Finite Group Databasebeta

← #44560159

Subgroups — #44560159

Order 32128.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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
1#30632240C211yesmaximal, characteristic, complemented
2C2×D4×Dic251C211yesmaximal, characteristic
3#30631989C212yesmaximal, complemented
4#30631985C212yesmaximal, complemented
5#30631975C212yesmaximal, complemented
6#30631965C212yesmaximal, complemented
7#30631730C211yesmaximal, characteristic, complemented
8#30631729C211yesmaximal, characteristic, complemented
9#30631710C211yesmaximal, characteristic, complemented
10#30631692C211yesmaximal, characteristic
11#418×C251C211yesmaximal, characteristic
12C23×Dic251—21no—
13C2×#16860794C2211yescharacteristic
14#22548714C2211yescharacteristic, complemented
15#22548714C2211yescharacteristic, complemented
16#22548706C2211yescharacteristic, complemented
17#22548706C2211yescharacteristic, complemented
18#22548692—21no—
19#22548692—21no—
20C4×Dic502—21no—
21C4×Dic502—21no—
22D4×Dic251—21no—
23D4×Dic251—21no—
24D4×Dic251C2212yes—
25D4×Dic251C2212yes—
26#22548674C2211yescharacteristic
27C2×C4⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251—21no—
29C2×C4×Dic251C2211yescharacteristic
30#22548649C2212yescomplemented
31#22548648C2212yescomplemented
32#22548647C2212yescomplemented
33#22548646C2212yescomplemented
34Dic251⋊C8C2212yescomplemented
35Dic251⋊C8C2212yescomplemented
36#22548636C2212yes—
37#22548636C2212yes—
38C22×D4×C251C2211yescharacteristic
39C2×M16×C251C2211yescharacteristic
40#117×C251C2211yescharacteristic
41#102×C251C2212yes—
42#102×C251C2212yes—
43C2×Dic502—21no—
44C2×Dic502—21no—
45C22×Dic251—22no—
46C22×Dic251—22no—
47C22×Dic251—21no—
48C22×Dic251—44no—
49C22×Dic251D411yescharacteristic
50C22×Dic251D411yescharacteristic
51#16860794—22no—
52C2×C251⋊C8C2312yes—
53C4⋊Dic251—21no—
54C4⋊Dic251—21no—
55C4⋊Dic251—21no—
56C4⋊Dic251C2311yescharacteristic
57C4⋊Dic251C2311yescharacteristic
58C4⋊Dic251C2312yes—
59Dic251⋊C4—21no—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62Dic251⋊C4—21no—
63C22⋊Dic251—22no—
64C22⋊Dic251—22no—
65C22⋊Dic251—21no—
66C4×Dic251—21no—
67C4×Dic251D411yescharacteristic
68C4×Dic251D411yescharacteristic
69C4×Dic251C2311yescharacteristic
70C4×Dic251C2311yescharacteristic
71C24×C251—21no—
72C2×D4×C251—21no—
73C2×D4×C251—22no—
74C2×D4×C251Q814yescomplemented
75C2×D4×C251C2312yes—
76C22×C4×C251C2311yescharacteristic
77M16×C251—22no—
78C2×C8×C251C2312yes—
79C4⋊C4×C251C2311yescharacteristic
80C4⋊C4×C251C2311yescharacteristic
81C22⋊C4×C251—21no—
82C42×C251—21no—
83Dic502—42no—
84Dic502—42no—
85C2×Dic251—22no—
86C2×Dic251—21no—
87C2×Dic251—21no—
88C2×Dic251—22no—
89C2×Dic251—41no—
90C2×Dic251—44no—
91C2×Dic251—42no—
92C2×Dic251—44no—
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C251⋊C8—22no—
98C23×C251—24no—
99C23×C251—22no—
100C23×C251—42no—
101C23×C251—42no—
102C23×C251C2×D411yescharacteristic
103D4×C251—24no—
104D4×C251—22no—
105D4×C251C2×Q814yes—
106C2×C4×C251—21no—
107C2×C4×C251—21no—
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2×Q812yes—
110C2×C4×C251#4112yes—
111C2×C4×C251C2411yescharacteristic
112C8×C251—22no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251—24no—
116C22×C251—48no—
117C22×C251—42no—
118C22×C251—42no—
119C22×C251C22⋊Q812yes—
120C22×C251C22×D411yescharacteristic
121C4×C251—41no—
122C4×C251—41no—
123C4×C251C22⋊Q812yes—
124C4×C251C22×Q811yescharacteristic
125C4×C251C2×#4111yescharacteristic
126Dic251—21no—
127Dic251—21no—
128Dic251—21no—
129Dic251—42no—
130Dic251—41no—
131Dic251—41no—
132Dic251AGL(1,ℤ/8)12yes—
133C2×C251—22no—
134C2×C251—44no—
135C2×C251C2×C22⋊Q811yescharacteristic
136C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C251#57511yescharacteristic
138C251#307711yescharacteristic, complemented, Sylow
139#3077—2511nomaximal, Sylow
140#533—2511no—
141C2×C4×D4—2511no—
142#478—2512no—
143#478—2512no—
144#476—2512no—
145#474—2512no—
146#428—2511no—
147#428—2511no—
148#423—2511no—
149#418—2511no—
150#418D25111yescharacteristic
151C22×D4C2×D25111yescharacteristic
152C23×C4—5021no—
153C2×M16—2511no—
154C2×M16C2×D25111yescharacteristic
155C4⋊Q8—2511no—
156C4⋊Q8—2511no—
157#125—2511no—
158#125—2511no—
159C22⋊Q8—5021no—
160C22⋊Q8—5021no—
161C4×Q8—5021no—
162C4×Q8—5021no—
163C4×D4—5021no—
164C4×D4—5021no—
165C4×D4—2512no—
166C4×D4—2512no—
167#117—2511no—
168#117C2×D25111yescharacteristic
169C2×C4⋊C4—2511no—
170C2×C22⋊C4—5021no—
171C2×C42—2511no—
172#107—2512no—
173#107—2512no—
174#106—2512no—
175#106—2512no—
176C4⋊C8—2512no—
177C4⋊C8—2512no—
178#102—2512no—
179#102—2512no—
180#102C2×D25112yes—
181#102C2×D25112yes—
182C24—21no—
183C2×Q8—5021no—
184C2×Q8—5021no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic50214yescomplemented
188C2×D4C22×D25112yes—
189C22×C4—10044no—
190C22×C4—5021no—
191C22×C4—5022no—
192C22×C4—5022no—
193C22×C4—2511no—
194C22×C4—2511no—
195C22×C4C22×D25111yescharacteristic
196M16—22no—
197M16—5022no—
198C2×C8—2512no—
199C2×C8C22×D25112yes—
200C4⋊C4—5021no—
201C4⋊C4—5021no—
202C4⋊C4—5021no—
203C4⋊C4—5021no—
204C4⋊C4—5021no—
205C4⋊C4—5021no—
206C4⋊C4—5021no—
207C4⋊C4—2511no—
208C4⋊C4—2511no—
209C4⋊C4—2512no—
210C4⋊C4C22×D25111yescharacteristic
211C4⋊C4C22×D25111yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—5021no—
214C22⋊C4—5022no—
215C22⋊C4—5022no—
216C42—21no—
217C42—5021no—
218C42—2511no—
219C42—2511no—
220C42—2511no—
221C42—2511no—
222C23—22no—
223C23—24no—
224C23—42no—
225C23—42no—
226C23D4×D25111yescharacteristic
227Q8—10042no—
228Q8—10042no—
229D4—22no—
230D4—24no—
231D4C2×Dic50214yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—10042no—
235C2×C4—10044no—
236C2×C4—10044no—
237C2×C4—10041no—
238C2×C4—5021no—
239C2×C4—5021no—
240C2×C4—5022no—
241C2×C4—5022no—
242C2×C4—2511no—
243C2×C4—2511no—
244C2×C4—2511no—
245C2×C4—2511no—
246C2×C4D4×D25111yescharacteristic
247C2×C4C2×Dic50212yes—
248C2×C4#1686080912yes—
249C2×C4C23×D25111yescharacteristic
250C8—22no—
251C8—5022no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#2254869212yes—
259C22C2×D4×D25111yescharacteristic
260C4—41no—
261C4—41no—
262C4—10042no—
263C4—10041no—
264C4—10041no—
265C4—5021no—
266C4—5021no—
267C4—5021no—
268C4—2512no—
269C4#2254869212yes—
270C4C22×Dic50211yescharacteristic
271C4C2×#1686080911yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#2254869211yescharacteristic
275C2AGL(1,ℤ/8)×D25111yescharacteristic
276C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#418×C251C2
socleC22×C251C22×D4
prime radicalC22×D4×C251C22
hypercenter#418D251
hypocenterC251#3077
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #30632240C2semidirect
3. #30631989C2semidirect
4. #30631985C2semidirect
5. #30631975C2semidirect
6. #30631965C2semidirect
7. #30631730C2semidirect
8. #30631729C2semidirect
9. #30631710C2semidirect
14. #22548714C22semidirect
15. #22548714C22semidirect
16. #22548706C22semidirect
17. #22548706C22semidirect
30. #22548649C22semidirect
31. #22548648C22semidirect
32. #22548647C22semidirect
33. #22548646C22semidirect
34. Dic251⋊C8C22semidirect
35. Dic251⋊C8C22semidirect
74. C2×D4×C251Q8semidirect
138. C251#3077semidirect
187. C2×D4Dic502semidirect

Maximal subgroups

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

subgroupindex
#306322402
C2×D4×Dic2512
#306319892
#306319852
#306319752
#306319652
#306317302
#306317292
#306317102
#306316922
#418×C2512
#3077251