2×#16860794C22semidirect
20. C2×C8×D251C22semidirect
23. #22548706C22semidirect
26. #22548691C22semidirect
32. Q8×Dic251C22semidirect
35. #22548674C22semidirect
37. C4⋊C4×D251C22semidirect
38. #22548649C22semidirect
39. #22548648C22semidirect
40. #22548647C22semidirect
41. Dic251⋊C8C22semidirect
42. #22548639C22semidirect
43. #22548637C22semidirect
44. #22548637C22semidirect
47. #22548636C22semidirect
48. #22548636C22semidirect
49. #22548635C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. #22548625C22semidirect
54. C2×SD16×C251C22semidirect
55. C22⋊Q8×C251C22semidirect
57. #102×C251C22semidirect
58. C22⋊C8×C251C22semidirect
195. C251#3310semidirect
207. #450D251semidirect
217. C2×SD16C2×D251semidirect
226. C22⋊Q8C2×D251semidirect
251. #102C2×D251semidirect
253. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323712
D4×C22⋊D2512
#306320502
#306320122
#306319992
#306319892
#306319352
#306319132
#306318852
#306318362
#306317882
#306317612
#306316922
#102×D2512
#450×C2512
#3310251