Finite Group Databasebeta

← #44562200

Subgroups — #44562200

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#30632377C211yesmaximal, characteristic, complemented
2#30632041C211yesmaximal, characteristic, complemented
3#30632015C211yesmaximal, characteristic, complemented
4#30632008C211yesmaximal, characteristic, complemented
5#30631987C211yesmaximal, characteristic, complemented
6#30631926C211yesmaximal, characteristic, complemented
7#30631914C211yesmaximal, characteristic, complemented
8#30631906C211yesmaximal, characteristic, complemented
9#30631849C211yesmaximal, characteristic, complemented
10#30631838C211yesmaximal, characteristic, complemented
11C4×#16860798C211yesmaximal, characteristic, complemented
12D4×C251⋊C8C211yesmaximal, characteristic, complemented
13#30631683C211yesmaximal, characteristic, complemented
14C2×#22548636C211yesmaximal, characteristic, complemented
15#546×C251C211yesmaximal, characteristic
16C2×#16860800C2211yescharacteristic, complemented
17C2×#16860798C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C22×C251⋊C8C2211yescharacteristic
20C22×C251⋊C8C2211yescharacteristic, complemented
21#22548708—21no—
22#22548707C2211yescharacteristic, complemented
23#22548705—21no—
24#22548696—21no—
25#22548694C2211yescharacteristic
26#22548694C2211yescharacteristic, complemented
27C4×Dic502C2211yescharacteristic, complemented
28#22548674C2211yescharacteristic, complemented
29#22548671—21no—
30C2×C4⋊Dic251C2211yescharacteristic
31#22548648C2211yescharacteristic, complemented
32#22548646C2211yescharacteristic, complemented
33#22548646C2211yescharacteristic, complemented
34C4⋊(C251⋊C8)C2211yescharacteristic, complemented
35#22548639C2211yescharacteristic, complemented
36#22548639C2211yescharacteristic, complemented
37#22548639C2211yescharacteristic, complemented
38#22548638C2211yescharacteristic, complemented
39#22548636C2211yescharacteristic
40#22548636C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548636C2211yescharacteristic
43#22548636C2211yescharacteristic
44#22548636C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic, complemented
47C4×C251⋊C8C2211yescharacteristic, complemented
48C2×#41×C251C2211yescharacteristic
49C22×D4×C251C2211yescharacteristic
50#124×C251C2211yescharacteristic
51#123×C251—21no—
52C22⋊Q8×C251C2211yescharacteristic
53#121×C251—21no—
54#121×C251C2211yescharacteristic
55#120×C251—21no—
56C4×D4×C251C2211yescharacteristic
57C4×D4×C251C2211yescharacteristic
58C2×C22⋊C4×C251—21no—
59C2×Dic502C2311yescharacteristic
60C22×Dic251—21no—
61#16860800—41no—
62#16860798—22no—
63#16860798—41no—
64C2×C251⋊C8—41no—
65C2×C251⋊C8—41no—
66C2×C251⋊C8C2311yescharacteristic
67C2×C251⋊C8C2311yescharacteristic
68C2×C251⋊C8C2311yescharacteristic
69C2×C251⋊C8C2311yescharacteristic
70C4⋊Dic251—21no—
71C4⋊Dic251C2311yescharacteristic
72C4⋊Dic251C2311yescharacteristic
73C4⋊Dic251C2311yescharacteristic
74Dic251⋊C4—21no—
75Dic251⋊C4—21no—
76Dic251⋊C4—21no—
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79C22⋊Dic251—21no—
80C22⋊Dic251—21no—
81C22⋊Dic251—21no—
82C4×Dic251—21no—
83C4×Dic251—21no—
84C24×C251—21no—
85#41×C251—22no—
86#41×C251—41no—
87C2×Q8×C251C2311yescharacteristic
88C2×D4×C251—22no—
89C2×D4×C251—21no—
90C2×D4×C251—21no—
91C2×D4×C251—21no—
92C2×D4×C251—41no—
93C2×D4×C251D411yescharacteristic
94C2×D4×C251C2311yescharacteristic
95C2×D4×C251C2311yescharacteristic
96C22×C4×C251—21no—
97C22×C4×C251—21no—
98C22×C4×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C4⋊C4×C251—21no—
101C4⋊C4×C251D411yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C22⋊C4×C251—21no—
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C22⋊C4×C251—21no—
108C22⋊C4×C251D411yescharacteristic
109C22⋊C4×C251D411yescharacteristic
110C42×C251C2311yescharacteristic
111Dic502—21no—
112Dic502—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—42no—
118C251⋊C8—21no—
119C251⋊C8—21no—
120C251⋊C8—21no—
121C251⋊C8—21no—
122C23×C251—21no—
123C23×C251—21no—
124C23×C251—42no—
125C23×C251—42no—
126C23×C251—81no—
127C23×C251C2×D411yescharacteristic
128C23×C251C2×D411yescharacteristic
129Q8×C251—21no—
130Q8×C251—21no—
131D4×C251—22no—
132D4×C251—21no—
133D4×C251—21no—
134D4×C251—21no—
135D4×C251—42no—
136D4×C251—42no—
137D4×C251—81no—
138D4×C251#4112yes—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—42no—
146C2×C4×C251—81no—
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2×D411yescharacteristic
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2411yescharacteristic
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—22no—
156C22×C251—21no—
157C22×C251—21no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—84no—
164C22×C251#13512yes—
165C22×C251C22×D411yescharacteristic
166C4×C251—21no—
167C4×C251—21no—
168C4×C251—21no—
169C4×C251—21no—
170C4×C251—41no—
171C4×C251—41no—
172C4×C251C2×#4111yescharacteristic
173C4×C251H(5,𝔽2)11yescharacteristic
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177Dic251—41no—
178C2×C251—22no—
179C2×C251—41no—
180C2×C251—42no—
181C2×C251—81no—
182C2×C251#54611yescharacteristic
183C2×C251C2×#13511yescharacteristic
184C2×C251#57711yescharacteristic
185C251#329611yescharacteristic, complemented, Sylow
186#3296—2511nomaximal, Sylow
187#552—2511no—
188#546D25111yescharacteristic
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#416—2511no—
201C2×#102—2511no—
202C2×#41C2×D25111yescharacteristic
203C22×D4C2×D25111yescharacteristic
204C2×Q16—2511no—
205C2×SD16—2511no—
206C2×SD16—2511no—
207C22×C8—2511no—
208C22×C8—2511no—
209#126—5021no—
210#125—5021no—
211#125—2511no—
212#124C2×D25111yescharacteristic
213#123—21no—
214#123—5021no—
215C22⋊Q8—2511no—
216C22⋊Q8—2511no—
217C22⋊Q8C2×D25111yescharacteristic
218#121—21no—
219#121C2×D25111yescharacteristic
220#120—21no—
221C4×Q8—2511no—
222C4×D4C2×D25111yescharacteristic
223C4×D4C2×D25111yescharacteristic
224#117—5021no—
225#117—2511no—
226C2×C4⋊C4—2511no—
227C2×C22⋊C4—21no—
228#107—2511no—
229#106—2511no—
230#106—2511no—
231C4⋊C8—2511no—
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103—2511no—
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241#102—2511no—
242C22⋊C8—2511no—
243C22⋊C8—2511no—
244C4×C8—2511no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—2511no—
249C2×Q8C22×D25111yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D25111yescharacteristic
256C2×D4C22×D25111yescharacteristic
257C2×D4C22×D25111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—5021no—
261C22×C4C22×D25111yescharacteristic
262C22×C4C22×D25111yescharacteristic
263Q16—10041no—
264SD16—10041no—
265SD16—5022no—
266C2×C8—10041no—
267C2×C8—10041no—
268C2×C8—2511no—
269C2×C8—2511no—
270C2×C8—2511no—
271C2×C8—2511no—
272C4⋊C4—21no—
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—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4C22⋊D25111yescharacteristic
293C22⋊C4C22⋊D25111yescharacteristic
294C42—5021no—
295C42—5021no—
296C42C22×D25111yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D25111yescharacteristic
303C23C2×C22⋊D25111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5021no—
307Q8—5021no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1686080912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—10042no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4C2×C22⋊D25111yescharacteristic
330C2×C4C2×C22⋊D25111yescharacteristic
331C2×C4C2×C22⋊D25111yescharacteristic
332C2×C4C2×C22⋊D25111yescharacteristic
333C2×C4C23×D25111yescharacteristic
334C8—5021no—
335C8—5021no—
336C8—5021no—
337C8—5021no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#2254873512yes—
351C22C22×C22⋊D25111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—10041no—
359C4—10041no—
360C4—10041no—
361C4—10041no—
362C4C2×#1686080911yescharacteristic
363C4#2254876011yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#3063233111yescharacteristic
369C2C2×#2254873511yescharacteristic
370C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting, prime radical#546×C251C2
socleC22×C251C22×D4
hypercenter#546D251
hypocenterC251#3296
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #30632377C2semidirect
2. #30632041C2semidirect
3. #30632015C2semidirect
4. #30632008C2semidirect
5. #30631987C2semidirect
6. #30631926C2semidirect
7. #30631914C2semidirect
8. #30631906C2semidirect
9. #30631849C2semidirect
10. #30631838C2semidirect
11. C4×#16860798C2semidirect
12. D4×C251⋊C8C2semidirect
13. #30631683C2semidirect
14. C2×#22548636C2semidirect
16. C2×#16860800C22semidirect
20. C22×C251⋊C8C22semidirect
22. #22548707C22semidirect
26. #22548694C22semidirect
27. C4×Dic502C22semidirect
28. #22548674C22semidirect
31. #22548648C22semidirect
32. #22548646C22semidirect
33. #22548646C22semidirect
34. C4⋊(C251⋊C8)C22semidirect
35. #22548639C22semidirect
36. #22548639C22semidirect
37. #22548639C22semidirect
38. #22548638C22semidirect
46. C22⋊(C251⋊C8)C22semidirect
47. C4×C251⋊C8C22semidirect
185. C251#3296semidirect

Maximal subgroups

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

subgroupindex
#306323772
#306320412
#306320152
#306320082
#306319872
#306319262
#306319142
#306319062
#306318492
#306318382
C4×#168607982
D4×C251⋊C82
#306316832
C2×#225486362
#546×C2512
#3296251