Finite Group Databasebeta

← #30649089

Subgroups — #30649089

Order 16256.

404 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550445C211yesmaximal, characteristic
2#22550242C212yesmaximal, complemented
3#22550238C212yesmaximal, complemented
4#22550233C212yesmaximal, complemented
5#22550224C212yesmaximal, complemented
6#22549880C211yesmaximal, characteristic, complemented
7#22549875C211yesmaximal, characteristic, complemented
8#22549870C211yesmaximal, characteristic, complemented
9C2×#16861129C211yesmaximal, characteristic, complemented
10#22549832C211yesmaximal, characteristic, complemented
11#544×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16#16861206C2212yes—
17#16861197C2212yes—
18#16861184—21no—
19#16861184—21no—
20#16861179—22no—
21#16861132C2211yescharacteristic, complemented
22#16861132C2211yescharacteristic, complemented
23#16861132C2212yescomplemented
24#16861129C2212yes—
25#16861129C2212yes—
26#16861129C2212yes—
27#16861129C2212yes—
28#16861129C2211yescharacteristic, complemented
29#16861129C2211yescharacteristic, complemented
30#16861129C2212yescomplemented
31#16861120C2212yescomplemented
32C4×C127⋊C8C2212yescomplemented
33C4⋊Q8×C127—21no—
34C4⋊Q8×C127C2212yescomplemented
35#125×C127—21no—
36#125×C127C2212yescomplemented
37C22⋊Q8×C127—22no—
38#117×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41C23×D127—21no—
42#14523248—22no—
43#14523248—22no—
44C2×Dic254C2311yescharacteristic
45C2×D508—21no—
46C2×D508—22no—
47C2×D508—22no—
48C2×D508—22no—
49C2×D508—22no—
50C2×D508C2312yes—
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—21no—
53C2×C4×D127—21no—
54#14523232—22no—
55C2×C127⋊C8—22no—
56C2×C127⋊C8C2311yescharacteristic
57C2×C127⋊C8C2311yescharacteristic
58C2×C127⋊C8C2312yes—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—22no—
61C2×Q8×C127—22no—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64C4⋊C4×C127—21no—
65C4⋊C4×C127—22no—
66C4⋊C4×C127—22no—
67C4⋊C4×C127—22no—
68C4⋊C4×C127D411yescharacteristic
69C4⋊C4×C127D411yescharacteristic
70C4⋊C4×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—21no—
74C22⋊C4×C127D412yes—
75C42×C127—21no—
76C42×C127C2312yes—
77C22×D127—22no—
78C22×D127—21no—
79C22×D127—42no—
80C22×D127—81no—
81C22×D127—81no—
82Dic254—21no—
83Dic254—21no—
84D508—22no—
85D508—22no—
86D508—21no—
87D508—21no—
88D508—42no—
89D508—82no—
90D508—82no—
91C22⋊D127—42no—
92C2×Dic127—21no—
93C4×D127—42no—
94C127⋊C8—21no—
95C127⋊C8—21no—
96C127⋊C8—22no—
97C23×C127C2×D411yescharacteristic
98Q8×C127—82no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—22no—
102C2×C4×C127—81no—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127#4112yes—
107C2×C4×C127#4112yes—
108C2×C4×C127C2411yescharacteristic
109C22×C127—21no—
110C22×C127—21no—
111C22×C127#12412yes—
112C22×C127C22×D411yescharacteristic
113C2×D127—41no—
114C2×D127—42no—
115C2×D127—41no—
116C2×D127—42no—
117C2×D127—41no—
118C2×D127—42no—
119C2×D127—84no—
120C4×C127—22no—
121C4×C127—22no—
122C4×C127—42no—
123C4×C127—41no—
124C4×C127—41no—
125C4×C127#12412yes—
126C4×C127C2×#4111yescharacteristic
127C4×C127C2×#4111yescharacteristic
128Dic127—41no—
129C2×C127—22no—
130C2×C127C2×#12411yescharacteristic
131C2×C127#57611yescharacteristic
132C2×C127#57711yescharacteristic
133D127—81no—
134D127—82no—
135#3136—1271nomaximal, Sylow
136C127#313611yescharacteristic, complemented, Sylow
137#544D12711yescharacteristic, complemented
138#535—1271no—
139#490—1272no—
140#489—1272no—
141#488—1272no—
142#486—1272no—
143#418—1271no—
144#417—1271no—
145#416—1271no—
146C2×#102—1271no—
147#405—1271no—
148C2×#41—1271no—
149C22×D4—1271no—
150C2×M16—1271no—
151C22×C8—1271no—
152C4⋊Q8—21no—
153C4⋊Q8C2×D12712yescomplemented
154#127—1272no—
155#125—21no—
156#125C2×D12712yescomplemented
157#124—1272no—
158C22⋊Q8—22no—
159#121—2541no—
160#121—2541no—
161#120—2542no—
162#117C2×D12711yescharacteristic
163#117C2×D12711yescharacteristic
164C2×C4⋊C4C2×D12711yescharacteristic
165#103—1271no—
166#103—1271no—
167#103—1272no—
168#102—1272no—
169#102—1272no—
170#102—1271no—
171#102—1271no—
172#102—1272no—
173#102—1272no—
174#102—1272no—
175#97—1272no—
176C4×C8—1272no—
177C24—2541no—
178#41—2542no—
179#41—2542no—
180C2×Q8—22no—
181C2×Q8—1271no—
182C2×D4—1272no—
183C2×D4—1271no—
184C2×D4—2541no—
185C2×D4—2542no—
186C2×D4—2542no—
187C2×D4—2541no—
188C2×D4—2542no—
189C2×D4—2542no—
190C22×C4—21no—
191C22×C4—2541no—
192C22×C4C22×D12711yescharacteristic
193M16—2542no—
194C2×C8—1271no—
195C2×C8—1272no—
196C2×C8—1271no—
197C2×C8—2542no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4C22⋊D12711yescharacteristic
203C4⋊C4C22⋊D12711yescharacteristic
204C4⋊C4C22×D12712yes—
205C4⋊C4C22×D12711yescharacteristic
206C4⋊C4C22×D12711yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—2542no—
209C22⋊C4—2542no—
210C22⋊C4C22⋊D12712yes—
211C42—21no—
212C42C22×D12712yes—
213C23—10161no—
214C23—10161no—
215C23—5082no—
216C23—2542no—
217C23—2541no—
218C23C2×C22⋊D12711yescharacteristic
219Q8—82no—
220Q8—2541no—
221Q8—2541no—
222D4—10162no—
223D4—10162no—
224D4—5082no—
225D4—5082no—
226D4—2542no—
227D4—2542no—
228D4—2541no—
229D4—2541no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—81no—
234C2×C4—5082no—
235C2×C4—2541no—
236C2×C4C2×C22⋊D12712yes—
237C2×C4C2×C22⋊D12712yes—
238C2×C4C2×C22⋊D12711yescharacteristic
239C2×C4#1452324912yes—
240C2×C4#1452324912yes—
241C2×C4C23×D12711yescharacteristic
242C8—2541no—
243C8—2541no—
244C8—2542no—
245C22—21no—
246C22—21no—
247C22—10164no—
248C22—5081no—
249C22—5082no—
250C22—5081no—
251C22—5082no—
252C22—5081no—
253C22—5082no—
254C22#1686119812yes—
255C22C22×C22⋊D12711yescharacteristic
256C4—22no—
257C4—22no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—5081no—
262C4#1686119812yes—
263C4C2×#1452324911yescharacteristic
264C4C2×#1452324911yescharacteristic
265C2—22no—
266C2—10162no—
267C2—10161no—
268C2C2×#1686119811yescharacteristic
269C2#2255067911yescharacteristic
270C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#544×C127C2
socleC22×C127C22×D4
prime radical#22550445C2
hypercenter#544D127
hypocenterC127#3136
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
2. #22550242C2semidirect
3. #22550238C2semidirect
4. #22550233C2semidirect
5. #22550224C2semidirect
6. #22549880C2semidirect
7. #22549875C2semidirect
8. #22549870C2semidirect
9. C2×#16861129C2semidirect
10. #22549832C2semidirect
11. #544×C127C2semidirect
21. #16861132C22semidirect
22. #16861132C22semidirect
23. #16861132C22semidirect
28. #16861129C22semidirect
29. #16861129C22semidirect
30. #16861129C22semidirect
31. #16861120C22semidirect
32. C4×C127⋊C8C22semidirect
34. C4⋊Q8×C127C22semidirect
36. #125×C127C22semidirect
136. C127#3136semidirect
137. #544D127semidirect
153. C4⋊Q8C2×D127semidirect
156. #125C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504452
#225502422
#225502382
#225502332
#225502242
#225498802
#225498752
#225498702
C2×#168611292
#225498322
#544×C1272
#3136127