Finite Group Databasebeta

← #44562201

Subgroups — #44562201

Order 32128.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632376C211yesmaximal, characteristic, complemented
2#30632325C211yesmaximal, characteristic
3#30632043C211yesmaximal, characteristic, complemented
4#30632016C211yesmaximal, characteristic, complemented
5#30632011C211yesmaximal, characteristic, complemented
6#30631989C211yesmaximal, characteristic, complemented
7#30631927C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631912C211yesmaximal, characteristic, complemented
10#30631849C211yesmaximal, characteristic, complemented
11#30631839C211yesmaximal, characteristic, complemented
12#30631778C211yesmaximal, characteristic, complemented
13C8×C22⋊D251C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#416×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C2×#16860800C2211yescharacteristic, complemented
18C2×#16860799C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21#22548708—21no—
22#22548707—21no—
23#22548706C2211yescharacteristic, complemented
24#22548704C2211yescharacteristic
25#22548698—21no—
26#22548697—21no—
27#22548695C2211yescharacteristic
28#22548694C2211yescharacteristic, complemented
29#22548691C2211yescharacteristic
30#22548689C2211yescharacteristic
31#22548687—21no—
32#22548683—21no—
33#22548680C2211yescharacteristic, complemented
34D4×Dic251C2211yescharacteristic
35C4×C22⋊D251C2211yescharacteristic
36#22548672—21no—
37C4⋊C4×D251C2211yescharacteristic
38C2×(C2×D251)⋊C4—21no—
39#22548648C2211yescharacteristic, complemented
40#22548647C2211yescharacteristic, complemented
41#22548646C2211yescharacteristic, complemented
42Dic251⋊C8C2211yescharacteristic, complemented
43#22548640C2211yescharacteristic, complemented
44#22548639C2211yescharacteristic, complemented
45#22548637C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic, complemented
52C2×#41×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic, complemented
54#117×C251C2211yescharacteristic, complemented
55#103×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C23×D251—21no—
60C2×Dic502C2311yescharacteristic
61C2×D1004C2311yescharacteristic
62C2×C22⋊D251—21no—
63C2×C22⋊D251—21no—
64C2×C22⋊D251D411yescharacteristic
65D4×D251—22no—
66D4×D251—41no—
67C22×Dic251—21no—
68C2×C4×D251—21no—
69C2×C4×D251C2311yescharacteristic
70#16860800—41no—
71#16860799—41no—
72#16860798—22no—
73C2×C251⋊C8C2311yescharacteristic
74C2×C251⋊C8C2311yescharacteristic
75C8×D251—41no—
76C4⋊Dic251—21no—
77C4⋊Dic251—21no—
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4D411yescharacteristic
86C22⋊Dic251—21no—
87C22⋊Dic251—21no—
88C22⋊Dic251D411yescharacteristic
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4D411yescharacteristic
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97#41×C251—22no—
98#41×C251—41no—
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C22×C4×C251C2311yescharacteristic
104C2×C8×C251—41no—
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C22⋊C4×C251—21no—
110C42×C251—21no—
111C22×D251—21no—
112C22×D251—42no—
113C22×D251—42no—
114C22×D251—81no—
115C22×D251C2×D411yescharacteristic
116Dic502—21no—
117Dic502—21no—
118D1004—21no—
119D1004—21no—
120C22⋊D251—22no—
121C22⋊D251—42no—
122C22⋊D251—81no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—81no—
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C4×D251—21no—
131C4×D251—21no—
132C4×D251—42no—
133C251⋊C8—21no—
134C251⋊C8—21no—
135C23×C251—21no—
136C23×C251C2×D411yescharacteristic
137Q8×C251—21no—
138Q8×C251—21no—
139D4×C251—21no—
140D4×C251—42no—
141D4×C251#4112yes—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—42no—
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—21no—
153C22×C251—22no—
154C22×C251—21no—
155C22×C251—21no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—84no—
165C2×D251#13512yes—
166C4×C251—21no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251C2×#4111yescharacteristic
171C4×C251H(5,𝔽2)11yescharacteristic
172Dic251—21no—
173Dic251—21no—
174Dic251—21no—
175Dic251—41no—
176Dic251—41no—
177Dic251—41no—
178C2×C251—42no—
179C2×C251—41no—
180C2×C251#54611yescharacteristic
181C2×C251C2×#13511yescharacteristic
182C2×C251#57711yescharacteristic
183D251—22no—
184D251—81no—
185C251#329611yescharacteristic, complemented, Sylow
186#3296—2511nomaximal, Sylow
187#552—2511no—
188#546—2511no—
189#488—2511no—
190#483—2511no—
191#482—2511no—
192#478—2511no—
193#467—2511no—
194#465—2511no—
195#464—2511no—
196#452—2511no—
197#450—2511no—
198C4×SD16—2511no—
199C8×D4—2511no—
200#416D25111yescharacteristic, complemented
201C2×#102—2511no—
202C2×#41C2×D25111yescharacteristic
203C22×D4—2511no—
204C2×Q16—2511no—
205C2×SD16—2511no—
206C2×SD16—2511no—
207C22×C8—2511no—
208C22×C8C2×D25111yescharacteristic, complemented
209#126—5021no—
210#125—5021no—
211#125—2511no—
212#124—2511no—
213#123—5021no—
214#123—5021no—
215C22⋊Q8—2511no—
216C22⋊Q8—2511no—
217C22⋊Q8—2511no—
218#121—5021no—
219#121—2511no—
220#120—5021no—
221C4×Q8—2511no—
222C4×D4—2511no—
223C4×D4—2511no—
224#117—5021no—
225#117C2×D25111yescharacteristic, complemented
226C2×C4⋊C4—2511no—
227C2×C22⋊C4—5021no—
228#107—2511no—
229#106—2511no—
230#106—2511no—
231C4⋊C8—2511no—
232#103—2511no—
233#103—2511no—
234#103C2×D25111yescharacteristic, complemented
235#103C2×D25111yescharacteristic, complemented
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102C2×D25111yescharacteristic
241#102C2×D25111yescharacteristic
242C22⋊C8—2511no—
243C22⋊C8—2511no—
244C4×C8—2511no—
245C24—5021no—
246#41—22no—
247#41—41no—
248C2×Q8—2511no—
249C2×Q8C22×D25111yescharacteristic
250C2×D4—21no—
251C2×D4—10041no—
252C2×D4—5021no—
253C2×D4—5022no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4C22×D25111yescharacteristic
258C22×C4—21no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—2511no—
262C22×C4C22×D25111yescharacteristic
263Q16—10041no—
264SD16—10041no—
265SD16—5022no—
266C2×C8—41no—
267C2×C8—10041no—
268C2×C8—2511no—
269C2×C8—2511no—
270C2×C8C22×D25111yescharacteristic
271C2×C8C22×D25111yescharacteristic
272C4⋊C4—5021no—
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—5021no—
279C4⋊C4—2511no—
280C4⋊C4—2511no—
281C4⋊C4—2511no—
282C4⋊C4C22×D25111yescharacteristic
283C4⋊C4C22×D25111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4—2511no—
293C22⋊C4—2511no—
294C42—21no—
295C42—5021no—
296C42—2511no—
297C23—21no—
298C23—20081no—
299C23—10042no—
300C23—10042no—
301C23—5021no—
302C23—2511no—
303C23D4×D25111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5021no—
307Q8—5021no—
308D4—21no—
309D4—42no—
310D4—20081no—
311D4—10042no—
312D4—5022no—
313D4—5021no—
314D4—5021no—
315D4#1686081012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—20081no—
323C2×C4—10042no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4—2511no—
332C2×C4D4×D25111yescharacteristic
333C2×C4C23×D25111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5021no—
337C8—5021no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—20084no—
345C22—10041no—
346C22—10041no—
347C22—10041no—
348C22—5021no—
349C22—5021no—
350C22—2512no—
351C22C2×D4×D25111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—10041no—
357C4—10041no—
358C4—10041no—
359C4—5021no—
360C4—5021no—
361C4—5021no—
362C4C2×#1686081011yescharacteristic
363C4#2254876011yescharacteristic
364C2—41no—
365C2—42no—
366C2—20081no—
367C2—5022no—
368C2#3063232711yescharacteristic
369C2#135×D25111yescharacteristic
370C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
prime radical#30632325C2
hypercenter#416D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #30632376C2semidirect
3. #30632043C2semidirect
4. #30632016C2semidirect
5. #30632011C2semidirect
6. #30631989C2semidirect
7. #30631927C2semidirect
8. #30631916C2semidirect
9. #30631912C2semidirect
10. #30631849C2semidirect
11. #30631839C2semidirect
12. #30631778C2semidirect
13. C8×C22⋊D251C2semidirect
14. #102×D251C2semidirect
15. #416×C251C2semidirect
17. C2×#16860800C22semidirect
23. #22548706C22semidirect
28. #22548694C22semidirect
33. #22548680C22semidirect
39. #22548648C22semidirect
40. #22548647C22semidirect
41. #22548646C22semidirect
42. Dic251⋊C8C22semidirect
43. #22548640C22semidirect
44. #22548639C22semidirect
49. C22⋊(C251⋊C8)C22semidirect
51. C8×Dic251C22semidirect
53. C22×C8×C251C22semidirect
54. #117×C251C22semidirect
55. #103×C251C22semidirect
56. #103×C251C22semidirect
185. C251#3296semidirect
200. #416D251semidirect
208. C22×C8C2×D251semidirect
225. #117C2×D251semidirect
234. #103C2×D251semidirect
235. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323762
#306323252
#306320432
#306320162
#306320112
#306319892
#306319272
#306319162
#306319122
#306318492
#306318392
#306317782
C8×C22⋊D2512
#102×D2512
#416×C2512
#3296251