Finite Group Databasebeta

← #30647976

Subgroups — #30647976

Order 16256.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#14523249C211yesmaximal, characteristic, complemented
2C2×#16861240C211yesmaximal, characteristic, complemented
3C2×#16861237C211yesmaximal, characteristic, complemented
4#22550388C211yesmaximal, characteristic, complemented
5#22550041C214yesmaximal, complemented
6#22550024C214yesmaximal, complemented
7#22549880C211yesmaximal, characteristic, complemented
8#22549839C211yesmaximal, characteristic, complemented
9#419×C127C211yesmaximal, characteristic, complemented
10C2×#14523249—24no—
11C2×#14523249—22no—
12C2×#14523249—24no—
13C2×#14523249C2214yes—
14C2×#14523248C2211yescharacteristic
15C22×D508—21no—
16C22×D508C2211yescharacteristic
17C22×C4×D127—21no—
18C22×C4×D127C2211yescharacteristic
19#16861240—24no—
20#16861237—24no—
21C2×#14523238C2212yescomplemented
22C2×#14523237C2212yescomplemented
23C2×#14523235C2212yescomplemented
24C2×D1016C2212yescomplemented
25C2×#14523232C2211yescharacteristic
26#16861193—21no—
27#16861189C2212yescomplemented
28#16861184C2212yescomplemented
29#16861179—21no—
30C4×D508—22no—
31#16861129C2214yescomplemented
32(C2×D127)⋊C8C2214yescomplemented
33C22×Q8×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic
35#117×C127C2211yescharacteristic
36#103×C127C2214yescomplemented
37C23×D127—21no—
38C23×D127D411yescharacteristic
39#14523249—216no—
40#14523249—28no—
41#14523249—28no—
42#14523248—22no—
43#14523248—22no—
44C2×Dic254C2311yescharacteristic
45C2×D508—28no—
46C2×D508—24no—
47C2×D508—22no—
48C2×D508—21no—
49C2×D508—22no—
50C2×D508D414yescomplemented
51C2×D508C2311yescharacteristic
52C2×D508C2312yes—
53C2×C22⋊D127—21no—
54C22×Dic127D411yescharacteristic
55C2×C4×D127—28no—
56C2×C4×D127—24no—
57C2×C4×D127—22no—
58C2×C4×D127—22no—
59C2×C4×D127—21no—
60C2×C4×D127D412yes—
61C2×C4×D127C2312yes—
62#14523238—24no—
63#14523237—24no—
64#14523235—24no—
65D1016—24no—
66#14523232—22no—
67C2×C127⋊C8C2312yes—
68C4⋊Dic127—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71C2×Q8×C127—22no—
72C2×Q8×C127—21no—
73C2×Q8×C127D414yescomplemented
74C2×Q8×C127C2312yes—
75C22×C4×C127—21no—
76C22×C4×C127C2311yescharacteristic
77M16×C127—22no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C42×C127—21no—
82C22×D127—22no—
83C22×D127—24no—
84C22×D127—22no—
85C22×D127—21no—
86C22×D127—42no—
87C22×D127—44no—
88C22×D127C2×D412yes—
89Dic254—21no—
90Dic254—21no—
91D508—216no—
92D508—24no—
93D508—22no—
94D508—21no—
95D508—21no—
96D508—44no—
97D508C2×D414yes—
98C22⋊D127—42no—
99C2×Dic127—22no—
100C2×Dic127—21no—
101C2×Dic127C2×D412yes—
102C4×D127—216no—
103C4×D127—24no—
104C4×D127—22no—
105C4×D127—22no—
106C4×D127—42no—
107C127⋊C8—22no—
108C23×C127C2×D411yescharacteristic
109Q8×C127—24no—
110Q8×C127—22no—
111Q8×C127C2×D414yes—
112C2×C4×C127—24no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2411yescharacteristic
119C8×C127—22no—
120C22×C127—22no—
121C22×C127#12012yes—
122C22×C127C22×D411yescharacteristic
123C2×D127—22no—
124C2×D127—24no—
125C2×D127—22no—
126C2×D127—22no—
127C2×D127—44no—
128C2×D127—48no—
129C2×D127—44no—
130C2×D127—41no—
131C2×D127—42no—
132C4×C127—24no—
133C4×C127—42no—
134C4×C127#12012yes—
135C4×C127C22×D411yescharacteristic
136C4×C127C22×D411yescharacteristic
137Dic127—22no—
138Dic127—41no—
139C2×C127—22no—
140C2×C127C2×#12011yescharacteristic
141C2×C127#57512yes—
142D127—42no—
143D127—44no—
144D127—81no—
145#3005—1271nomaximal, Sylow
146C127#300511yescharacteristic, complemented, Sylow
147C22×#41—1271no—
148C2×#137—1271no—
149C2×AGL(1,ℤ/8)—1271no—
150#525—1271no—
151#452—1274no—
152#449—1274no—
153#419D12711yescharacteristic, complemented
154#418—1271no—
155#407—1271no—
156C2×#41—1274no—
157C2×#41—1271no—
158C2×#41—2542no—
159C2×#41—2544no—
160C2×#41—2544no—
161C22×Q8C2×D12711yescharacteristic
162C22×D4—1271no—
163C22×D4—2541no—
164C23×C4—1271no—
165C23×C4—2541no—
166#137—2544no—
167AGL(1,ℤ/8)—2544no—
168C2×Q16—1272no—
169C2×SD16—1272no—
170C2×SD16—1272no—
171C2×D8—1272no—
172C2×M16—1271no—
173C2×M16C2×D12711yescharacteristic
174#123—2541no—
175C22⋊Q8—1272no—
176#121—1272no—
177#120—2541no—
178C4×D4—2542no—
179#117C2×D12711yescharacteristic
180#103C2×D12714yescomplemented
181#102—1274no—
182C22⋊C8—1274no—
183C24—1271no—
184C24—2541no—
185#41—2542no—
186#41—2542no—
187#41—2548no—
188#41—25416no—
189#41—2548no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—1271no—
193C2×Q8D50814yescomplemented
194C2×Q8C22×D12712yes—
195C2×D4—1274no—
196C2×D4—1272no—
197C2×D4—1271no—
198C2×D4—2542no—
199C2×D4—2544no—
200C2×D4—2548no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2542no—
204C22×C4—21no—
205C22×C4—1271no—
206C22×C4—1272no—
207C22×C4—1272no—
208C22×C4—2544no—
209C22×C4—2548no—
210C22×C4—2541no—
211C22×C4—2542no—
212C22×C4—2542no—
213C22×C4C22×D12711yescharacteristic
214Q16—2544no—
215SD16—2544no—
216SD16—2544no—
217D8—2544no—
218M16—22no—
219M16—2542no—
220C2×C8—1272no—
221C2×C8C22×D12712yes—
222C4⋊C4—2542no—
223C4⋊C4C22×D12712yes—
224C22⋊C4—21no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C42—21no—
228C23—1272no—
229C23—5084no—
230C23—5082no—
231C23—2541no—
232C23—2542no—
233C23—2544no—
234C23—2542no—
235C23D4×D12711yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—2541no—
239Q8—2541no—
240Q8C2×D50814yes—
241D4—1274no—
242D4—5084no—
243D4—5082no—
244D4—25416no—
245D4—2541no—
246D4—2541no—
247D4—2542no—
248D4—2544no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—1272no—
253C2×C4—5082no—
254C2×C4—25416no—
255C2×C4—2541no—
256C2×C4—2542no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4—2544no—
260C2×C4D4×D12712yes—
261C2×C4D4×D12711yescharacteristic
262C2×C4C2×D50812yes—
263C2×C4C23×D12711yescharacteristic
264C8—22no—
265C8—2542no—
266C22—22no—
267C22—5088no—
268C22—5084no—
269C22—5081no—
270C22—5082no—
271C22—5084no—
272C22—2544no—
273C22—2542no—
274C22—2542no—
275C22—2542no—
276C22#1686117912yes—
277C22C2×D4×D12711yescharacteristic
278C4—24no—
279C4—42no—
280C4—5081no—
281C4—2542no—
282C4#1686117912yes—
283C4C2×D4×D12711yescharacteristic
284C4C22×D50811yescharacteristic
285C2—22no—
286C2—10161no—
287C2—5084no—
288C2—5082no—
289C2C2×#1686117911yescharacteristic
290C2#2255067512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
hypercenter#419D127
hypocenterC127#3005
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14523249C2semidirect
2. C2×#16861240C2semidirect
3. C2×#16861237C2semidirect
4. #22550388C2semidirect
5. #22550041C2semidirect
6. #22550024C2semidirect
7. #22549880C2semidirect
8. #22549839C2semidirect
9. #419×C127C2semidirect
21. C2×#14523238C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523235C22semidirect
24. C2×D1016C22semidirect
27. #16861189C22semidirect
28. #16861184C22semidirect
31. #16861129C22semidirect
32. (C2×D127)⋊C8C22semidirect
36. #103×C127C22semidirect
50. C2×D508D4semidirect
73. C2×Q8×C127D4semidirect
146. C127#3005semidirect
153. #419D127semidirect
180. #103C2×D127semidirect
193. C2×Q8D508semidirect

Maximal subgroups

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

subgroupindex
C22×#145232492
C2×#168612402
C2×#168612372
#225503882
#225500412
#225500242
#225498802
#225498392
#419×C1272
#3005127