Finite Group Databasebeta

← #44560507

Subgroups — #44560507

Order 32128.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632224C211yesmaximal, characteristic, complemented
2C4×D4×D251C211yesmaximal, characteristic, complemented
3#30632016C211yesmaximal, characteristic, complemented
4#30632002C211yesmaximal, characteristic, complemented
5#30631990C211yesmaximal, characteristic, complemented
6#30631989C211yesmaximal, characteristic, complemented
7#30631905C211yesmaximal, characteristic, complemented
8#30631904C211yesmaximal, characteristic, complemented
9#30631836C211yesmaximal, characteristic, complemented
10#30631822C211yesmaximal, characteristic
11#30631800C211yesmaximal, characteristic, complemented
12#30631797C211yesmaximal, characteristic, complemented
13#30631785C211yesmaximal, characteristic, complemented
14#30631742C211yesmaximal, characteristic, complemented
15#440×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C22×C4×D251—21no—
18C2×#16860797C2211yescharacteristic
19C2×D2008C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21#22548710—21no—
22#22548706C2211yescharacteristic, complemented
23#22548704C2211yescharacteristic, complemented
24#22548700—21no—
25#22548691C2211yescharacteristic
26#22548690C2211yescharacteristic
27C4×D1004—21no—
28C4×D1004C2211yescharacteristic
29C4×D1004C2211yescharacteristic, complemented
30D4×Dic251C2211yescharacteristic
31#22548677C2211yescharacteristic
32C4×C22⋊D251—21no—
33(C22⋊D251)⋊C4—21no—
34#22548672—21no—
35C4⋊C4×D251C2211yescharacteristic
36C22⋊C4×D251—21no—
37C42×D251C2211yescharacteristic
38#22548648C2211yescharacteristic, complemented
39#22548647C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41Dic251⋊C8C2211yescharacteristic, complemented
42Dic251⋊C8C2211yescharacteristic, complemented
43#22548639C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic, complemented
46#22548636C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic, complemented
49(C2×D251)⋊C8C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51#22548626C2211yescharacteristic, complemented
52C2×SD16×C251C2211yescharacteristic
53C4×Q8×C251C2211yescharacteristic, complemented
54C4×D4×C251C2211yescharacteristic
55#107×C251C2211yescharacteristic, complemented
56#103×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
58#97×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C2×D1004—21no—
61C2×D1004C2311yescharacteristic
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64D4×D251—22no—
65D4×D251—22no—
66C22×Dic251—21no—
67C2×C4×D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—44no—
72C2×C4×D251D411yescharacteristic
73C2×C4×D251D411yescharacteristic
74C2×C4×D251C2311yescharacteristic
75#16860797—22no—
76D2008—22no—
77#16860795—22no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C4⋊Dic251—21no—
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86C22⋊Dic251—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
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—
94C4×Dic251D411yescharacteristic
95C4×Dic251D411yescharacteristic
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251C2311yescharacteristic
98C2×D4×C251C2311yescharacteristic
99C22×C4×C251—21no—
100SD16×C251—22no—
101C2×C8×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C4⋊C4×C251—21no—
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C42×C251—21no—
108C42×C251C2311yescharacteristic
109C22×D251—24no—
110C22×D251—21no—
111C22×D251—21no—
112C22×D251—42no—
113C22×D251—42no—
114C22×D251C2×D411yescharacteristic
115D1004—21no—
116D1004—21no—
117D1004—21no—
118D1004—42no—
119D1004#4112yes—
120C22⋊D251—24no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—42no—
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C4×D251—22no—
128C4×D251—21no—
129C4×D251—21no—
130C4×D251—41no—
131C4×D251—42no—
132C4×D251—42no—
133C4×D251—42no—
134C4×D251—42no—
135C251⋊C8—21no—
136C251⋊C8—21no—
137C23×C251—21no—
138Q8×C251—21no—
139Q8×C251—21no—
140D4×C251—21no—
141D4×C251#4112yes—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—42no—
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2411yescharacteristic
149C8×C251—21no—
150C8×C251—21no—
151C22×C251—22no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251C22×D411yescharacteristic
155C2×D251—21no—
156C2×D251—21no—
157C2×D251—22no—
158C2×D251—48no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251—41no—
164C2×D251AGL(1,ℤ/8)12yes—
165C4×C251—21no—
166C4×C251—21no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251C2×#4111yescharacteristic
171C4×C251C2×#4111yescharacteristic
172Dic251—21no—
173Dic251—21no—
174Dic251—21no—
175Dic251—41no—
176Dic251—41no—
177C2×C251—42no—
178C2×C251#52511yescharacteristic
179C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C251#57511yescharacteristic
181D251—22no—
182D251—42no—
183D251—81no—
184C251#311411yescharacteristic, complemented, Sylow
185#3114—2511nomaximal, Sylow
186#529—2511no—
187C2×C4×D4—2511no—
188#483—2511no—
189#480—2511no—
190#478—2511no—
191#478—2511no—
192#463—2511no—
193#463—2511no—
194#450—2511no—
195#447—2511no—
196#442—2511no—
197#442—2511no—
198#440—2511no—
199#440D25111yescharacteristic, complemented
200#432—2511no—
201C22×D4—2511no—
202C23×C4—5021no—
203C2×SD16—2511no—
204C2×SD16C2×D25111yescharacteristic
205C2×D8—2511no—
206C2×D8—2511no—
207#126—5021no—
208#125—2511no—
209#124—2511no—
210#123—5021no—
211C22⋊Q8—2511no—
212#121—2511no—
213C4×Q8C2×D25111yescharacteristic, complemented
214C4×D4—5021no—
215C4×D4—5021no—
216C4×D4—5021no—
217C4×D4—2511no—
218C4×D4—2511no—
219C4×D4—2511no—
220C4×D4—2511no—
221C4×D4C2×D25111yescharacteristic
222#117—5021no—
223C2×C4⋊C4—2511no—
224C2×C22⋊C4—5021no—
225C2×C42—2511no—
226#107—2511no—
227#107C2×D25111yescharacteristic, complemented
228#106—2511no—
229#106—2511no—
230C4⋊C8—2511no—
231C4⋊C8—2511no—
232#103—2511no—
233#103C2×D25111yescharacteristic, complemented
234#102—2511no—
235#102—2511no—
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102C2×D25111yescharacteristic
240C22⋊C8—2511no—
241C22⋊C8—2511no—
242#97—2511no—
243#97C2×D25111yescharacteristic, complemented
244C24—5021no—
245C2×Q8C22×D25111yescharacteristic
246C2×D4—5021no—
247C2×D4—5022no—
248C2×D4—5022no—
249C2×D4—5021no—
250C2×D4—2511no—
251C2×D4—2511no—
252C2×D4C22×D25111yescharacteristic
253C22×C4—21no—
254C22×C4—10044no—
255C22×C4—5021no—
256C22×C4—5021no—
257C22×C4—5021no—
258C22×C4—5021no—
259C22×C4—5021no—
260C22×C4—2511no—
261C22×C4—2511no—
262C22×C4—2511no—
263SD16—22no—
264SD16—5022no—
265D8—5022no—
266D8—5022no—
267C2×C8—2511no—
268C2×C8—2511no—
269C2×C8C22×D25111yescharacteristic
270C2×C8C22×D25111yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—5021no—
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—2511no—
277C4⋊C4—2511no—
278C4⋊C4C22×D25111yescharacteristic
279C4⋊C4C22×D25111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C42—21no—
290C42—2511no—
291C42—2511no—
292C42—2511no—
293C42C22×D25111yescharacteristic
294C23—21no—
295C23—10042no—
296C23—10042no—
297C23—5024no—
298C23—5021no—
299C23—5021no—
300C23—2511no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—10042no—
305D4—5024no—
306D4—5021no—
307D4—5021no—
308D4—5021no—
309D4—2512no—
310D4#1686081012yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—10042no—
316C2×C4—10042no—
317C2×C4—10041no—
318C2×C4—10042no—
319C2×C4—10042no—
320C2×C4—10042no—
321C2×C4—5022no—
322C2×C4—5021no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—2511no—
327C2×C4—2511no—
328C2×C4—2511no—
329C2×C4D4×D25111yescharacteristic
330C2×C4D4×D25111yescharacteristic
331C2×C4C23×D25111yescharacteristic
332C8—21no—
333C8—21no—
334C8—5021no—
335C8—5021no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—10048no—
340C22—10041no—
341C22—10041no—
342C22—10041no—
343C22—10041no—
344C22—10041no—
345C22—5022no—
346C22—5021no—
347C22—5021no—
348C22—2512no—
349C22C2×D4×D25111yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—10041no—
356C4—10041no—
357C4—5021no—
358C4—5021no—
359C4—5021no—
360C4#41×D25111yescharacteristic
361C4C2×#1686081011yescharacteristic
362C2—42no—
363C2—20081no—
364C2—10042no—
365C2—5022no—
366C2#3063219611yescharacteristic
367C2AGL(1,ℤ/8)×D25111yescharacteristic
368C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#440×C251C2
socleC22×C251C22×D4
prime radical#30631822C2
hypercenter#440D251
hypocenterC251#3114
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #30632224C2semidirect
2. C4×D4×D251C2semidirect
3. #30632016C2semidirect
4. #30632002C2semidirect
5. #30631990C2semidirect
6. #30631989C2semidirect
7. #30631905C2semidirect
8. #30631904C2semidirect
9. #30631836C2semidirect
11. #30631800C2semidirect
12. #30631797C2semidirect
13. #30631785C2semidirect
14. #30631742C2semidirect
15. #440×C251C2semidirect
22. #22548706C22semidirect
23. #22548704C22semidirect
29. C4×D1004C22semidirect
38. #22548648C22semidirect
39. #22548647C22semidirect
40. #22548646C22semidirect
41. Dic251⋊C8C22semidirect
42. Dic251⋊C8C22semidirect
43. #22548639C22semidirect
45. #22548637C22semidirect
48. #22548635C22semidirect
51. #22548626C22semidirect
53. C4×Q8×C251C22semidirect
55. #107×C251C22semidirect
56. #103×C251C22semidirect
58. #97×C251C22semidirect
184. C251#3114semidirect
199. #440D251semidirect
213. C4×Q8C2×D251semidirect
227. #107C2×D251semidirect
233. #103C2×D251semidirect
243. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322242
C4×D4×D2512
#306320162
#306320022
#306319902
#306319892
#306319052
#306319042
#306318362
#306318222
#306318002
#306317972
#306317852
#306317422
#440×C2512
#3114251