Finite Group Databasebeta

← #486×D251 #44560668

Subgroups — #486×D251 #44560668

Order 32128.

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

normal subgroups only

This group has 528 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
1C4⋊Q8×D251C211yesmaximal, characteristic, complemented
2#127×D251C211yesmaximal, characteristic
3#30632035C211yesmaximal, characteristic, complemented
4#30632034C211yesmaximal, characteristic, complemented
5#30632033C211yesmaximal, characteristic, complemented
6#30632032C214yesmaximal, complemented
7#102×D251C214yesmaximal, complemented
8C4×C8×D251C211yesmaximal, characteristic, complemented
9#486×C251C211yesmaximal, characteristic, complemented
10C2×Q8×D251—21no—
11C2×D4×D251—21no—
12C2×D4×D251—21no—
13C2×D4×D251C2212yes—
14C2×C8×D251C2212yes—
15#22548715C2211yescharacteristic, complemented
16#22548714—21no—
17#22548714C2212yescomplemented
18#22548713—21no—
19#22548712C2211yescharacteristic
20#22548711—21no—
21#22548711—21no—
22#22548711C2212yes—
23C4⋊C4×D251—21no—
24C4⋊C4×D251C2212yes—
25C42×D251C2211yescharacteristic
26#22548637C2214yescomplemented
27#22548636C2214yescomplemented
28#22548635C2214yescomplemented
29C4×C251⋊C8C2211yescharacteristic, complemented
30C8×Dic251C2212yescomplemented
31C4⋊Q8×C251C2211yescharacteristic, complemented
32#127×C251C2211yescharacteristic
33#102×C251C2214yescomplemented
34C4×C8×C251C2211yescharacteristic, complemented
35C23×D251—22no—
36C2×Dic502—22no—
37C2×Dic502—21no—
38Q8×D251—44no—
39C2×D1004—21no—
40C2×D1004—21no—
41C2×D1004C2312yes—
42C2×C22⋊D251—22no—
43C2×C22⋊D251—22no—
44C2×C22⋊D251—22no—
45D4×D251—28no—
46D4×D251—44no—
47D4×D251—44no—
48C2×C4×D251—22no—
49C2×C4×D251D411yescharacteristic
50C2×C4×D251D411yescharacteristic
51C2×C4×D251C2311yescharacteristic
52C2×C251⋊C8C2312yes—
53C8×D251—24no—
54C4⋊Dic251—21no—
55C4⋊Dic251C2312yes—
56Dic251⋊C4—22no—
57Dic251⋊C4—22no—
58C4×Dic251D411yescharacteristic
59C4×Dic251D411yescharacteristic
60C4×Dic251C2311yescharacteristic
61C2×Q8×C251—21no—
62C2×D4×C251—21no—
63C2×D4×C251—21no—
64C2×D4×C251C2312yes—
65C2×C8×C251C2312yes—
66C4⋊C4×C251—21no—
67C4⋊C4×C251C2312yes—
68C42×C251C2311yescharacteristic
69C22×D251—22no—
70C22×D251—44no—
71C22×D251—44no—
72C22×D251—44no—
73C22×D251C2×D411yescharacteristic
74Dic502—42no—
75Dic502—44no—
76D1004—24no—
77D1004—42no—
78D1004—42no—
79C22⋊D251—44no—
80C22⋊D251—44no—
81C22⋊D251—44no—
82C2×Dic251—22no—
83C2×Dic251C2×D411yescharacteristic
84C2×Dic251C2×D411yescharacteristic
85C2×Dic251C2×D411yescharacteristic
86C4×D251—44no—
87C4×D251D814yes—
88C4×D251SD1614yes—
89C4×D251#4114yes—
90C251⋊C8—22no—
91C23×C251—22no—
92Q8×C251—42no—
93D4×C251—24no—
94D4×C251—42no—
95D4×C251—42no—
96C2×C4×C251—22no—
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251C2×D411yescharacteristic
99C2×C4×C251C2411yescharacteristic
100C8×C251—22no—
101C22×C251—42no—
102C22×C251—42no—
103C22×C251—42no—
104C22×C251C22×D411yescharacteristic
105C2×D251—42no—
106C2×D251—42no—
107C2×D251—42no—
108C2×D251—88no—
109C2×D251#12412yes—
110C2×D251C2×D812yes—
111C2×D251C2×SD1612yes—
112C4×C251—42no—
113C4×C251C2×D812yes—
114C4×C251C2×SD1612yes—
115C4×C251C2×#4112yes—
116Dic251—42no—
117Dic251#12412yes—
118Dic251C2×D812yes—
119Dic251C2×SD1612yes—
120C2×C251—82no—
121C2×C251C2×#12411yescharacteristic
122C2×C251C22×D811yescharacteristic
123C2×C251C22×SD1611yescharacteristic
124D251—82no—
125D251#48614yesdirect product factor
126C251C2×#48611yescharacteristic, complemented, Sylow
127C2×#486—2511nomaximal, Sylow
128C2×C4⋊Q8—2511no—
129C2×#127—2511no—
130#486—2511no—
131#486—2511no—
132#486—2514no—
133#486—2511no—
134#486D25111yescharacteristic, direct product factor
135C2×#102—2514no—
136C2×C4×C8—2511no—
137C22×Q8—5021no—
138C22×D4—5021no—
139C22×D4—5021no—
140C22×D4—2512no—
141C22×C8—2512no—
142C4⋊Q8—5021no—
143C4⋊Q8—5021no—
144C4⋊Q8—2512no—
145C4⋊Q8—2511no—
146C4⋊Q8C2×D25111yescharacteristic, complemented
147#127—5021no—
148#127—5021no—
149#127—2512no—
150#127—2511no—
151#127C2×D25111yescharacteristic
152C2×C4⋊C4—5021no—
153C2×C4⋊C4—2512no—
154C2×C42—2511no—
155#102—2514no—
156#102—2514no—
157#102—2514no—
158#102C2×D25114yescomplemented
159C4×C8—2512no—
160C4×C8—2511no—
161C4×C8C2×D25111yescharacteristic, complemented
162C24—5022no—
163C2×Q8—21no—
164C2×Q8—10044no—
165C2×Q8—5022no—
166C2×Q8—5021no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—10044no—
170C2×D4—10044no—
171C2×D4—5028no—
172C2×D4—5022no—
173C2×D4—5022no—
174C2×D4—5022no—
175C2×D4—5021no—
176C2×D4—5021no—
177C2×D4—2512no—
178C2×D4C22×D25112yes—
179C22×C4—5022no—
180C22×C4—2511no—
181C22×C4—2511no—
182C22×C4—2511no—
183C2×C8—5024no—
184C2×C8—2512no—
185C2×C8C22×D25112yes—
186C4⋊C4—21no—
187C4⋊C4—5022no—
188C4⋊C4—5022no—
189C4⋊C4—5021no—
190C4⋊C4—2512no—
191C4⋊C4C22×D25112yes—
192C42—2511no—
193C42—2511no—
194C42—2511no—
195C42C22×D25111yescharacteristic
196C23—22no—
197C23—10044no—
198C23—10044no—
199C23—10044no—
200C23—5022no—
201C23—2511no—
202Q8—42no—
203Q8—10044no—
204Q8—10042no—
205D4—24no—
206D4—42no—
207D4—42no—
208D4—10044no—
209D4—10044no—
210D4—10044no—
211D4—10042no—
212D4—10042no—
213D4—5024no—
214C2×C4—22no—
215C2×C4—10044no—
216C2×C4—5022no—
217C2×C4—2511no—
218C2×C4—2511no—
219C2×C4—2511no—
220C2×C4—2514no—
221C2×C4—2514no—
222C2×C4—2514no—
223C2×C4D4×D25111yescharacteristic
224C2×C4D4×D25111yescharacteristic
225C2×C4C23×D25111yescharacteristic
226C8—22no—
227C8—5022no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—20088no—
232C22—10042no—
233C22—10042no—
234C22—10042no—
235C22—2512no—
236C22—2512no—
237C22—2512no—
238C22C2×D4×D25111yescharacteristic
239C4—42no—
240C4—10042no—
241C4—2512no—
242C4—2512no—
243C4—2512no—
244C4D8×D25112yes—
245C4SD16×D25112yes—
246C4#41×D25112yes—
247C2—82no—
248C2—20082no—
249C2—2514no—
250C2#124×D25111yescharacteristic
251C2C2×D8×D25111yescharacteristic
252C2C2×SD16×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#486×C251C2
socleC22×C251C22×D4
prime radical#127×D251C2
hypercenter#486D251
hypocenterC251C2×#486
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D251C2semidirect
3. #30632035C2semidirect
4. #30632034C2semidirect
5. #30632033C2semidirect
6. #30632032C2semidirect
7. #102×D251C2semidirect
8. C4×C8×D251C2semidirect
9. #486×C251C2semidirect
15. #22548715C22semidirect
17. #22548714C22semidirect
26. #22548637C22semidirect
27. #22548636C22semidirect
28. #22548635C22semidirect
29. C4×C251⋊C8C22semidirect
30. C8×Dic251C22semidirect
31. C4⋊Q8×C251C22semidirect
33. #102×C251C22semidirect
34. C4×C8×C251C22semidirect
125. D251#486direct (N × H)
126. C251C2×#486semidirect
134. #486D251direct (N × H)
146. C4⋊Q8C2×D251semidirect
158. #102C2×D251semidirect
161. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2512
#127×D2512
#306320352
#306320342
#306320332
#306320322
#102×D2512
C4×C8×D2512
#486×C2512
C2×#486251