Finite Group Databasebeta

← #30649208

Subgroups — #30649208

Order 16256.

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

normal subgroups only

This group has 564 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
1C2×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic, complemented
3C2×#16861229C211yesmaximal, characteristic, complemented
4C22×#14523235C211yesmaximal, characteristic, complemented
5#22550445C211yesmaximal, characteristic, complemented
6#22550268C212yesmaximal, complemented
7#22550266C212yesmaximal, complemented
8#22550258C212yesmaximal, complemented
9#22550250C212yesmaximal, complemented
10#22549832C211yesmaximal, characteristic, complemented
11#536×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14#16861240—24no—
15#16861234—24no—
16#16861229—24no—
17C2×#14523238C2211yescharacteristic, complemented
18C2×#14523238C2212yescomplemented
19C2×#14523236C2211yescharacteristic, complemented
20C2×#14523235—24no—
21C2×#14523235C2212yes—
22C2×#14523235C2212yes—
23C2×#14523235C2212yes—
24C2×#14523235C2211yescharacteristic, complemented
25C2×#14523235C2212yescomplemented
26C2×#14523233C2212yes—
27C2×#14523233C2211yescharacteristic, complemented
28C2×#14523232C2211yescharacteristic
29C22×C127⋊C8C2211yescharacteristic
30#16861206C2212yes—
31#16861197C2212yes—
32#16861184—22no—
33#16861179—22no—
34#16861120C2212yescomplemented
35C4×C127⋊C8C2212yescomplemented
36C2×#41×C127C2211yescharacteristic
37C22×Q8×C127C2211yescharacteristic
38C4⋊Q8×C127C2212yescomplemented
39#124×C127C2212yescomplemented
40#123×C127—22no—
41C22⋊Q8×C127—22no—
42#117×C127C2211yescharacteristic
43C23×D127—21no—
44#14523248—22no—
45#14523248—22no—
46C2×Dic254C2311yescharacteristic
47C2×D508—22no—
48C2×D508—22no—
49C2×D508—21no—
50C2×D508—24no—
51C2×D508C2312yes—
52C2×D508C2311yescharacteristic
53C2×C22⋊D127—21no—
54C2×C4×D127—21no—
55#14523238—22no—
56#14523238—24no—
57#14523236—22no—
58#14523235—24no—
59#14523235—24no—
60#14523235—24no—
61#14523235—22no—
62#14523235—24no—
63#14523233—24no—
64#14523233—22no—
65#14523232D414yescomplemented
66C2×C127⋊C8D414yescomplemented
67C2×C127⋊C8C2311yescharacteristic
68C2×C127⋊C8C2311yescharacteristic
69C2×C127⋊C8C2312yes—
70(C2×D127)⋊C4—24no—
71#41×C127—22no—
72#41×C127—22no—
73C2×Q8×C127—21no—
74C2×Q8×C127—22no—
75C2×Q8×C127—22no—
76C2×Q8×C127C2312yes—
77C2×Q8×C127C2311yescharacteristic
78C2×D4×C127—21no—
79C2×D4×C127C2311yescharacteristic
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127C2311yescharacteristic
83C4⋊C4×C127—24no—
84C4⋊C4×C127D412yes—
85C22⋊C4×C127—24no—
86C22⋊C4×C127D412yes—
87C42×C127C2312yes—
88C22×D127—22no—
89C22×D127—21no—
90C22×D127—42no—
91C22×D127—82no—
92Dic254—21no—
93Dic254—21no—
94D508—22no—
95D508—21no—
96D508—22no—
97D508—21no—
98D508—42no—
99D508—84no—
100C22⋊D127—42no—
101C2×Dic127—21no—
102C4×D127—42no—
103C127⋊C8C2×D412yes—
104C127⋊C8C2×D412yes—
105C127⋊C8C2×D414yes—
106C23×C127—21no—
107C23×C127C2×D411yescharacteristic
108Q8×C127—22no—
109Q8×C127—22no—
110Q8×C127—21no—
111Q8×C127—21no—
112Q8×C127—42no—
113D4×C127—21no—
114D4×C127—21no—
115D4×C127—42no—
116C2×C4×C127—22no—
117C2×C4×C127—21no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D414yes—
123C2×C4×C127C2×D411yescharacteristic
124C2×C4×C127C2411yescharacteristic
125C22×C127—22no—
126C22×C127—41no—
127C22×C127—42no—
128C22×C127#12712yes—
129C22×C127C22×D411yescharacteristic
130C2×D127—41no—
131C2×D127—42no—
132C2×D127—42no—
133C2×D127—44no—
134C2×D127—84no—
135C4×C127—24no—
136C4×C127—41no—
137C4×C127—42no—
138C4×C127#12712yes—
139C4×C127C22×D411yescharacteristic
140C4×C127C22×D411yescharacteristic
141Dic127—41no—
142C2×C127—22no—
143C2×C127—81no—
144C2×C127C2×#12711yescharacteristic
145C2×C127#57712yes—
146D127—81no—
147D127—82no—
148#3154—1271nomaximal, Sylow
149C127#315411yescharacteristic, complemented, Sylow
150C2×#137—1271no—
151C2×AGL(1,ℤ/8)—1271no—
152C2×#135—1271no—
153C22×SD16—1271no—
154#536D12711yescharacteristic, complemented
155#535—1271no—
156#497—1272no—
157#496—1272no—
158#495—1272no—
159#492—1272no—
160#405—1271no—
161C2×#41—1271no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8C2×D12711yescharacteristic
164C22×D4—1271no—
165#137—2544no—
166AGL(1,ℤ/8)—2544no—
167#135—2544no—
168C2×Q16—1271no—
169C2×Q16—1272no—
170C2×SD16—1272no—
171C2×SD16—1272no—
172C2×SD16—1272no—
173C2×SD16—1271no—
174C2×SD16—1272no—
175C2×SD16—1271no—
176C2×SD16—2544no—
177C2×D8—1272no—
178C2×D8—1271no—
179C2×M16—1271no—
180C22×C8—1271no—
181C4⋊Q8C2×D12712yescomplemented
182#127—1272no—
183#124—1272no—
184#124C2×D12712yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—2542no—
188#120—2542no—
189#117C2×D12711yescharacteristic
190#97—1272no—
191C4×C8—1272no—
192C24—2541no—
193#41—22no—
194#41—22no—
195#41—2542no—
196#41—2542no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1271no—
201C2×Q8C22×D12712yes—
202C2×Q8C22×D12711yescharacteristic
203C2×D4—21no—
204C2×D4—1272no—
205C2×D4—1271no—
206C2×D4—2541no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4—2544no—
210C2×D4—2541no—
211C2×D4C22×D12711yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—2541no—
215C22×C4C22×D12711yescharacteristic
216Q16—2542no—
217Q16—2544no—
218SD16—2542no—
219SD16—2544no—
220SD16—2544no—
221SD16—2542no—
222SD16—2544no—
223SD16—2544no—
224D8—2542no—
225D8—2544no—
226M16—1274no—
227C2×C8—1274no—
228C2×C8—1271no—
229C2×C8—1272no—
230C2×C8—1271no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D12712yes—
233C22⋊C4—24no—
234C22⋊C4—2544no—
235C22⋊C4C22⋊D12712yes—
236C42C22×D12712yes—
237C23—21no—
238C23—10162no—
239C23—5082no—
240C23—2542no—
241C23—2541no—
242C23C2×C22⋊D12711yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—2541no—
249Q8—2541no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—10164no—
254D4—5082no—
255D4—5082no—
256D4—2542no—
257D4—2541no—
258D4—2542no—
259D4—2541no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—5082no—
265C2×C4—2541no—
266C2×C4D4×D12712yes—
267C2×C4D4×D12712yes—
268C2×C4C2×C22⋊D12714yes—
269C2×C4C2×C22⋊D12711yescharacteristic
270C2×C4C23×D12711yescharacteristic
271C8—1272no—
272C8—1272no—
273C8—1274no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—10164no—
278C22—5081no—
279C22—5082no—
280C22—5082no—
281C22—5084no—
282C22#1686120512yes—
283C22C22×C22⋊D12711yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—5081no—
288C4#1686120512yes—
289C4C2×D4×D12711yescharacteristic
290C4C2×D4×D12711yescharacteristic
291C2—22no—
292C2—81no—
293C2—10162no—
294C2—10161no—
295C2C2×#1686120511yescharacteristic
296C2#2255068412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
hypercenter#536D127
hypocenterC127#3154
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C2×#16861234C2semidirect
3. C2×#16861229C2semidirect
4. C22×#14523235C2semidirect
5. #22550445C2semidirect
6. #22550268C2semidirect
7. #22550266C2semidirect
8. #22550258C2semidirect
9. #22550250C2semidirect
10. #22549832C2semidirect
11. #536×C127C2semidirect
17. C2×#14523238C22semidirect
18. C2×#14523238C22semidirect
19. C2×#14523236C22semidirect
24. C2×#14523235C22semidirect
25. C2×#14523235C22semidirect
27. C2×#14523233C22semidirect
34. #16861120C22semidirect
35. C4×C127⋊C8C22semidirect
38. C4⋊Q8×C127C22semidirect
39. #124×C127C22semidirect
65. #14523232D4semidirect
66. C2×C127⋊C8D4semidirect
149. C127#3154semidirect
154. #536D127semidirect
181. C4⋊Q8C2×D127semidirect
184. #124C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612342
C2×#168612292
C22×#145232352
#225504452
#225502682
#225502662
#225502582
#225502502
#225498322
#536×C1272
#3154127