Finite Group Databasebeta

← #44560508

Subgroups — #44560508

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#30632222C211yesmaximal, characteristic, complemented
2C4×D4×D251C211yesmaximal, characteristic, complemented
3#30632019C211yesmaximal, characteristic, complemented
4#30632002C211yesmaximal, characteristic, complemented
5#30631984C211yesmaximal, characteristic, complemented
6#30631903C211yesmaximal, characteristic, complemented
7#30631902C211yesmaximal, characteristic, complemented
8#30631839C211yesmaximal, characteristic, complemented
9#30631822C211yesmaximal, characteristic
10#30631798C211yesmaximal, characteristic, complemented
11#30631796C211yesmaximal, characteristic, complemented
12#30631786C211yesmaximal, characteristic, complemented
13#30631784C211yesmaximal, characteristic, complemented
14#30631743C211yesmaximal, characteristic, complemented
15#478×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C22×C4×D251—21no—
18C2×#16860799C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×D2008C2211yescharacteristic
21C2×#16860795C2211yescharacteristic
22#22548710—21no—
23#22548703C2211yescharacteristic, complemented
24#22548700—21no—
25#22548695C2211yescharacteristic
26#22548690C2211yescharacteristic
27#22548680C2211yescharacteristic, complemented
28C4×D1004C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30#22548677—21no—
31#22548677C2211yescharacteristic
32#22548677C2211yescharacteristic, complemented
33C4×C22⋊D251—21no—
34(C22⋊D251)⋊C4—21no—
35#22548673—21no—
36C4⋊C4×D251C2211yescharacteristic
37C22⋊C4×D251—21no—
38C42×D251C2211yescharacteristic
39#22548648C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41C4⋊(C251⋊C8)C2211yescharacteristic, complemented
42#22548640C2211yescharacteristic, complemented
43#22548638C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic, complemented
45#22548635C2211yescharacteristic
46#22548635C2211yescharacteristic
47#22548635C2211yescharacteristic, complemented
48(C2×D251)⋊C8C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50#22548625C2211yescharacteristic, complemented
51#22548625C2211yescharacteristic, complemented
52#125×C251C2211yescharacteristic, complemented
53C4×D4×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic, complemented
55#106×C251C2211yescharacteristic, complemented
56C4⋊C8×C251C2211yescharacteristic, complemented
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C23×D251—21no—
60C2×Dic502C2311yescharacteristic
61C2×D1004—21no—
62C2×D1004C2311yescharacteristic
63C2×D1004C2311yescharacteristic
64C2×C22⋊D251—21no—
65D4×D251—22no—
66D4×D251—22no—
67C22×Dic251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—44no—
73C2×C4×D251D411yescharacteristic
74C2×C4×D251D411yescharacteristic
75C2×C4×D251C2311yescharacteristic
76#16860799—22no—
77#16860798—22no—
78D2008—22no—
79#16860795—22no—
80C2×C251⋊C8C2311yescharacteristic
81C2×C251⋊C8C2311yescharacteristic
82C4⋊Dic251—21no—
83C4⋊Dic251C2311yescharacteristic
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×Dic251—21no—
95C4×Dic251D411yescharacteristic
96C4×Dic251D411yescharacteristic
97C4×Dic251C2311yescharacteristic
98C2×D4×C251C2311yescharacteristic
99C22×C4×C251—21no—
100C2×C8×C251C2311yescharacteristic
101C2×C8×C251C2311yescharacteristic
102C4⋊C4×C251—21no—
103C4⋊C4×C251—21no—
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108C42×C251C2311yescharacteristic
109C22×D251—24no—
110C22×D251—21no—
111C22×D251—21no—
112C22×D251—42no—
113C22×D251—42no—
114C22×D251C2×D411yescharacteristic
115Dic502—21no—
116Dic502—21no—
117D1004—21no—
118D1004—21no—
119D1004—21no—
120D1004—42no—
121D1004#4112yes—
122C22⋊D251—24no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—42no—
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—22no—
130C4×D251—21no—
131C4×D251—21no—
132C4×D251—41no—
133C4×D251—42no—
134C4×D251—42no—
135C4×D251—42no—
136C4×D251—42no—
137C251⋊C8—21no—
138C251⋊C8—21no—
139C23×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#478D25111yescharacteristic, complemented
192#463—2511no—
193#463—2511no—
194#450—2511no—
195#447—2511no—
196#442—2511no—
197#442—2511no—
198#440—2511no—
199#440—2511no—
200#432—2511no—
201C22×D4—2511no—
202C23×C4—5021no—
203C2×SD16—2511no—
204C2×SD16—2511no—
205C2×D8—2511no—
206C2×D8—2511no—
207#126—5021no—
208#125C2×D25111yescharacteristic, complemented
209#124—2511no—
210#123—5021no—
211C22⋊Q8—2511no—
212#121—2511no—
213C4×Q8—2511no—
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#106C2×D25111yescharacteristic, complemented
230C4⋊C8—2511no—
231C4⋊C8C2×D25111yescharacteristic, complemented
232#103—2511no—
233#103—2511no—
234#102—2511no—
235#102—2511no—
236#102—2511no—
237#102—2511no—
238#102C2×D25111yescharacteristic
239#102C2×D25111yescharacteristic
240C22⋊C8—2511no—
241C22⋊C8—2511no—
242#97—2511no—
243#97—2511no—
244C24—5021no—
245C2×Q8—2511no—
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—5022no—
264SD16—5022no—
265D8—5022no—
266D8—5022no—
267C2×C8—2511no—
268C2×C8—2511no—
269C2×C8C22×D25111yescharacteristic
270C2×C8C22×D25111yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—5021no—
274C4⋊C4—5021no—
275C4⋊C4—5021no—
276C4⋊C4—2511no—
277C4⋊C4C22×D25111yescharacteristic
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—5021no—
290C42—2511no—
291C42—2511no—
292C42—2511no—
293C42C22×D25111yescharacteristic
294C23—21no—
295C23—10042no—
296C23—10042no—
297C23—5024no—
298C23—5021no—
299C23—5021no—
300C23—2511no—
301Q8—5021no—
302Q8—5021no—
303D4—21no—
304D4—10042no—
305D4—5024no—
306D4—5021no—
307D4—5021no—
308D4—5021no—
309D4—2512no—
310D4#1686081112yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—10042no—
316C2×C4—10041no—
317C2×C4—10042no—
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×#1686081111yescharacteristic
362C2—42no—
363C2—20081no—
364C2—10042no—
365C2—5022no—
366C2#3063220011yescharacteristic
367C2AGL(1,ℤ/8)×D25111yescharacteristic
368C2#3063248411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#478×C251C2
socleC22×C251C22×D4
prime radical#30631822C2
hypercenter#478D251
hypocenterC251#3114
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #30632222C2semidirect
2. C4×D4×D251C2semidirect
3. #30632019C2semidirect
4. #30632002C2semidirect
5. #30631984C2semidirect
6. #30631903C2semidirect
7. #30631902C2semidirect
8. #30631839C2semidirect
10. #30631798C2semidirect
11. #30631796C2semidirect
12. #30631786C2semidirect
13. #30631784C2semidirect
14. #30631743C2semidirect
15. #478×C251C2semidirect
23. #22548703C22semidirect
27. #22548680C22semidirect
32. #22548677C22semidirect
39. #22548648C22semidirect
40. #22548646C22semidirect
41. C4⋊(C251⋊C8)C22semidirect
42. #22548640C22semidirect
43. #22548638C22semidirect
44. #22548637C22semidirect
47. #22548635C22semidirect
50. #22548625C22semidirect
51. #22548625C22semidirect
52. #125×C251C22semidirect
54. #107×C251C22semidirect
55. #106×C251C22semidirect
56. C4⋊C8×C251C22semidirect
184. C251#3114semidirect
191. #478D251semidirect
208. #125C2×D251semidirect
227. #107C2×D251semidirect
229. #106C2×D251semidirect
231. C4⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322222
C4×D4×D2512
#306320192
#306320022
#306319842
#306319032
#306319022
#306318392
#306318222
#306317982
#306317962
#306317862
#306317842
#306317432
#478×C2512
#3114251