Finite Group Databasebeta

← #496×D251 #44560795

Subgroups — #496×D251 #44560795

Order 32128.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D251C212yesmaximal, complemented
2C2×D8×D251C212yesmaximal, complemented
3#127×D251C211yesmaximal, characteristic, complemented
4#124×D251C211yesmaximal, characteristic, complemented
5#30632077C211yesmaximal, characteristic, complemented
6#30632076C211yesmaximal, characteristic, complemented
7#30632075C212yesmaximal, complemented
8#30632074C211yesmaximal, characteristic, complemented
9#30632073C212yesmaximal, complemented
10#97×D251C211yesmaximal, characteristic, complemented
11#496×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251C2211yescharacteristic
13C2×D4×D251—22no—
14C2×D4×D251C2211yescharacteristic
15C2×D4×D251C2212yes—
16C2×#16860799C2212yescomplemented
17C2×#16860798C2212yescomplemented
18C2×#16860797C2212yescomplemented
19SD16×D251—28no—
20C2×D2008C2212yescomplemented
21C2×#16860795C2212yescomplemented
22C2×#16860795C2212yescomplemented
23D8×D251—28no—
24C2×C8×D251C2212yes—
25#22548712C2211yescharacteristic, complemented
26#22548711—22no—
27#22548711C2212yescomplemented
28#22548704C2211yescharacteristic, complemented
29#22548703C2211yescharacteristic, complemented
30#22548702C2211yescharacteristic, complemented
31#22548701—22no—
32C22⋊C4×D251—22no—
33C42×D251C2211yescharacteristic
34#22548626C2211yescharacteristic, complemented
35#22548625C2212yescomplemented
36C2×SD16×C251C2212yescomplemented
37C2×D8×C251C2212yescomplemented
38#127×C251C2211yescharacteristic, complemented
39#124×C251C2211yescharacteristic, complemented
40#97×C251C2211yescharacteristic, complemented
41C23×D251—22no—
42C23×D251—21no—
43C2×Dic502—21no—
44C2×Dic502C2311yescharacteristic
45Q8×D251—24no—
46C2×D1004—22no—
47C2×D1004C2311yescharacteristic
48C2×D1004C2312yes—
49C2×C22⋊D251—24no—
50C2×C22⋊D251—21no—
51C2×C22⋊D251—22no—
52D4×D251—24no—
53D4×D251—24no—
54D4×D251—24no—
55D4×D251—48no—
56C2×C4×D251—21no—
57C2×C4×D251D412yes—
58C2×C4×D251C2311yescharacteristic
59#16860799—24no—
60#16860798—24no—
61#16860797—24no—
62D2008—24no—
63#16860795—24no—
64#16860795—24no—
65C2×C251⋊C8C2312yes—
66C8×D251D418yescomplemented
67C22⋊Dic251—22no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4—22no—
70C4×Dic251D412yes—
71C4×Dic251C2311yescharacteristic
72C2×Q8×C251C2311yescharacteristic
73C2×D4×C251—22no—
74C2×D4×C251C2311yescharacteristic
75C2×D4×C251C2312yes—
76SD16×C251—24no—
77D8×C251—24no—
78C2×C8×C251C2312yes—
79C22⋊C4×C251—22no—
80C42×C251C2311yescharacteristic
81C22×D251—21no—
82C22×D251—22no—
83C22×D251—42no—
84C22×D251—44no—
85C22×D251—48no—
86C22×D251—44no—
87C22×D251C2×D411yescharacteristic
88Dic502—22no—
89Dic502—42no—
90D1004—22no—
91D1004—22no—
92D1004—22no—
93D1004—44no—
94C22⋊D251—48no—
95C22⋊D251—42no—
96C22⋊D251—44no—
97C2×Dic251—21no—
98C2×Dic251C2×D412yes—
99C2×Dic251C2×D411yescharacteristic
100C4×D251—24no—
101C4×D251—42no—
102C4×D251C2×D414yes—
103C251⋊C8C2×D414yescomplemented
104C23×C251—21no—
105C23×C251—22no—
106Q8×C251—22no—
107D4×C251—22no—
108D4×C251—22no—
109D4×C251—22no—
110D4×C251—44no—
111C2×C4×C251—21no—
112C2×C4×C251C2×D412yes—
113C2×C4×C251C2411yescharacteristic
114C8×C251C2×D414yescomplemented
115C22×C251—41no—
116C22×C251—42no—
117C22×C251—44no—
118C22×C251—42no—
119C22×C251C22×D411yescharacteristic
120C2×D251—41no—
121C2×D251—42no—
122C2×D251—44no—
123C2×D251—42no—
124C2×D251—88no—
125C2×D251—84no—
126C2×D251#12712yes—
127C2×D251AGL(1,ℤ/8)14yes—
128C4×C251—22no—
129C4×C251—41no—
130C4×C251C22×D412yes—
131Dic251—22no—
132Dic251—41no—
133Dic251#12712yes—
134C2×C251—82no—
135C2×C251—81no—
136C2×C251C2×#12711yescharacteristic
137C2×C251C2×AGL(1,ℤ/8)12yes—
138D251—82no—
139D251—81no—
140D251#49614yesdirect product factor
141C251C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—2511nomaximal, Sylow
143C22×SD16—2512no—
144C22×D8—2512no—
145C2×#127—2511no—
146C2×#124—2511no—
147#496—2512no—
148#496—2512no—
149#496—2511no—
150#496—2511no—
151#496—2511no—
152#496D25111yescharacteristic, direct product factor
153C2×#97—2511no—
154C22×Q8—2511no—
155C22×D4—5022no—
156C22×D4—2511no—
157C22×D4—2512no—
158C2×SD16—5028no—
159C2×SD16—2512no—
160C2×SD16—2512no—
161C2×SD16—2512no—
162C2×SD16C2×D25112yescomplemented
163C2×D8—5028no—
164C2×D8—2512no—
165C2×D8—2512no—
166C2×D8—2512no—
167C2×D8C2×D25112yescomplemented
168C22×C8—2512no—
169#127—5022no—
170#127—2512no—
171#127—2511no—
172#127C2×D25111yescharacteristic, complemented
173#124—5022no—
174#124—2511no—
175#124—2511no—
176#124—2511no—
177#124C2×D25111yescharacteristic, complemented
178C2×C22⋊C4—5022no—
179C2×C42—2511no—
180#97—2512no—
181#97—2511no—
182#97C2×D25111yescharacteristic, complemented
183C24—5021no—
184C24—5022no—
185C2×Q8—5021no—
186C2×Q8—5024no—
187C2×Q8—2511no—
188C2×Q8C22×D25111yescharacteristic
189C2×D4—22no—
190C2×D4—10048no—
191C2×D4—5024no—
192C2×D4—5024no—
193C2×D4—5024no—
194C2×D4—5021no—
195C2×D4—5022no—
196C2×D4—5024no—
197C2×D4—5022no—
198C2×D4—2511no—
199C2×D4—2512no—
200C2×D4C22×D25111yescharacteristic
201C2×D4C22×D25112yes—
202C22×C4—5021no—
203C22×C4—2512no—
204C22×C4—2511no—
205SD16—24no—
206SD16—5024no—
207SD16—5024no—
208SD16—5024no—
209D8—24no—
210D8—5024no—
211D8—5024no—
212D8—5024no—
213C2×C8—2512no—
214C2×C8—2518no—
215C2×C8C22×D25112yes—
216C22⋊C4—22no—
217C22⋊C4—5022no—
218C22⋊C4—5022no—
219C22⋊C4—5022no—
220C42—2512no—
221C42—2511no—
222C42C22×D25111yescharacteristic
223C23—21no—
224C23—22no—
225C23—10048no—
226C23—10042no—
227C23—10044no—
228C23—10044no—
229C23—5021no—
230C23—5022no—
231C23—2511no—
232Q8—22no—
233Q8—10042no—
234Q8—5022no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—10044no—
240D4—10048no—
241D4—10042no—
242D4—10044no—
243D4—5022no—
244D4—5022no—
245D4—5022no—
246C2×C4—21no—
247C2×C4—10042no—
248C2×C4—5024no—
249C2×C4—5021no—
250C2×C4—2512no—
251C2×C4—2511no—
252C2×C4—2514no—
253C2×C4D4×D25112yes—
254C2×C4C23×D25111yescharacteristic
255C8—2514no—
256C8D4×D25114yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—20088no—
262C22—20084no—
263C22—10044no—
264C22—10042no—
265C22—10041no—
266C22—10042no—
267C22—2514no—
268C22—2512no—
269C22C2×D4×D25111yescharacteristic
270C4—22no—
271C4—41no—
272C4—10041no—
273C4—5022no—
274C4—2512no—
275C4C2×D4×D25112yes—
276C2—82no—
277C2—81no—
278C2—20082no—
279C2—20081no—
280C2—2514no—
281C2#127×D25111yescharacteristic
282C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#496×C251C2
socleC22×C251C22×D4
hypercenter#496D251
hypocenterC251C2×#496
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D251C2semidirect
2. C2×D8×D251C2semidirect
3. #127×D251C2semidirect
4. #124×D251C2semidirect
5. #30632077C2semidirect
6. #30632076C2semidirect
7. #30632075C2semidirect
8. #30632074C2semidirect
9. #30632073C2semidirect
10. #97×D251C2semidirect
11. #496×C251C2semidirect
16. C2×#16860799C22semidirect
17. C2×#16860798C22semidirect
18. C2×#16860797C22semidirect
20. C2×D2008C22semidirect
21. C2×#16860795C22semidirect
22. C2×#16860795C22semidirect
25. #22548712C22semidirect
27. #22548711C22semidirect
28. #22548704C22semidirect
29. #22548703C22semidirect
30. #22548702C22semidirect
34. #22548626C22semidirect
35. #22548625C22semidirect
36. C2×SD16×C251C22semidirect
37. C2×D8×C251C22semidirect
38. #127×C251C22semidirect
39. #124×C251C22semidirect
40. #97×C251C22semidirect
66. C8×D251D4semidirect
103. C251⋊C8C2×D4semidirect
114. C8×C251C2×D4semidirect
140. D251#496direct (N × H)
141. C251C2×#496semidirect
152. #496D251direct (N × H)
162. C2×SD16C2×D251semidirect
167. C2×D8C2×D251semidirect
172. #127C2×D251semidirect
177. #124C2×D251semidirect
182. #97C2×D251semidirect
256. C8D4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2512
C2×D8×D2512
#127×D2512
#124×D2512
#306320772
#306320762
#306320752
#306320742
#306320732
#97×D2512
#496×C2512
C2×#496251