Finite Group Databasebeta

← #44562199

Subgroups — #44562199

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#30632375C211yesmaximal, characteristic, complemented
2#30632330C211yesmaximal, characteristic
3#30632016C211yesmaximal, characteristic, complemented
4#30632010C211yesmaximal, characteristic, complemented
5#30631984C211yesmaximal, characteristic, complemented
6#30631927C211yesmaximal, characteristic, complemented
7#30631915C211yesmaximal, characteristic, complemented
8#30631906C211yesmaximal, characteristic, complemented
9#30631852C211yesmaximal, characteristic, complemented
10#30631839C211yesmaximal, characteristic, complemented
11C4×#16860799C211yesmaximal, characteristic, complemented
12C8×D1004C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#488×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×#16860800C2211yescharacteristic, complemented
19C2×#16860799C2211yescharacteristic
20C2×#16860798C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic, complemented
23#22548710—21no—
24#22548706—21no—
25#22548700—21no—
26#22548697—21no—
27#22548695C2211yescharacteristic
28#22548695C2211yescharacteristic, complemented
29#22548691C2211yescharacteristic
30#22548688—21no—
31#22548686C2211yescharacteristic
32#22548685—21no—
33C4×Dic502C2211yescharacteristic, complemented
34C4×D1004C2211yescharacteristic
35C4×D1004C2211yescharacteristic
36#22548673—21no—
37#22548673C2211yescharacteristic, complemented
38C4⋊C4×D251C2211yescharacteristic
39C22⋊C4×D251—21no—
40#22548648C2211yescharacteristic, complemented
41#22548647C2211yescharacteristic, complemented
42#22548646C2211yescharacteristic, complemented
43C4⋊(C251⋊C8)C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548639C2211yescharacteristic, complemented
46#22548637C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548635C2211yescharacteristic
49#22548635C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52#125×C251C2211yescharacteristic, complemented
53#124×C251C2211yescharacteristic
54#103×C251C2211yescharacteristic, complemented
55#103×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C4×C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860811—22no—
61#16860811—41no—
62C2×Dic502C2311yescharacteristic
63C2×D1004—21no—
64C2×D1004D411yescharacteristic
65C2×D1004C2311yescharacteristic
66C2×C22⋊D251—21no—
67C2×C22⋊D251—21no—
68D4×D251—22no—
69D4×D251—41no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74C2×C4×D251C2311yescharacteristic
75#16860800—41no—
76#16860799—22no—
77#16860798—41no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C8×D251—41no—
81C8×D251—41no—
82C4⋊Dic251—21no—
83C4⋊Dic251—21no—
84C4⋊Dic251D411yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86C4⋊Dic251C2311yescharacteristic
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4—21no—
90C22⋊Dic251—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4D411yescharacteristic
97(C2×D251)⋊C4D411yescharacteristic
98C4×Dic251—21no—
99C4×Dic251—21no—
100C2×Q8×C251C2311yescharacteristic
101C2×D4×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C4⋊C4×C251—21no—
105C4⋊C4×C251—21no—
106C4⋊C4×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
108C22⋊C4×C251—21no—
109C22⋊C4×C251—21no—
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112C22×D251—42no—
113C22×D251—42no—
114C22×D251—81no—
115C22×D251C2×D411yescharacteristic
116C22×D251C2×D411yescharacteristic
117Dic502—21no—
118Dic502—21no—
119D1004—22no—
120D1004—21no—
121D1004—42no—
122D1004#4112yes—
123C22⋊D251—42no—
124C22⋊D251—81no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C4×D251—21no—
131C4×D251—21no—
132C4×D251—21no—
133C4×D251—21no—
134C4×D251—42no—
135C4×D251—42no—
136C4×D251—81no—
137C251⋊C8—21no—
138C251⋊C8—21no—
139C23×C251—21no—
140Q8×C251—21no—
141Q8×C251—21no—
142D4×C251—21no—
143D4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251C22×D411yescharacteristic
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—21no—
159C2×D251—22no—
160C2×D251—21no—
161C2×D251—21no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251—84no—
165C2×D251#13512yes—
166C4×C251—21no—
167C4×C251—21no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251—41no—
171C4×C251C2×#4111yescharacteristic
172C4×C251H(5,𝔽2)11yescharacteristic
173Dic251—21no—
174Dic251—21no—
175Dic251—41no—
176Dic251—41no—
177Dic251—41no—
178C2×C251—81no—
179C2×C251#54611yescharacteristic
180C2×C251C2×#13511yescharacteristic
181C2×C251#57711yescharacteristic
182D251—22no—
183D251—41no—
184D251—42no—
185C251#329611yescharacteristic, complemented, Sylow
186#3296—2511nomaximal, Sylow
187#552—2511no—
188#546—2511no—
189#488D25111yescharacteristic, complemented
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#416—2511no—
201C2×#102—2511no—
202C2×#41—2511no—
203C22×D4—2511no—
204C2×Q16—2511no—
205C2×SD16—2511no—
206C2×SD16—2511no—
207C22×C8—2511no—
208C22×C8—2511no—
209#126—5021no—
210#125—5021no—
211#125C2×D25111yescharacteristic, complemented
212#124C2×D25111yescharacteristic
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#117—2511no—
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×C8C2×D25111yescharacteristic, complemented
245C24—5021no—
246#41—10041no—
247#41—5022no—
248C2×Q8—2511no—
249C2×Q8C22×D25111yescharacteristic
250C2×D4—10041no—
251C2×D4—5021no—
252C2×D4—5022no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—2511no—
256C2×D4—2511no—
257C2×D4C22×D25111yescharacteristic
258C22×C4—5021no—
259C22×C4—5021no—
260C22×C4—5021no—
261C22×C4—2511no—
262C22×C4—2511no—
263Q16—10041no—
264SD16—10041no—
265SD16—5022no—
266C2×C8—10041no—
267C2×C8—10041no—
268C2×C8—2511no—
269C2×C8—2511no—
270C2×C8C22×D25111yescharacteristic
271C2×C8C22×D25111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
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—21no—
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—5021no—
295C42—5021no—
296C42C22×D25111yescharacteristic
297C23—21no—
298C23—20081no—
299C23—10042no—
300C23—10042no—
301C23—5021no—
302C23—2511no—
303C23—2511no—
304Q8—21no—
305Q8—21no—
306Q8—5021no—
307Q8—5021no—
308D4—21no—
309D4—21no—
310D4—20081no—
311D4—10042no—
312D4—10042no—
313D4—5022no—
314D4—5021no—
315D4—2512no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—20081no—
320C2×C4—10042no—
321C2×C4—10042no—
322C2×C4—5021no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4D4×D25111yescharacteristic
332C2×C4D4×D25111yescharacteristic
333C2×C4C23×D25111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5021no—
337C8—5021no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—20084no—
342C22—10041no—
343C22—10041no—
344C22—5021no—
345C22—5021no—
346C22—5022no—
347C22—5021no—
348C22—5021no—
349C22—5021no—
350C22—2512no—
351C22C2×D4×D25111yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—10041no—
358C4—10041no—
359C4—10041no—
360C4—5021no—
361C4—5021no—
362C4#41×D25111yescharacteristic
363C4#2254876111yescharacteristic
364C2—81no—
365C2—10042no—
366C2—10041no—
367C2—5022no—
368C2#3063233011yescharacteristic
369C2#135×D25111yescharacteristic
370C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#488×C251C2
socleC22×C251C22×D4
prime radical#30632330C2
hypercenter#488D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #30632375C2semidirect
3. #30632016C2semidirect
4. #30632010C2semidirect
5. #30631984C2semidirect
6. #30631927C2semidirect
7. #30631915C2semidirect
8. #30631906C2semidirect
9. #30631852C2semidirect
10. #30631839C2semidirect
11. C4×#16860799C2semidirect
12. C8×D1004C2semidirect
13. #30631682C2semidirect
14. #102×D251C2semidirect
15. #488×C251C2semidirect
18. C2×#16860800C22semidirect
22. C2×C8×D251C22semidirect
28. #22548695C22semidirect
33. C4×Dic502C22semidirect
37. #22548673C22semidirect
40. #22548648C22semidirect
41. #22548647C22semidirect
42. #22548646C22semidirect
43. C4⋊(C251⋊C8)C22semidirect
44. #22548640C22semidirect
45. #22548639C22semidirect
51. (C2×D251)⋊C8C22semidirect
52. #125×C251C22semidirect
54. #103×C251C22semidirect
55. #103×C251C22semidirect
58. C4×C8×C251C22semidirect
185. C251#3296semidirect
189. #488D251semidirect
211. #125C2×D251semidirect
234. #103C2×D251semidirect
235. #103C2×D251semidirect
244. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323752
#306323302
#306320162
#306320102
#306319842
#306319272
#306319152
#306319062
#306318522
#306318392
C4×#168607992
C8×D10042
#306316822
#102×D2512
#488×C2512
#3296251