Finite Group Databasebeta

← #30650537

Subgroups — #30650537

Order 16256.

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

normal subgroups only

This group has 494 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, complemented
2#22550224C211yesmaximal, characteristic, complemented
3#22550189C212yesmaximal, complemented
4#22550153C211yesmaximal, characteristic, complemented
5#22550110C212yesmaximal, complemented
6#22550071C211yesmaximal, characteristic, complemented
7#22550009C212yesmaximal, complemented
8C4×#14523233C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10C2×#16861129C212yesmaximal, complemented
11#547×C127C211yesmaximal, characteristic, complemented
12C22×D508C2212yes—
13C2×D4×D127—21no—
14C2×#14523233C2211yescharacteristic
15C2×#14523233C2212yescomplemented
16C22×C127⋊C8C2212yescomplemented
17#16861206C2211yescharacteristic
18#16861184—21no—
19#16861183C2212yes—
20#16861179—22no—
21C4×D508C2211yescharacteristic
22#16861142C2212yescomplemented
23#16861142C2211yescharacteristic, complemented
24C4⋊(C127⋊C8)C2211yescharacteristic, complemented
25#16861130C2211yescharacteristic, complemented
26#16861129C2211yescharacteristic
27#16861129C2212yes—
28#16861129C2212yescomplemented
29#16861129C2212yescomplemented
30#16861129C2212yescomplemented
31C22⋊(C127⋊C8)C2212yescomplemented
32C4×C127⋊C8C2211yescharacteristic, complemented
33C2×#41×C127—21no—
34C4⋊Q8×C127C2211yescharacteristic, complemented
35#123×C127—22no—
36C22⋊Q8×C127—21no—
37#121×C127C2212yescomplemented
38C4×D4×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic, complemented
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#14523233—22no—
53#14523233—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—
61#41×C127—44no—
62C2×Q8×C127—21no—
63C2×D4×C127—22no—
64C2×D4×C127D411yescharacteristic
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—22no—
67C22×C4×C127—21no—
68C22×C4×C127C2312yes—
69C4⋊C4×C127—22no—
70C4⋊C4×C127—21no—
71C4⋊C4×C127D411yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C4⋊C4×C127C2312yes—
74C22⋊C4×C127—21no—
75C22⋊C4×C127—22no—
76C22⋊C4×C127D412yes—
77C42×C127C2311yescharacteristic
78C22×D127—21no—
79C22×D127—22no—
80C22×D127—44no—
81C22×D127—44no—
82C22×D127—82no—
83D508—22no—
84D508—21no—
85D508—22no—
86D508—44no—
87D508—42no—
88D508—81no—
89D508#4112yescomplemented
90C22⋊D127—44no—
91C2×Dic127—21no—
92C4×D127—42no—
93C127⋊C8—21no—
94C127⋊C8—21no—
95C127⋊C8—22no—
96C23×C127—21no—
97C23×C127C2×D412yes—
98Q8×C127—42no—
99D4×C127—21no—
100D4×C127—21no—
101D4×C127—44no—
102D4×C127D814yescomplemented
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—44no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D412yes—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C22×C127—22no—
114C22×C127—22no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127C2×D814yes—
119C22×C127C22×D411yescharacteristic
120C2×D127—22no—
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—42no—
124C2×D127—42no—
125C2×D127—42no—
126C2×D127—88no—
127C4×C127—22no—
128C4×C127—21no—
129C4×C127—41no—
130C4×C127—42no—
131C4×C127C2×D812yes—
132C4×C127C2×#4111yescharacteristic
133C4×C127H(5,𝔽2)11yescharacteristic
134Dic127—41no—
135C2×C127—24no—
136C2×C127—81no—
137C2×C127#54611yescharacteristic
138C2×C127C22×D811yescharacteristic
139C2×C127#57711yescharacteristic
140D127—42no—
141D127—82no—
142#3294—1271nomaximal, Sylow
143C127#329411yescharacteristic, complemented, Sylow
144#547D12711yescharacteristic, complemented
145D42—1271no—
146#486—1271no—
147#480—1272no—
148#474—1271no—
149#466—1272no—
150#459—1271no—
151#447—1272no—
152C4×D8—1271no—
153C8×D4—1271no—
154C2×#102—1272no—
155C2×#41—21no—
156C22×D4—1272no—
157C22×D4—2541no—
158C2×D8—1272no—
159C2×D8—1271no—
160C22×C8—1272no—
161C4⋊Q8C2×D12711yescharacteristic, complemented
162#127—1271no—
163#123—22no—
164C22⋊Q8—21no—
165#121—1272no—
166#121—2541no—
167#121C2×D12712yescomplemented
168#120—2542no—
169C4×D4—1271no—
170C4×D4C2×D12711yescharacteristic
171C4×D4C2×D12711yescharacteristic, complemented
172C2×C4⋊C4C2×D12712yescomplemented
173#107—1271no—
174#107—1272no—
175C4⋊C8—1271no—
176#102—1271no—
177#102—1271no—
178#102—1272no—
179#102—1272no—
180#102—1272no—
181#102—1272no—
182C22⋊C8—1272no—
183C4×C8—1271no—
184C24—2542no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—1271no—
189C2×D4—1272no—
190C2×D4—5082no—
191C2×D4—5084no—
192C2×D4—2544no—
193C2×D4—2542no—
194C2×D4—2541no—
195C2×D4—2542no—
196C2×D4—2541no—
197C2×D4C22⋊D12711yescharacteristic
198C2×D4C22×D12711yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—2541no—
202C22×C4C22×D12712yes—
203D8—5082no—
204D8—2542no—
205C2×C8—1271no—
206C2×C8—1271no—
207C2×C8—1272no—
208C2×C8—5082no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1271no—
212C4⋊C4C22⋊D12711yescharacteristic
213C4⋊C4C22×D12711yescharacteristic
214C4⋊C4C22×D12712yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2541no—
218C22⋊C4—2542no—
219C22⋊C4C22⋊D12712yes—
220C42C22×D12711yescharacteristic
221C23—21no—
222C23—10162no—
223C23—5084no—
224C23—5084no—
225C23—2542no—
226C23—2541no—
227C23C2×C22⋊D12712yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—1272no—
233D4—10161no—
234D4—5084no—
235D4—5084no—
236D4—5082no—
237D4—2542no—
238D4—2542no—
239D4—2541no—
240D4#1452323314yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—5082no—
248C2×C4—2541no—
249C2×C4C2×C22⋊D12711yescharacteristic
250C2×C4C2×C22⋊D12712yes—
251C2×C4C2×C22⋊D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—2541no—
254C8—2541no—
255C8—2542no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—10168no—
262C22—5082no—
263C22—5082no—
264C22—5082no—
265C22—5081no—
266C22—5081no—
267C22—2542no—
268C22C2×#1452323314yes—
269C22C22×C22⋊D12711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—5081no—
275C4C2×#1452323312yes—
276C4#41×D12711yescharacteristic
277C4#1686125511yescharacteristic
278C2—24no—
279C2—81no—
280C2—10162no—
281C2—5082no—
282C2#2255051411yescharacteristic
283C2C22×#1452323311yescharacteristic
284C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#547×C127C2
socleC22×C127C22×D4
hypercenter#547D127
hypocenterC127#3294
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D508C2semidirect
2. #22550224C2semidirect
3. #22550189C2semidirect
4. #22550153C2semidirect
5. #22550110C2semidirect
6. #22550071C2semidirect
7. #22550009C2semidirect
8. C4×#14523233C2semidirect
9. D4×C127⋊C8C2semidirect
10. C2×#16861129C2semidirect
11. #547×C127C2semidirect
15. C2×#14523233C22semidirect
16. C22×C127⋊C8C22semidirect
22. #16861142C22semidirect
23. #16861142C22semidirect
24. C4⋊(C127⋊C8)C22semidirect
25. #16861130C22semidirect
28. #16861129C22semidirect
29. #16861129C22semidirect
30. #16861129C22semidirect
31. C22⋊(C127⋊C8)C22semidirect
32. C4×C127⋊C8C22semidirect
34. C4⋊Q8×C127C22semidirect
37. #121×C127C22semidirect
39. C4×D4×C127C22semidirect
40. C2×C4⋊C4×C127C22semidirect
89. D508#41semidirect
102. D4×C127D8semidirect
143. C127#3294semidirect
144. #547D127semidirect
161. C4⋊Q8C2×D127semidirect
167. #121C2×D127semidirect
171. C4×D4C2×D127semidirect
172. C2×C4⋊C4C2×D127semidirect
240. D4#14523233semidirect

Maximal subgroups

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

subgroupindex
D4×D5082
#225502242
#225501892
#225501532
#225501102
#225500712
#225500092
C4×#145232332
D4×C127⋊C82
C2×#168611292
#547×C1272
#3294127