Finite Group Databasebeta

← #30650548

Subgroups — #30650548

Order 16256.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D508C211yesmaximal, characteristic
2#22550224C211yesmaximal, characteristic, complemented
3#22550194C212yesmaximal, complemented
4#22550163C211yesmaximal, characteristic, complemented
5#22550107C212yesmaximal, complemented
6#22550076C211yesmaximal, characteristic, complemented
7#22550026C212yesmaximal, complemented
8C4×#14523235C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10C2×#16861129C212yesmaximal, complemented
11#554×C127C211yesmaximal, characteristic, complemented
12C22×D508C2212yes—
13C2×D4×D127—21no—
14C2×#14523235C2211yescharacteristic
15C2×#14523235C2212yescomplemented
16C22×C127⋊C8C2212yescomplemented
17#16861206C2211yescharacteristic
18#16861184—21no—
19#16861183C2212yes—
20#16861179—22no—
21C4×D508C2211yescharacteristic
22#16861140C2212yescomplemented
23#16861140C2211yescharacteristic, complemented
24C4⋊(C127⋊C8)C2211yescharacteristic, complemented
25#16861133C2211yescharacteristic, complemented
26#16861129C2212yes—
27#16861129C2211yescharacteristic
28#16861129C2212yescomplemented
29#16861129C2212yescomplemented
30#16861129C2212yescomplemented
31C22⋊(C127⋊C8)C2212yescomplemented
32C4×C127⋊C8C2211yescharacteristic, complemented
33C4⋊Q8×C127C2211yescharacteristic, complemented
34#125×C127—21no—
35C22⋊Q8×C127—22no—
36C22⋊Q8×C127C2212yescomplemented
37C4×Q8×C127C2211yescharacteristic, complemented
38C4×D4×C127—21no—
39C4×D4×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2212yescomplemented
41C23×D127—22no—
42C2×D508—24no—
43C2×D508—21no—
44C2×D508—22no—
45C2×D508—21no—
46C2×D508—42no—
47C2×D508C2312yes—
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—22no—
50D4×D127—44no—
51C2×C4×D127—21no—
52#14523235—22no—
53#14523235—42no—
54C2×C127⋊C8—42no—
55C2×C127⋊C8C2311yescharacteristic
56C2×C127⋊C8C2312yes—
57C2×C127⋊C8C2311yescharacteristic
58C4⋊Dic127C2311yescharacteristic
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—21no—
61C2×Q8×C127—21no—
62C2×Q8×C127C2311yescharacteristic
63C2×D4×C127D411yescharacteristic
64C22×C4×C127—22no—
65C22×C4×C127C2312yes—
66C4⋊C4×C127—22no—
67C4⋊C4×C127—21no—
68C4⋊C4×C127—21no—
69C4⋊C4×C127—22no—
70C4⋊C4×C127D411yescharacteristic
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2312yes—
73C22⋊C4×C127—22no—
74C22⋊C4×C127D412yes—
75C42×C127—21no—
76C42×C127C2311yescharacteristic
77C22×D127—21no—
78C22×D127—22no—
79C22×D127—44no—
80C22×D127—44no—
81C22×D127—82no—
82D508—22no—
83D508—21no—
84D508—22no—
85D508—44no—
86D508—42no—
87D508—81no—
88D508#4112yes—
89C22⋊D127—44no—
90C2×Dic127—21no—
91C4×D127—42no—
92C127⋊C8—21no—
93C127⋊C8—21no—
94C127⋊C8—22no—
95C23×C127C2×D412yes—
96Q8×C127—21no—
97Q8×C127—21no—
98Q8×C127—81no—
99D4×C127SD1614yescomplemented
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—22no—
105C2×C4×C127—82no—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2411yescharacteristic
110C22×C127—22no—
111C22×C127—22no—
112C22×C127C2×SD1614yes—
113C22×C127C22×D411yescharacteristic
114C2×D127—22no—
115C2×D127—41no—
116C2×D127—41no—
117C2×D127—42no—
118C2×D127—42no—
119C2×D127—42no—
120C2×D127—88no—
121C4×C127—22no—
122C4×C127—21no—
123C4×C127—41no—
124C4×C127—41no—
125C4×C127—42no—
126C4×C127C2×SD1612yes—
127C4×C127C2×#4111yescharacteristic
128C4×C127H(5,𝔽2)11yescharacteristic
129Dic127—41no—
130C2×C127—24no—
131C2×C127#54611yescharacteristic
132C2×C127C22×SD1611yescharacteristic
133C2×C127#57611yescharacteristic
134D127—42no—
135D127—82no—
136#3295—1271nomaximal, Sylow
137C127#329511yescharacteristic, complemented, Sylow
138#554D12711yescharacteristic, complemented
139D42—1271no—
140#486—1271no—
141#481—1272no—
142#476—1271no—
143#465—1272no—
144#460—1271no—
145#450—1272no—
146C4×SD16—1271no—
147C8×D4—1271no—
148C2×#102—1272no—
149C22×D4—1272no—
150C22×D4—2541no—
151C2×SD16—1272no—
152C2×SD16—1271no—
153C22×C8—1272no—
154C4⋊Q8C2×D12711yescharacteristic, complemented
155#127—1271no—
156#125—21no—
157C22⋊Q8—22no—
158C22⋊Q8C2×D12712yescomplemented
159#121—1272no—
160#121—2541no—
161#120—2542no—
162C4×Q8C2×D12711yescharacteristic, complemented
163C4×D4—21no—
164C4×D4—1271no—
165C4×D4C2×D12711yescharacteristic
166C2×C4⋊C4C2×D12712yescomplemented
167#106—1271no—
168#106—1272no—
169C4⋊C8—1271no—
170#103—1271no—
171#102—1271no—
172#102—1272no—
173#102—1272no—
174#102—1272no—
175#102—1272no—
176C22⋊C8—1272no—
177C4×C8—1271no—
178C24—2542no—
179C2×Q8—21no—
180C2×Q8C22×D12711yescharacteristic
181C2×D4—1271no—
182C2×D4—1272no—
183C2×D4—5082no—
184C2×D4—5084no—
185C2×D4—2544no—
186C2×D4—2542no—
187C2×D4—2541no—
188C2×D4—2542no—
189C2×D4—2541no—
190C2×D4C22⋊D12711yescharacteristic
191C22×C4—22no—
192C22×C4—2541no—
193C22×C4C22×D12712yes—
194SD16—5082no—
195SD16—2542no—
196C2×C8—1271no—
197C2×C8—1271no—
198C2×C8—1272no—
199C2×C8—5082no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—1271no—
205C4⋊C4C22⋊D12711yescharacteristic
206C4⋊C4C22×D12711yescharacteristic
207C4⋊C4C22×D12712yes—
208C22⋊C4—22no—
209C22⋊C4—2541no—
210C22⋊C4—2542no—
211C22⋊C4C22⋊D12712yes—
212C42—21no—
213C42C22×D12711yescharacteristic
214C23—10162no—
215C23—5084no—
216C23—5084no—
217C23—2542no—
218C23—2541no—
219C23C2×C22⋊D12712yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—1272no—
224D4—10161no—
225D4—5084no—
226D4—5084no—
227D4—5082no—
228D4—2542no—
229D4—2542no—
230D4—2541no—
231D4#1452323514yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—5082no—
239C2×C4—2541no—
240C2×C4C2×C22⋊D12711yescharacteristic
241C2×C4C2×C22⋊D12712yes—
242C2×C4C2×C22⋊D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—2541no—
245C8—2541no—
246C8—2542no—
247C22—22no—
248C22—22no—
249C22—10168no—
250C22—5082no—
251C22—5082no—
252C22—5082no—
253C22—5081no—
254C22—5081no—
255C22—2542no—
256C22C2×#1452323514yes—
257C22C22×C22⋊D12711yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—5081no—
264C4C2×#1452323512yes—
265C4#41×D12711yescharacteristic
266C4#1686125511yescharacteristic
267C2—24no—
268C2—10162no—
269C2—5082no—
270C2#2255051411yescharacteristic
271C2C22×#1452323511yescharacteristic
272C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#554×C127C2
socleC22×C127C22×D4
prime radicalD4×D508C2
hypercenter#554D127
hypocenterC127#3295
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #22550224C2semidirect
3. #22550194C2semidirect
4. #22550163C2semidirect
5. #22550107C2semidirect
6. #22550076C2semidirect
7. #22550026C2semidirect
8. C4×#14523235C2semidirect
9. D4×C127⋊C8C2semidirect
10. C2×#16861129C2semidirect
11. #554×C127C2semidirect
15. C2×#14523235C22semidirect
16. C22×C127⋊C8C22semidirect
22. #16861140C22semidirect
23. #16861140C22semidirect
24. C4⋊(C127⋊C8)C22semidirect
25. #16861133C22semidirect
28. #16861129C22semidirect
29. #16861129C22semidirect
30. #16861129C22semidirect
31. C22⋊(C127⋊C8)C22semidirect
32. C4×C127⋊C8C22semidirect
33. C4⋊Q8×C127C22semidirect
36. C22⋊Q8×C127C22semidirect
37. C4×Q8×C127C22semidirect
40. C2×C4⋊C4×C127C22semidirect
99. D4×C127SD16semidirect
137. C127#3295semidirect
138. #554D127semidirect
154. C4⋊Q8C2×D127semidirect
158. C22⋊Q8C2×D127semidirect
162. C4×Q8C2×D127semidirect
166. C2×C4⋊C4C2×D127semidirect
231. D4#14523235semidirect

Maximal subgroups

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

subgroupindex
D4×D5082
#225502242
#225501942
#225501632
#225501072
#225500762
#225500262
C4×#145232352
D4×C127⋊C82
C2×#168611292
#554×C1272
#3295127