Finite Group Databasebeta

← #30650415

Subgroups — #30650415

Order 16256.

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

normal subgroups only

This group has 526 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
1C2×#16861228C211yesmaximal, characteristic, complemented
2#22550520C212yesmaximal, complemented
3#22550259C211yesmaximal, characteristic, complemented
4#22550117C211yesmaximal, characteristic, complemented
5#22550109C211yesmaximal, characteristic, complemented
6#22550089C212yesmaximal, complemented
7#22550027C212yesmaximal, complemented
8#22550018C212yesmaximal, complemented
9D8×Dic127C211yesmaximal, characteristic, complemented
10C8×C22⋊D127C211yesmaximal, characteristic, complemented
11C22×D8×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yescomplemented
13#16861228—24no—
14C2×Dic508C2211yescharacteristic, complemented
15C2×#14523236C2212yescomplemented
16C2×#14523236C2212yescomplemented
17C2×C8×D127C2211yescharacteristic, complemented
18#16861196C2212yescomplemented
19#16861190—22no—
20#16861188C2212yes—
21#16861181—22no—
22#16861180C2212yescomplemented
23#16861178—22no—
24D4×Dic127C2212yescomplemented
25C4×C22⋊D127C2211yescharacteristic
26C2×C22⋊Dic127—22no—
27#16861143C2211yescharacteristic, complemented
28Dic127⋊C8C2211yescharacteristic, complemented
29#16861134C2212yescomplemented
30#16861130C2212yescomplemented
31#16861130C2212yescomplemented
32C22⋊(C127⋊C8)C2211yescharacteristic
33(C2×D127)⋊C8C2211yescharacteristic, complemented
34C8×Dic127C2211yescharacteristic, complemented
35C22×D4×C127C2212yes—
36C2×D8×C127—24no—
37C2×D8×C127C2212yes—
38C2×D8×C127C2211yescharacteristic
39C2×D8×C127C2211yescharacteristic
40C22×C8×C127C2211yescharacteristic
41#14523247—24no—
42#14523247—42no—
43C2×Dic254C2312yes—
44C2×C22⋊D127—22no—
45C2×C22⋊D127—22no—
46C2×C22⋊D127D411yescharacteristic
47C22×Dic127—22no—
48C22×Dic127—22no—
49C2×C4×D127C2311yescharacteristic
50Dic508—22no—
51#14523236—24no—
52#14523236—42no—
53C2×C127⋊C8C2311yescharacteristic
54C2×C127⋊C8C2311yescharacteristic
55C8×D127—22no—
56C4⋊Dic127C2312yes—
57Dic127⋊C4—22no—
58Dic127⋊C4D411yescharacteristic
59C22⋊Dic127—22no—
60C22⋊Dic127—22no—
61C22⋊Dic127—22no—
62C22⋊Dic127—22no—
63C22⋊Dic127—22no—
64C22⋊Dic127D411yescharacteristic
65(C2×D127)⋊C4D411yescharacteristic
66C4×Dic127C2311yescharacteristic
67C24×C127—22no—
68C2×D4×C127—24no—
69C2×D4×C127—22no—
70C2×D4×C127—42no—
71C2×D4×C127C2312yes—
72C2×D4×C127C2312yes—
73C22×C4×C127C2311yescharacteristic
74D8×C127—24no—
75D8×C127—22no—
76D8×C127D414yescomplemented
77C2×C8×C127—22no—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C22×D127C2×D411yescharacteristic
81Dic254—22no—
82Dic254—22no—
83C22⋊D127—22no—
84C22⋊D127—44no—
85C22⋊D127—82no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—44no—
89C2×Dic127—82no—
90C2×Dic127C2×D411yescharacteristic
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C4×D127—21no—
94C4×D127—21no—
95C127⋊C8—21no—
96C127⋊C8—21no—
97C23×C127—22no—
98C23×C127—22no—
99C23×C127—44no—
100C23×C127—44no—
101C23×C127—82no—
102C23×C127C2×D411yescharacteristic
103D4×C127—22no—
104D4×C127—22no—
105D4×C127—22no—
106D4×C127—44no—
107D4×C127C2×D414yes—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—21no—
113C8×C127C2×D412yes—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127—24no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127—42no—
122C22×C127—88no—
123C22×C127#13512yescomplemented
124C22×C127C22×D411yescharacteristic
125C2×D127—21no—
126C2×D127—21no—
127C2×D127—21no—
128C4×C127—21no—
129C4×C127C22×D411yescharacteristic
130C4×C127H(5,𝔽2)11yescharacteristic
131Dic127—21no—
132Dic127—21no—
133Dic127—21no—
134Dic127—42no—
135Dic127—42no—
136C2×C127—22no—
137C2×C127—44no—
138C2×C127—82no—
139C2×C127D4211yescharacteristic
140C2×C127C2×#13511yescharacteristic
141C2×C127#57611yescharacteristic
142D127—41no—
143#3280—1271nomaximal, Sylow
144C127#328011yescharacteristic, complemented, Sylow
145C2×#135—1271no—
146C22×D8D12711yescharacteristic, complemented
147#546—1272no—
148#495—1271no—
149#467—1271no—
150#466—1271no—
151#463—1272no—
152#450—1272no—
153#449—1272no—
154C4×D8—1271no—
155C8×D4—1271no—
156C2×#41—1272no—
157C22×D4C2×D12712yes—
158#135—2544no—
159C2×Q16—1271no—
160C2×SD16—1272no—
161C2×SD16—1272no—
162C2×D8—24no—
163C2×D8C2×D12712yes—
164C2×D8C2×D12711yescharacteristic
165C2×D8C2×D12711yescharacteristic
166C22×C8—1271no—
167C22×C8C2×D12711yescharacteristic
168#124—1272no—
169#123—2542no—
170C22⋊Q8—1272no—
171#121—1272no—
172#121—2542no—
173#120—2542no—
174C4×D4—1271no—
175C4×D4—1272no—
176C2×C22⋊C4—2542no—
177#107—1271no—
178C4⋊C8—1271no—
179#103—1272no—
180#102—1272no—
181#102—1272no—
182C22⋊C8—1271no—
183C22⋊C8—1271no—
184C4×C8—1271no—
185C24—22no—
186#41—5082no—
187#41—2544no—
188C2×Q8—1272no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—1271no—
193C2×D4—2542no—
194C2×D4—2542no—
195C2×D4C22×D12712yes—
196C2×D4C22×D12712yes—
197C22×C4—1271no—
198C22×C4—2542no—
199C22×C4—2542no—
200C22×C4C22×D12711yescharacteristic
201Q16—2542no—
202SD16—5082no—
203SD16—2544no—
204D8—24no—
205D8—22no—
206D8C22⋊D12714yescomplemented
207C2×C8—22no—
208C2×C8—1271no—
209C2×C8—1271no—
210C2×C8—2542no—
211C2×C8C22×D12711yescharacteristic
212C2×C8C22×D12711yescharacteristic
213C4⋊C4—1271no—
214C4⋊C4—1272no—
215C4⋊C4—2542no—
216C22⋊C4—1271no—
217C22⋊C4—1271no—
218C22⋊C4—2542no—
219C22⋊C4—2542no—
220C22⋊C4—2542no—
221C22⋊C4—2542no—
222C22⋊C4—2542no—
223C42—1271no—
224C23—22no—
225C23—22no—
226C23—44no—
227C23—44no—
228C23—82no—
229C23—1271no—
230C23D4×D12711yescharacteristic
231Q8—2542no—
232Q8—2542no—
233D4—22no—
234D4—22no—
235D4—22no—
236D4—44no—
237D4—10162no—
238D4—5084no—
239D4—2542no—
240D4C2×C22⋊D12714yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—1271no—
244C2×C4—1271no—
245C2×C4—1271no—
246C2×C4—10162no—
247C2×C4—5084no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4D4×D12711yescharacteristic
253C2×C4C23×D12711yescharacteristic
254C8—21no—
255C8—2541no—
256C8—2541no—
257C8C2×C22⋊D12712yes—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—88no—
267C22—2541no—
268C22—2541no—
269C22—2541no—
270C22#1686122812yescomplemented
271C22C2×D4×D12711yescharacteristic
272C4—21no—
273C4—5082no—
274C4—5082no—
275C4—2541no—
276C4—2541no—
277C4—2541no—
278C4C22×C22⋊D12711yescharacteristic
279C4#1686125411yescharacteristic
280C2—22no—
281C2—44no—
282C2—82no—
283C2—5081no—
284C2D4×C22⋊D12711yescharacteristic
285C2C2×#1686122811yescharacteristic
286C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC22×C127C22×D4
hypercenterC22×D8D127
hypocenterC127#3280
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861228C2semidirect
2. #22550520C2semidirect
3. #22550259C2semidirect
4. #22550117C2semidirect
5. #22550109C2semidirect
6. #22550089C2semidirect
7. #22550027C2semidirect
8. #22550018C2semidirect
9. D8×Dic127C2semidirect
10. C8×C22⋊D127C2semidirect
11. C22×D8×C127C2semidirect
12. C2×#14523247C22semidirect
14. C2×Dic508C22semidirect
15. C2×#14523236C22semidirect
16. C2×#14523236C22semidirect
17. C2×C8×D127C22semidirect
18. #16861196C22semidirect
22. #16861180C22semidirect
24. D4×Dic127C22semidirect
27. #16861143C22semidirect
28. Dic127⋊C8C22semidirect
29. #16861134C22semidirect
30. #16861130C22semidirect
31. #16861130C22semidirect
33. (C2×D127)⋊C8C22semidirect
34. C8×Dic127C22semidirect
76. D8×C127D4semidirect
123. C22×C127#135semidirect
144. C127#3280semidirect
146. C22×D8D127semidirect
206. D8C22⋊D127semidirect
270. C22#16861228semidirect

Maximal subgroups

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

subgroupindex
C2×#168612282
#225505202
#225502592
#225501172
#225501092
#225500892
#225500272
#225500182
D8×Dic1272
C8×C22⋊D1272
C22×D8×C1272
#3280127