Finite Group Databasebeta

← #44562330

Subgroups — #44562330

Order 32128.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632337C211yesmaximal, characteristic, complemented
2#30632331C211yesmaximal, characteristic, complemented
3#30632045C211yesmaximal, characteristic, complemented
4#30632018C211yesmaximal, characteristic, complemented
5#30632004C211yesmaximal, characteristic, complemented
6#30631975C211yesmaximal, characteristic, complemented
7#30631937C211yesmaximal, characteristic, complemented
8#30631914C211yesmaximal, characteristic, complemented
9#30631900C211yesmaximal, characteristic, complemented
10#30631829C211yesmaximal, characteristic, complemented
11#30631799C211yesmaximal, characteristic, complemented
12#30631755C211yesmaximal, characteristic, complemented
13#30631685C211yesmaximal, characteristic, complemented
14C2×#22548636C211yesmaximal, characteristic, complemented
15#447×C251C211yesmaximal, characteristic, complemented
16C2×#16860809—21no—
17C2×#16860809C2211yescharacteristic
18C2×#16860798C2211yescharacteristic, complemented
19C2×#16860793C2211yescharacteristic, complemented
20C22×C251⋊C8C2211yescharacteristic
21#22548714C2211yescharacteristic, complemented
22#22548702C2211yescharacteristic
23#22548697—21no—
24#22548696—21no—
25#22548696—21no—
26#22548694C2211yescharacteristic
27#22548692—21no—
28#22548687—21no—
29#22548686C2211yescharacteristic
30#22548686C2211yescharacteristic, complemented
31#22548684—21no—
32D4×Dic251C2211yescharacteristic
33D4×Dic251C2211yescharacteristic, complemented
34C4×C22⋊D251C2211yescharacteristic
35C2×C4⋊Dic251C2211yescharacteristic
36C4⋊C4×D251C2211yescharacteristic, complemented
37C2×C22⋊Dic251—21no—
38#22548647C2211yescharacteristic, complemented
39#22548647C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548640C2211yescharacteristic, complemented
44#22548638C2211yescharacteristic, complemented
45#22548636C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51#22548625C2211yescharacteristic, complemented
52C22×D4×C251C2211yescharacteristic
53C2×D8×C251C2211yescharacteristic
54C2×D8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59#16860809—22no—
60#16860809—41no—
61#16860809—44no—
62C2×Dic502—21no—
63C2×Dic502C2311yescharacteristic
64C2×C22⋊D251—21no—
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251D411yescharacteristic
68C22×Dic251—21no—
69C22×Dic251—21no—
70C22×Dic251—21no—
71C22×Dic251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74#16860798—41no—
75#16860793—41no—
76C2×C251⋊C8—41no—
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
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⋊Dic251—21no—
91C22⋊Dic251—21no—
92C22⋊Dic251—21no—
93C22⋊Dic251—21no—
94C22⋊Dic251D411yescharacteristic
95(C2×D251)⋊C4D411yescharacteristic
96C4×Dic251C2311yescharacteristic
97C24×C251—21no—
98C2×D4×C251—22no—
99C2×D4×C251—21no—
100C2×D4×C251—21no—
101C2×D4×C251—41no—
102C2×D4×C251C2311yescharacteristic
103C2×D4×C251C2311yescharacteristic
104C2×D4×C251C2311yescharacteristic
105C22×C4×C251C2311yescharacteristic
106D8×C251—22no—
107D8×C251—41no—
108C2×C8×C251C2311yescharacteristic
109C2×C8×C251C2311yescharacteristic
110C4⋊C4×C251C2311yescharacteristic
111C22⋊C4×C251—21no—
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—21no—
115Dic502—42no—
116C22⋊D251—22no—
117C22⋊D251—42no—
118C22⋊D251—42no—
119C22⋊D251—81no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—42no—
125C2×Dic251—42no—
126C2×Dic251—42no—
127C2×Dic251—81no—
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C2×Dic251C2×D411yescharacteristic
131C4×D251—21no—
132C4×D251—21no—
133C4×D251—42no—
134C251⋊C8—21no—
135C251⋊C8—21no—
136C23×C251—21no—
137C23×C251—21no—
138C23×C251—21no—
139C23×C251—42no—
140C23×C251—42no—
141C23×C251—81no—
142C23×C251C2×D411yescharacteristic
143D4×C251—21no—
144D4×C251—21no—
145D4×C251—21no—
146D4×C251—21no—
147D4×C251—21no—
148D4×C251—42no—
149D4×C251—42no—
150D4×C251#4112yescomplemented
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—21no—
160C22×C251—22no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251—41no—
168C22×C251—41no—
169C22×C251—84no—
170C22×C251AGL(1,ℤ/8)12yes—
171C22×C251C22×D411yescharacteristic
172C2×D251—21no—
173C2×D251—21no—
174C2×D251—21no—
175C4×C251—21no—
176C4×C251—41no—
177C4×C251C2×#4111yescharacteristic
178C4×C251H(5,𝔽2)11yescharacteristic
179Dic251—21no—
180Dic251—21no—
181Dic251—21no—
182Dic251—41no—
183Dic251—41no—
184Dic251—41no—
185Dic251—41no—
186C2×C251—22no—
187C2×C251—42no—
188C2×C251—81no—
189C2×C251—81no—
190C2×C251#54611yescharacteristic
191C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C251#57711yescharacteristic
193D251—41no—
194C251#330611yescharacteristic, complemented, Sylow
195#3306—2511nomaximal, Sylow
196#547—2511no—
197#546—2511no—
198#489—2511no—
199#483—2511no—
200#481—2511no—
201#476—2511no—
202#469—2511no—
203#465—2511no—
204#463—2511no—
205#449—2511no—
206#447D25111yescharacteristic, complemented
207#442—2511no—
208#435—2511no—
209#417—2511no—
210C2×#102—2511no—
211C2×#41—5021no—
212C2×#41—2511no—
213C22×D4C2×D25111yescharacteristic
214C2×SD16—2511no—
215C2×D8C2×D25111yescharacteristic
216C2×D8C2×D25111yescharacteristic
217C2×M16—2511no—
218C22×C8—2511no—
219C4⋊Q8—2511no—
220#124—2511no—
221#123—5021no—
222#123—5021no—
223#123—5021no—
224C22⋊Q8—5021no—
225C22⋊Q8—2511no—
226#121—5021no—
227#121—2511no—
228#121—2511no—
229#121C2×D25111yescharacteristic
230#120—5021no—
231C4×D4—2511no—
232C4×D4—2511no—
233C4×D4—2511no—
234C2×C4⋊C4—2511no—
235C2×C4⋊C4—2511no—
236C2×C22⋊C4—5021no—
237#106—2511no—
238#106—2511no—
239#106—2511no—
240C4⋊C8—2511no—
241#103—2511no—
242#103—2511no—
243#103—2511no—
244#102—2511no—
245#102—2511no—
246#102—2511no—
247#102—2511no—
248#102—2511no—
249#102C2×D25111yescharacteristic
250#102C2×D25111yescharacteristic
251C22⋊C8—2511no—
252C22⋊C8C2×D25111yescharacteristic
253#97—2511no—
254C24—21no—
255#41—10044no—
256#41—10041no—
257#41—5022no—
258C2×Q8—5021no—
259C2×Q8—2511no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—5021no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—2511no—
268C2×D4C22×D25111yescharacteristic
269C2×D4C22×D25111yescharacteristic
270C2×D4C22×D25111yescharacteristic
271C22×C4—5021no—
272C22×C4—5021no—
273C22×C4—5021no—
274C22×C4—5021no—
275C22×C4—5021no—
276C22×C4—2511no—
277C22×C4C22×D25111yescharacteristic
278SD16—10041no—
279D8—22no—
280D8—41no—
281M16—10041no—
282C2×C8—10041no—
283C2×C8—2511no—
284C2×C8—2511no—
285C2×C8C22×D25111yescharacteristic
286C2×C8C22×D25111yescharacteristic
287C4⋊C4—5021no—
288C4⋊C4—5021no—
289C4⋊C4—5021no—
290C4⋊C4—5021no—
291C4⋊C4—2511no—
292C4⋊C4—2511no—
293C4⋊C4—2511no—
294C4⋊C4—2511no—
295C4⋊C4C22×D25111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—5021no—
298C22⋊C4—5021no—
299C22⋊C4—5021no—
300C22⋊C4—5021no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4—5021no—
304C22⋊C4—2511no—
305C22⋊C4—2511no—
306C42—2511no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—2511no—
314C23D4×D25111yescharacteristic
315Q8—10042no—
316Q8—5021no—
317Q8—5021no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—42no—
325D4—20081no—
326D4—10042no—
327D4—10042no—
328D4—5022no—
329D4#1686080912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—20081no—
334C2×C4—10042no—
335C2×C4—10042no—
336C2×C4—10042no—
337C2×C4—10042no—
338C2×C4—5021no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—2511no—
345C2×C4—2511no—
346C2×C4—2511no—
347C2×C4D4×D25111yescharacteristic
348C2×C4C23×D25111yescharacteristic
349C8—21no—
350C8—21no—
351C8—5021no—
352C8—5021no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—5021no—
366C22—5021no—
367C22—5021no—
368C22#2254873912yes—
369C22C2×D4×D25111yescharacteristic
370C4—21no—
371C4—41no—
372C4—10041no—
373C4—10041no—
374C4—10041no—
375C4—10041no—
376C4—5021no—
377C4—5021no—
378C4—5021no—
379C4C2×#1686080911yescharacteristic
380C4#2254876011yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—81no—
385C2—10041no—
386C2#3063232411yescharacteristic
387C2C2×#2254873911yescharacteristic
388C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#447×C251C2
socleC22×C251C22×D4
hypercenter#447D251
hypocenterC251#3306
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #30632337C2semidirect
2. #30632331C2semidirect
3. #30632045C2semidirect
4. #30632018C2semidirect
5. #30632004C2semidirect
6. #30631975C2semidirect
7. #30631937C2semidirect
8. #30631914C2semidirect
9. #30631900C2semidirect
10. #30631829C2semidirect
11. #30631799C2semidirect
12. #30631755C2semidirect
13. #30631685C2semidirect
14. C2×#22548636C2semidirect
15. #447×C251C2semidirect
18. C2×#16860798C22semidirect
19. C2×#16860793C22semidirect
21. #22548714C22semidirect
30. #22548686C22semidirect
33. D4×Dic251C22semidirect
36. C4⋊C4×D251C22semidirect
38. #22548647C22semidirect
39. #22548647C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. #22548640C22semidirect
43. #22548640C22semidirect
44. #22548638C22semidirect
49. #22548636C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. #22548625C22semidirect
150. D4×C251#41semidirect
194. C251#3306semidirect
206. #447D251semidirect
329. D4#16860809semidirect

Maximal subgroups

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

subgroupindex
#306323372
#306323312
#306320452
#306320182
#306320042
#306319752
#306319372
#306319142
#306319002
#306318292
#306317992
#306317552
#306316852
C2×#225486362
#447×C2512
#3306251