Finite Group Databasebeta

← #30648865

Subgroups — #30648865

Order 16256.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550407C211yesmaximal, characteristic, complemented
2C2×#16861171C211yesmaximal, characteristic, complemented
3#22550206C211yesmaximal, characteristic, complemented
4#22550179C212yesmaximal, complemented
5#22550090C212yesmaximal, complemented
6#22550029C211yesmaximal, characteristic, complemented
7#22550009C211yesmaximal, characteristic
8#22549985C212yesmaximal, complemented
9#22549975C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#480×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C4×D127—21no—
14C2×#14523237C2212yes—
15C2×#14523233C2212yes—
16#16861202—21no—
17#16861200C2211yescharacteristic, complemented
18#16861196C2211yescharacteristic, complemented
19#16861190—21no—
20#16861188C2211yescharacteristic
21#16861174C2211yescharacteristic, complemented
22D4×Dic127—21no—
23D4×Dic127C2211yescharacteristic, complemented
24#16861171—21no—
25#16861171—21no—
26#16861171C2212yes—
27#16861171C2212yes—
28#16861165—21no—
29C2×(C2×D127)⋊C4—21no—
30C2×C4×Dic127C2211yescharacteristic
31#16861140C2212yescomplemented
32Dic127⋊C8C2212yescomplemented
33#16861132C2212yescomplemented
34#16861131C2212yes—
35#16861129C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yescomplemented
38#121×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#107×C127C2212yescomplemented
41#102×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic
43C23×D127—21no—
44C2×Dic254C2311yescharacteristic
45C2×D508—24no—
46C2×D508—21no—
47C2×D508C2312yes—
48C22×Dic127—21no—
49C22×Dic127D411yescharacteristic
50C22×Dic127D411yescharacteristic
51C2×C4×D127—22no—
52C2×C4×D127—22no—
53C2×C4×D127—44no—
54#14523237—24no—
55#14523233—24no—
56C2×C127⋊C8C2312yes—
57C4⋊Dic127—21no—
58C4⋊Dic127C2311yescharacteristic
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62C22⋊Dic127—21no—
63C22⋊Dic127—21no—
64C22⋊Dic127—21no—
65C22⋊Dic127—21no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—22no—
68C4×Dic127—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C4×Dic127C2311yescharacteristic
73C2×D4×C127—21no—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127C2311yescharacteristic
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C22×D127—24no—
83C22×D127—22no—
84C22×D127—42no—
85C22×D127—42no—
86Dic254—22no—
87D508—24no—
88D508—22no—
89D508#4114yes—
90C2×Dic127—22no—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C2×Dic127—41no—
94C2×Dic127—42no—
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C4×D127—44no—
100C4×D127—44no—
101C127⋊C8—22no—
102C23×C127—21no—
103C23×C127C2×D411yescharacteristic
104D4×C127—22no—
105D4×C127—42no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—42no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127AGL(1,ℤ/8)12yes—
119C22×C127C22×D411yescharacteristic
120C2×D127—24no—
121C2×D127—48no—
122C2×D127—42no—
123C2×D127—42no—
124C4×C127—21no—
125C4×C127—42no—
126C4×C127—41no—
127C4×C127C2×#4112yes—
128Dic127—21no—
129Dic127—21no—
130Dic127—21no—
131Dic127—21no—
132Dic127—41no—
133Dic127—41no—
134C2×C127—22no—
135C2×C127—81no—
136C2×C127#52511yescharacteristic
137C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C127#57511yescharacteristic
139D127—44no—
140#3114—1271nomaximal, Sylow
141C127#311411yescharacteristic, complemented, Sylow
142#529—1271no—
143C2×C4×D4—1271no—
144#483—1271no—
145#480D12711yescharacteristic, complemented
146#478—1272no—
147#463—1272no—
148#450—1271no—
149#447—1271no—
150#442—1272no—
151#440—1272no—
152#432—1271no—
153C22×D4—1271no—
154C23×C4—2541no—
155C2×SD16—1272no—
156C2×D8—1272no—
157#126—2541no—
158#125—1271no—
159#124—1271no—
160#123—2541no—
161C22⋊Q8—1271no—
162#121C2×D12711yescharacteristic
163C4×Q8—1271no—
164C4×D4—1271no—
165C4×D4—1272no—
166C4×D4—1272no—
167C4×D4—2541no—
168C4×D4—2541no—
169C4×D4—2541no—
170#117—2541no—
171C2×C4⋊C4C2×D12711yescharacteristic
172C2×C22⋊C4—2541no—
173C2×C42—1271no—
174#107C2×D12712yescomplemented
175#106—1272no—
176C4⋊C8—1272no—
177#103—1272no—
178#102—1272no—
179#102—1272no—
180#102C2×D12712yescomplemented
181C22⋊C8—1271no—
182C22⋊C8C2×D12711yescharacteristic
183#97—1272no—
184C24—2541no—
185C2×Q8—1271no—
186C2×D4—21no—
187C2×D4—1272no—
188C2×D4—2544no—
189C2×D4—2541no—
190C2×D4C22×D12711yescharacteristic
191C22×C4—21no—
192C22×C4—1271no—
193C22×C4—1271no—
194C22×C4—5084no—
195C22×C4—2542no—
196C22×C4—2542no—
197C22×C4—2541no—
198C22×C4C22×D12711yescharacteristic
199SD16—2544no—
200D8—2544no—
201C2×C8—1272no—
202C2×C8C22×D12712yes—
203C4⋊C4—21no—
204C4⋊C4—1271no—
205C4⋊C4—2541no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4C22×D12711yescharacteristic
210C4⋊C4C22×D12712yes—
211C22⋊C4—21no—
212C22⋊C4—2542no—
213C22⋊C4—2542no—
214C22⋊C4—2541no—
215C22⋊C4—2541no—
216C22⋊C4—2541no—
217C22⋊C4—2541no—
218C42—1271no—
219C42—1271no—
220C42—1271no—
221C42—1271no—
222C42—2541no—
223C23—21no—
224C23—5082no—
225C23—5082no—
226C23—2544no—
227C23—2542no—
228C23D4×D12711yescharacteristic
229Q8—2542no—
230D4—22no—
231D4—42no—
232D4—1274no—
233D4—2544no—
234D4—2542no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—1271no—
240C2×C4—1271no—
241C2×C4—1271no—
242C2×C4—1271no—
243C2×C4—5084no—
244C2×C4—5084no—
245C2×C4—5082no—
246C2×C4—5081no—
247C2×C4—2541no—
248C2×C4—2541no—
249C2×C4—2542no—
250C2×C4D4×D12711yescharacteristic
251C2×C4C23×D12711yescharacteristic
252C8—22no—
253C8—2542no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—5088no—
260C22—5082no—
261C22—5082no—
262C22—2544no—
263C22#1686123312yes—
264C22C2×D4×D12711yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—5081no—
269C4—5081no—
270C4—2541no—
271C4—2541no—
272C4—2541no—
273C4—2541no—
274C4#41×D12712yes—
275C2—22no—
276C2—81no—
277C2—5084no—
278C2#2255038411yescharacteristic
279C2C2×#1686123311yescharacteristic
280C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#480×C127C2
socleC22×C127C22×D4
prime radical#22550009C2
hypercenter#480D127
hypocenterC127#3114
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #22550407C2semidirect
2. C2×#16861171C2semidirect
3. #22550206C2semidirect
4. #22550179C2semidirect
5. #22550090C2semidirect
6. #22550029C2semidirect
8. #22549985C2semidirect
9. #22549975C2semidirect
10. #22549930C2semidirect
11. #480×C127C2semidirect
17. #16861200C22semidirect
18. #16861196C22semidirect
21. #16861174C22semidirect
23. D4×Dic127C22semidirect
31. #16861140C22semidirect
32. Dic127⋊C8C22semidirect
33. #16861132C22semidirect
37. #16861119C22semidirect
40. #107×C127C22semidirect
41. #102×C127C22semidirect
141. C127#3114semidirect
145. #480D127semidirect
174. #107C2×D127semidirect
180. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504072
C2×#168611712
#225502062
#225501792
#225500902
#225500292
#225500092
#225499852
#225499752
#225499302
#480×C1272
#3114127