Finite Group Databasebeta

← #44562326

Subgroups — #44562326

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#30632341C211yesmaximal, characteristic, complemented
2#30632046C211yesmaximal, characteristic, complemented
3#30632015C211yesmaximal, characteristic, complemented
4#30632003C211yesmaximal, characteristic, complemented
5#30631977C211yesmaximal, characteristic, complemented
6#30631938C211yesmaximal, characteristic, complemented
7#30631918C211yesmaximal, characteristic, complemented
8#30631899C211yesmaximal, characteristic, complemented
9#30631832C211yesmaximal, characteristic, complemented
10#30631821C211yesmaximal, characteristic, complemented
11#30631797C211yesmaximal, characteristic, complemented
12#30631759C211yesmaximal, characteristic, complemented
13#30631688C211yesmaximal, characteristic, complemented
14C2×#22548636C211yesmaximal, characteristic, complemented
15#546×C251C211yesmaximal, characteristic, complemented
16C2×#16860809—21no—
17C2×#16860797C2211yescharacteristic, complemented
18C2×#16860795C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20C2×#16860794C2211yescharacteristic, complemented
21C22×C251⋊C8C2211yescharacteristic
22#22548715C2211yescharacteristic, complemented
23#22548699—21no—
24#22548699—21no—
25#22548695—21no—
26#22548689C2211yescharacteristic
27#22548689C2211yescharacteristic, complemented
28C4×D1004C2211yescharacteristic, complemented
29C2×C4⋊Dic251C2211yescharacteristic
30C2×C4⋊Dic251C2211yescharacteristic, complemented
31#22548646C2211yescharacteristic, complemented
32#22548646C2211yescharacteristic, complemented
33#22548646C2211yescharacteristic, complemented
34C4⋊(C251⋊C8)C2211yescharacteristic, complemented
35#22548639C2211yescharacteristic, complemented
36#22548639C2211yescharacteristic, complemented
37#22548639C2211yescharacteristic, complemented
38#22548636C2211yescharacteristic
39#22548636C2211yescharacteristic
40#22548636C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548636C2211yescharacteristic
43#22548636C2211yescharacteristic
44#22548635C2211yescharacteristic, complemented
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic, complemented
47#22548626C2211yescharacteristic, 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—
59#16860809—44no—
60C2×Dic502—21no—
61C2×D1004C2311yescharacteristic
62C2×C22⋊D251—21no—
63C2×C22⋊D251—21no—
64C22×Dic251—21no—
65C22×Dic251—21no—
66C2×C4×D251—21no—
67#16860797—41no—
68#16860795—22no—
69#16860795—41no—
70#16860794—41no—
71C2×C251⋊C8—41no—
72C2×C251⋊C8C2311yescharacteristic
73C2×C251⋊C8C2311yescharacteristic
74C2×C251⋊C8C2311yescharacteristic
75C2×C251⋊C8C2311yescharacteristic
76C4⋊Dic251—21no—
77C4⋊Dic251—21no—
78C4⋊Dic251—21no—
79C4⋊Dic251C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82(C2×D251)⋊C4—21no—
83(C2×D251)⋊C4—21no—
84(C2×D251)⋊C4—21no—
85C24×C251—21no—
86#41×C251—22no—
87#41×C251—41no—
88C2×Q8×C251C2311yescharacteristic
89C2×D4×C251—22no—
90C2×D4×C251—21no—
91C2×D4×C251—21no—
92C2×D4×C251—21no—
93C2×D4×C251—41no—
94C2×D4×C251D411yescharacteristic
95C2×D4×C251C2311yescharacteristic
96C2×D4×C251C2311yescharacteristic
97C22×C4×C251—21no—
98C22×C4×C251—21no—
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C4⋊C4×C251—21no—
102C4⋊C4×C251D411yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C22⋊C4×C251—21no—
108C22⋊C4×C251—21no—
109C22⋊C4×C251D411yescharacteristic
110C22⋊C4×C251D411yescharacteristic
111C42×C251C2311yescharacteristic
112C22×D251—21no—
113Dic502—42no—
114D1004—21no—
115D1004—21no—
116C22⋊D251—42no—
117C22⋊D251—42no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—42no—
122C2×Dic251—42no—
123C4×D251—42no—
124C251⋊C8—21no—
125C251⋊C8—21no—
126C251⋊C8—21no—
127C251⋊C8—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—42no—
131C23×C251—42no—
132C23×C251—81no—
133C23×C251C2×D411yescharacteristic
134C23×C251C2×D411yescharacteristic
135Q8×C251—21no—
136Q8×C251—21no—
137D4×C251—22no—
138D4×C251—21no—
139D4×C251—21no—
140D4×C251—21no—
141D4×C251—42no—
142D4×C251—42no—
143D4×C251—81no—
144D4×C251#4112yescomplemented
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—42no—
152C2×C4×C251—81no—
153C2×C4×C251C2×D411yescharacteristic
154C2×C4×C251C2×D411yescharacteristic
155C2×C4×C251C2×D411yescharacteristic
156C2×C4×C251C2×D411yescharacteristic
157C2×C4×C251C2411yescharacteristic
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—21no—
161C22×C251—22no—
162C22×C251—21no—
163C22×C251—21no—
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—41no—
173C2×D251—41no—
174C2×D251—41no—
175C4×C251—21no—
176C4×C251—21no—
177C4×C251—21no—
178C4×C251—21no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251C2×#4111yescharacteristic
182C4×C251H(5,𝔽2)11yescharacteristic
183Dic251—41no—
184Dic251—41no—
185Dic251—41no—
186C2×C251—22no—
187C2×C251—41no—
188C2×C251—42no—
189C2×C251—81no—
190C2×C251#54611yescharacteristic
191C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C251#57711yescharacteristic
193D251—81no—
194C251#330611yescharacteristic, complemented, Sylow
195#3306—2511nomaximal, Sylow
196#547—2511no—
197#546D25111yescharacteristic, complemented
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#447—2511no—
207#442—2511no—
208#435—2511no—
209#417—2511no—
210C2×#102—2511no—
211C2×#41—5021no—
212C2×#41C2×D25111yescharacteristic
213C22×D4C2×D25111yescharacteristic
214C2×SD16—2511no—
215C2×D8—2511no—
216C2×D8—2511no—
217C2×M16—2511no—
218C22×C8—2511no—
219C4⋊Q8—2511no—
220#124C2×D25111yescharacteristic
221#123—21no—
222#123—5021no—
223#123—5021no—
224C22⋊Q8—5021no—
225C22⋊Q8C2×D25111yescharacteristic
226#121—21no—
227#121—2511no—
228#121—2511no—
229#121C2×D25111yescharacteristic
230#120—21no—
231C4×D4—2511no—
232C4×D4C2×D25111yescharacteristic
233C4×D4C2×D25111yescharacteristic
234C2×C4⋊C4—2511no—
235C2×C4⋊C4—2511no—
236C2×C22⋊C4—21no—
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#102—2511no—
250#102—2511no—
251C22⋊C8—2511no—
252C22⋊C8—2511no—
253#97—2511no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—10044no—
258C2×Q8—5021no—
259C2×Q8C22×D25111yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—2511no—
268C2×D4C22⋊D25111yescharacteristic
269C2×D4C22×D25111yescharacteristic
270C2×D4C22×D25111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—5021no—
274C22×C4—5021no—
275C22×C4—5021no—
276C22×C4C22×D25111yescharacteristic
277C22×C4C22×D25111yescharacteristic
278SD16—10041no—
279D8—10041no—
280D8—5022no—
281M16—10041no—
282C2×C8—10041no—
283C2×C8—2511no—
284C2×C8—2511no—
285C2×C8—2511no—
286C2×C8—2511no—
287C4⋊C4—21no—
288C4⋊C4—5021no—
289C4⋊C4—5021no—
290C4⋊C4—5021no—
291C4⋊C4—2511no—
292C4⋊C4—2511no—
293C4⋊C4—2511no—
294C4⋊C4C22⋊D25111yescharacteristic
295C4⋊C4C22×D25111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—5021no—
302C22⋊C4—5021no—
303C22⋊C4—5021no—
304C22⋊C4C22⋊D25111yescharacteristic
305C22⋊C4C22⋊D25111yescharacteristic
306C42C22×D25111yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—5021no—
313C23C2×C22⋊D25111yescharacteristic
314C23C2×C22⋊D25111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—10042no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—10042no—
326D4—10042no—
327D4—5021no—
328D4—5021no—
329D4#1686080912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—10042no—
339C2×C4—10042no—
340C2×C4—10042no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4C2×C22⋊D25111yescharacteristic
345C2×C4C2×C22⋊D25111yescharacteristic
346C2×C4C2×C22⋊D25111yescharacteristic
347C2×C4C2×C22⋊D25111yescharacteristic
348C2×C4C23×D25111yescharacteristic
349C8—5021no—
350C8—5021no—
351C8—5021no—
352C8—5021no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—10041no—
366C22—10041no—
367C22—10041no—
368C22#2254874012yes—
369C22C22×C22⋊D25111yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—10041no—
377C4—10041no—
378C4—10041no—
379C4C2×#1686080911yescharacteristic
380C4#2254876011yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—20081no—
386C2#3063233111yescharacteristic
387C2C2×#2254874011yescharacteristic
388C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#546×C251C2
socleC22×C251C22×D4
hypercenter#546D251
hypocenterC251#3306
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. #30632341C2semidirect
2. #30632046C2semidirect
3. #30632015C2semidirect
4. #30632003C2semidirect
5. #30631977C2semidirect
6. #30631938C2semidirect
7. #30631918C2semidirect
8. #30631899C2semidirect
9. #30631832C2semidirect
10. #30631821C2semidirect
11. #30631797C2semidirect
12. #30631759C2semidirect
13. #30631688C2semidirect
14. C2×#22548636C2semidirect
15. #546×C251C2semidirect
17. C2×#16860797C22semidirect
20. C2×#16860794C22semidirect
22. #22548715C22semidirect
27. #22548689C22semidirect
28. C4×D1004C22semidirect
30. C2×C4⋊Dic251C22semidirect
31. #22548646C22semidirect
32. #22548646C22semidirect
33. #22548646C22semidirect
34. C4⋊(C251⋊C8)C22semidirect
35. #22548639C22semidirect
36. #22548639C22semidirect
37. #22548639C22semidirect
44. #22548635C22semidirect
46. C22⋊(C251⋊C8)C22semidirect
47. #22548626C22semidirect
144. D4×C251#41semidirect
194. C251#3306semidirect
197. #546D251semidirect
329. D4#16860809semidirect

Maximal subgroups

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

subgroupindex
#306323412
#306320462
#306320152
#306320032
#306319772
#306319382
#306319182
#306318992
#306318322
#306318212
#306317972
#306317592
#306316882
C2×#225486362
#546×C2512
#3306251