Finite Group Databasebeta

← #30648868

Subgroups — #30648868

Order 16256.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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#22550410C211yesmaximal, characteristic, complemented
2#22550206C211yesmaximal, characteristic, complemented
3#22550189C211yesmaximal, characteristic, complemented
4#22550176C212yesmaximal, complemented
5#22550091C212yesmaximal, complemented
6#22550027C211yesmaximal, characteristic, complemented
7#22550010C211yesmaximal, characteristic
8#22549986C212yesmaximal, complemented
9#22549976C212yesmaximal, complemented
10#22549933C211yesmaximal, characteristic, complemented
11C2×C4×D4×C127C211yesmaximal, characteristic, complemented
12C2×#14523236C2212yes—
13C2×#14523233C2212yes—
14#16861203—21no—
15#16861201C2211yescharacteristic, complemented
16#16861197C2211yescharacteristic, complemented
17#16861192—21no—
18#16861188C2211yescharacteristic
19#16861183C2211yescharacteristic
20C4×Dic254C2211yescharacteristic, complemented
21C4×D508C2211yescharacteristic, complemented
22C4×C22⋊D127—21no—
23#16861166—21no—
24#16861142C2212yescomplemented
25#16861140C2212yescomplemented
26C4⋊(C127⋊C8)C2212yescomplemented
27#16861132C2212yescomplemented
28#16861130C2212yes—
29#16861130C2212yes—
30#16861129C2212yescomplemented
31C22⋊(C127⋊C8)C2211yescharacteristic
32C22⋊(C127⋊C8)C2211yescharacteristic
33#16861120C2212yescomplemented
34C22×D4×C127C2211yescharacteristic
35C23×C4×C127—21no—
36C4×D4×C127—21no—
37C4×D4×C127—21no—
38C4×D4×C127C2212yes—
39C4×D4×C127C2212yes—
40C2×C4⋊C4×C127C2211yescharacteristic
41C2×C22⋊C4×C127—21no—
42C2×C42×C127C2211yescharacteristic
43C2×Dic254C2311yescharacteristic
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—21no—
46C2×C4×D127—21no—
47#14523236—24no—
48#14523233—24no—
49C2×C127⋊C8C2312yes—
50C2×C127⋊C8C2312yes—
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—21no—
54Dic127⋊C4—21no—
55Dic127⋊C4—21no—
56Dic127⋊C4—21no—
57C22⋊Dic127—21no—
58(C2×D127)⋊C4—21no—
59(C2×D127)⋊C4—21no—
60(C2×D127)⋊C4—21no—
61(C2×D127)⋊C4—21no—
62C4×Dic127—21no—
63C24×C127—21no—
64C2×D4×C127—24no—
65C2×D4×C127—21no—
66C2×D4×C127C2312yes—
67C22×C4×C127—21no—
68C22×C4×C127—22no—
69C22×C4×C127—22no—
70C22×C4×C127—44no—
71C22×C4×C127D411yescharacteristic
72C22×C4×C127D411yescharacteristic
73C22×C4×C127C2311yescharacteristic
74C4⋊C4×C127—21no—
75C4⋊C4×C127C2312yes—
76C22⋊C4×C127—22no—
77C22⋊C4×C127—22no—
78C42×C127D411yescharacteristic
79C42×C127D411yescharacteristic
80C42×C127C2311yescharacteristic
81C42×C127C2311yescharacteristic
82C22×D127—21no—
83Dic254—22no—
84D508—22no—
85C22⋊D127—42no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C4×D127—42no—
90C127⋊C8—22no—
91C127⋊C8—22no—
92C23×C127—24no—
93C23×C127—22no—
94C23×C127—42no—
95C23×C127—42no—
96C23×C127C2×D411yescharacteristic
97D4×C127—24no—
98D4×C127—22no—
99D4×C127#4114yes—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—41no—
104C2×C4×C127—44no—
105C2×C4×C127—44no—
106C2×C4×C127—42no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—24no—
116C22×C127—48no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127AGL(1,ℤ/8)12yes—
120C22×C127C22×D411yescharacteristic
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—41no—
124C4×C127—21no—
125C4×C127—21no—
126C4×C127—21no—
127C4×C127—21no—
128C4×C127—21no—
129C4×C127—42no—
130C4×C127C2×#4112yes—
131Dic127—41no—
132Dic127—41no—
133Dic127—41no—
134C2×C127—22no—
135C2×C127—44no—
136C2×C127#52511yescharacteristic
137C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C127#57511yescharacteristic
139D127—81no—
140#3114—1271nomaximal, Sylow
141C127#311411yescharacteristic, complemented, Sylow
142#529—1271no—
143C2×C4×D4D12711yescharacteristic, complemented
144#483—1271no—
145#480—1271no—
146#478—1272no—
147#463—1272no—
148#450—1271no—
149#447—1271no—
150#442—1272no—
151#440—1272no—
152#432—1271no—
153C22×D4C2×D12711yescharacteristic
154C23×C4—21no—
155C2×SD16—1272no—
156C2×D8—1272no—
157#126—2541no—
158#125—1271no—
159#124—1271no—
160#123—2541no—
161C22⋊Q8—1271no—
162#121—1271no—
163C4×Q8—1271no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—1271no—
167C4×D4—2541no—
168C4×D4C2×D12712yes—
169C4×D4C2×D12712yes—
170#117—2541no—
171C2×C4⋊C4C2×D12711yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D12711yescharacteristic
174#107—1272no—
175#106—1272no—
176C4⋊C8—1272no—
177#103—1272no—
178#102—1272no—
179#102—1272no—
180#102—1272no—
181C22⋊C8—1271no—
182C22⋊C8—1271no—
183#97—1272no—
184C24—21no—
185C2×Q8—1271no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—1271no—
189C2×D4—2541no—
190C2×D4C22×D12712yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—2541no—
196C22×C4C22⋊D12711yescharacteristic
197C22×C4C22⋊D12711yescharacteristic
198C22×C4C22×D12711yescharacteristic
199SD16—2544no—
200D8—2544no—
201C2×C8—1272no—
202C2×C8—1272no—
203C4⋊C4—21no—
204C4⋊C4—1271no—
205C4⋊C4—1271no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4—2541no—
210C4⋊C4C22×D12712yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—2541no—
214C22⋊C4—2541no—
215C22⋊C4—2541no—
216C22⋊C4—2541no—
217C22⋊C4—2541no—
218C42—2541no—
219C42C22⋊D12711yescharacteristic
220C42C22⋊D12711yescharacteristic
221C42C22×D12711yescharacteristic
222C42C22×D12711yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—2541no—
228C23C2×C22⋊D12711yescharacteristic
229Q8—2542no—
230D4—24no—
231D4—22no—
232D4—5082no—
233D4—2542no—
234D4#1452324814yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—5082no—
243C2×C4—2541no—
244C2×C4—2541no—
245C2×C4—2541no—
246C2×C4C2×C22⋊D12711yescharacteristic
247C2×C4C2×C22⋊D12711yescharacteristic
248C2×C4C2×C22⋊D12711yescharacteristic
249C2×C4C2×C22⋊D12711yescharacteristic
250C2×C4C2×C22⋊D12711yescharacteristic
251C2×C4C23×D12711yescharacteristic
252C8—2542no—
253C8—2542no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—5081no—
261C22—5081no—
262C22—5081no—
263C22#1686123512yes—
264C22C22×C22⋊D12711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—5081no—
272C4—5081no—
273C4—5081no—
274C4C2×#1452324812yes—
275C2—22no—
276C2—44no—
277C2—10161no—
278C2#2255038711yescharacteristic
279C2C2×#1686123511yescharacteristic
280C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×D4×C127C2
socleC22×C127C22×D4
prime radical#22550010C2
hypercenterC2×C4×D4D127
hypocenterC127#3114
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22550410C2semidirect
2. #22550206C2semidirect
3. #22550189C2semidirect
4. #22550176C2semidirect
5. #22550091C2semidirect
6. #22550027C2semidirect
8. #22549986C2semidirect
9. #22549976C2semidirect
10. #22549933C2semidirect
11. C2×C4×D4×C127C2semidirect
15. #16861201C22semidirect
16. #16861197C22semidirect
20. C4×Dic254C22semidirect
21. C4×D508C22semidirect
24. #16861142C22semidirect
25. #16861140C22semidirect
26. C4⋊(C127⋊C8)C22semidirect
27. #16861132C22semidirect
30. #16861129C22semidirect
33. #16861120C22semidirect
141. C127#3114semidirect
143. C2×C4×D4D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
#225502062
#225501892
#225501762
#225500912
#225500272
#225500102
#225499862
#225499762
#225499332
C2×C4×D4×C1272
#3114127