Finite Group Databasebeta

← #30650418

Subgroups — #30650418

Order 16256.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16861231C211yesmaximal, characteristic, complemented
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550523C212yesmaximal, complemented
4#22550259C211yesmaximal, characteristic, complemented
5#22550118C211yesmaximal, characteristic, complemented
6#22550090C212yesmaximal, complemented
7#22550028C212yesmaximal, complemented
8#22550019C212yesmaximal, complemented
9#22549958C211yesmaximal, characteristic, complemented
10C8×C22⋊D127C211yesmaximal, characteristic, complemented
11#466×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yescomplemented
13C2×D4×D127C2212yes—
14#16861231—24no—
15C2×Dic508C2211yescharacteristic, complemented
16C2×#14523237C2212yescomplemented
17C2×#14523236C2212yescomplemented
18C2×D1016C2211yescharacteristic
19C2×#14523233C2212yes—
20D8×D127—24no—
21C2×C8×D127C2211yescharacteristic
22#16861196C2212yescomplemented
23#16861191—22no—
24#16861189C2212yes—
25#16861180—22no—
26#16861177—22no—
27#16861171C2212yescomplemented
28C4×C22⋊D127C2211yescharacteristic
29C22⋊C4×D127—22no—
30Dic127⋊C8C2211yescharacteristic, complemented
31#16861132C2212yescomplemented
32#16861129C2212yescomplemented
33C22⋊(C127⋊C8)C2211yescharacteristic, complemented
34(C2×D127)⋊C8C2211yescharacteristic
35C8×Dic127C2211yescharacteristic, complemented
36C2×D8×C127C2211yescharacteristic
37C22×C8×C127C2211yescharacteristic, complemented
38#121×C127C2212yescomplemented
39#107×C127C2211yescharacteristic, complemented
40#102×C127C2212yescomplemented
41C23×D127—22no—
42#14523248—24no—
43#14523248—42no—
44C2×Dic254C2312yes—
45C2×D508C2312yes—
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127D411yescharacteristic
49D4×D127—24no—
50D4×D127—42no—
51C2×C4×D127—22no—
52C2×C4×D127—22no—
53C2×C4×D127C2311yescharacteristic
54Dic508—22no—
55#14523237—24no—
56#14523236—42no—
57D1016D414yescomplemented
58#14523233—24no—
59C2×C127⋊C8C2311yescharacteristic
60C2×C127⋊C8C2311yescharacteristic
61C8×D127—22no—
62C4⋊Dic127—22no—
63Dic127⋊C4D411yescharacteristic
64C22⋊Dic127—22no—
65C22⋊Dic127—22no—
66C22⋊Dic127D411yescharacteristic
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4D411yescharacteristic
70C4×Dic127C2311yescharacteristic
71C2×D4×C127—22no—
72C2×D4×C127C2312yes—
73C22×C4×C127C2311yescharacteristic
74D8×C127—22no—
75C2×C8×C127—22no—
76C2×C8×C127C2311yescharacteristic
77C2×C8×C127C2311yescharacteristic
78C4⋊C4×C127C2312yes—
79C22⋊C4×C127—22no—
80C22×D127—22no—
81C22×D127—44no—
82C22×D127—44no—
83C22×D127—82no—
84C22×D127C2×D411yescharacteristic
85Dic254—22no—
86Dic254—22no—
87D508—22no—
88D508C2×D414yes—
89C22⋊D127—22no—
90C22⋊D127—44no—
91C22⋊D127—44no—
92C2×Dic127—22no—
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—44no—
99C4×D127—82no—
100C127⋊C8—21no—
101C127⋊C8—21no—
102C23×C127—22no—
103C23×C127C2×D411yescharacteristic
104D4×C127—22no—
105D4×C127—22no—
106D4×C127—82no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—21no—
113C8×C127C2×D412yes—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127—21no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127C22×D411yescharacteristic
121C2×D127—21no—
122C2×D127—21no—
123C2×D127—24no—
124C2×D127—42no—
125C2×D127—42no—
126C2×D127—88no—
127C2×D127#13512yescomplemented
128C4×C127—21no—
129C4×C127—42no—
130C4×C127C22×D411yescharacteristic
131C4×C127H(5,𝔽2)11yescharacteristic
132Dic127—21no—
133Dic127—21no—
134Dic127—21no—
135Dic127—42no—
136C2×C127—41no—
137C2×C127—82no—
138C2×C127D4211yescharacteristic
139C2×C127C2×#13511yescharacteristic
140C2×C127#57611yescharacteristic
141D127—22no—
142D127—44no—
143#3280—1271nomaximal, Sylow
144C127#328011yescharacteristic, complemented, Sylow
145C2×#135—1271no—
146C22×D8—1271no—
147#546—1272no—
148#495—1271no—
149#467—1271no—
150#466D12711yescharacteristic, complemented
151#463—1272no—
152#450—1272no—
153#449—1272no—
154C4×D8—1271no—
155C8×D4—1271no—
156C2×#41—1272no—
157C22×D4—1272no—
158#135—2544no—
159C2×Q16—1271no—
160C2×SD16—1272no—
161C2×SD16—1272no—
162C2×D8—1272no—
163C2×D8—1271no—
164C2×D8—2544no—
165C2×D8C2×D12711yescharacteristic
166C22×C8—1271no—
167C22×C8C2×D12711yescharacteristic, complemented
168#124—1272no—
169#123—2542no—
170C22⋊Q8—1272no—
171#121—2542no—
172#121C2×D12712yescomplemented
173#120—2542no—
174C4×D4—1271no—
175C4×D4—1272no—
176C2×C22⋊C4—2542no—
177#107C2×D12711yescharacteristic, complemented
178C4⋊C8—1271no—
179#103—1272no—
180#102—1272no—
181#102C2×D12712yescomplemented
182C22⋊C8—1271no—
183C22⋊C8—1271no—
184C4×C8—1271no—
185C24—2542no—
186#41—5082no—
187#41—2544no—
188C2×Q8—1272no—
189C2×D4—22no—
190C2×D4—1271no—
191C2×D4—1272no—
192C2×D4—5082no—
193C2×D4—2542no—
194C2×D4—2544no—
195C2×D4—2542no—
196C2×D4C22×D12712yes—
197C22×C4—1271no—
198C22×C4—2542no—
199C22×C4—2542no—
200C22×C4C22×D12711yescharacteristic
201Q16—2542no—
202SD16—5082no—
203SD16—2544no—
204D8—22no—
205D8—1274no—
206D8—2544no—
207C2×C8—22no—
208C2×C8—1271no—
209C2×C8—1271no—
210C2×C8—2542no—
211C2×C8C22×D12711yescharacteristic
212C2×C8C22×D12711yescharacteristic
213C4⋊C4—1271no—
214C4⋊C4—2542no—
215C4⋊C4C22×D12712yes—
216C22⋊C4—22no—
217C22⋊C4—1271no—
218C22⋊C4—1271no—
219C22⋊C4—2542no—
220C22⋊C4—2542no—
221C22⋊C4—2542no—
222C22⋊C4—2542no—
223C42—1271no—
224C23—22no—
225C23—1271no—
226C23—10162no—
227C23—5084no—
228C23—5084no—
229C23—2542no—
230C23D4×D12711yescharacteristic
231Q8—2542no—
232Q8—2542no—
233D4—22no—
234D4—22no—
235D4—82no—
236D4—1274no—
237D4—5084no—
238D4—5084no—
239D4—2542no—
240D4—2542no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—1271no—
245C2×C4—1271no—
246C2×C4—1271no—
247C2×C4—10162no—
248C2×C4—5084no—
249C2×C4—2542no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4D4×D12711yescharacteristic
253C2×C4C23×D12711yescharacteristic
254C8—21no—
255C8—2541no—
256C8—2541no—
257C8D4×D12712yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—1272no—
265C22—10168no—
266C22—5082no—
267C22—5082no—
268C22—2544no—
269C22—2541no—
270C22—2541no—
271C22C2×D4×D12711yescharacteristic
272C4—21no—
273C4—42no—
274C4—5082no—
275C4—2541no—
276C4—2541no—
277C4—2541no—
278C4C2×D4×D12711yescharacteristic
279C4#1686125511yescharacteristic
280C2—41no—
281C2—82no—
282C2—5084no—
283C2—2542no—
284C2#2255051011yescharacteristic
285C2#135×D12711yescharacteristic
286C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#466×C127C2
socleC22×C127C22×D4
hypercenter#466D127
hypocenterC127#3280
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16861231C2semidirect
2. C2×D8×D127C2semidirect
3. #22550523C2semidirect
4. #22550259C2semidirect
5. #22550118C2semidirect
6. #22550090C2semidirect
7. #22550028C2semidirect
8. #22550019C2semidirect
9. #22549958C2semidirect
10. C8×C22⋊D127C2semidirect
11. #466×C127C2semidirect
12. C2×#14523248C22semidirect
15. C2×Dic508C22semidirect
16. C2×#14523237C22semidirect
17. C2×#14523236C22semidirect
22. #16861196C22semidirect
27. #16861171C22semidirect
30. Dic127⋊C8C22semidirect
31. #16861132C22semidirect
32. #16861129C22semidirect
33. C22⋊(C127⋊C8)C22semidirect
35. C8×Dic127C22semidirect
37. C22×C8×C127C22semidirect
38. #121×C127C22semidirect
39. #107×C127C22semidirect
40. #102×C127C22semidirect
57. D1016D4semidirect
127. C2×D127#135semidirect
144. C127#3280semidirect
150. #466D127semidirect
167. C22×C8C2×D127semidirect
172. #121C2×D127semidirect
177. #107C2×D127semidirect
181. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612312
C2×D8×D1272
#225505232
#225502592
#225501182
#225500902
#225500282
#225500192
#225499582
C8×C22⋊D1272
#466×C1272
#3280127