Finite Group Databasebeta

← #30649202

Subgroups — #30649202

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×#16861239C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C2×#16861229C211yesmaximal, characteristic, complemented
4C22×#14523236C211yesmaximal, characteristic, complemented
5#22550452C211yesmaximal, characteristic, complemented
6#22550271C212yesmaximal, complemented
7#22550263C212yesmaximal, complemented
8#22550258C212yesmaximal, complemented
9#22550248C212yesmaximal, complemented
10#22549832C211yesmaximal, characteristic, complemented
11#535×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14#16861239—24no—
15#16861235—24no—
16#16861229—24no—
17C2×#14523238C2211yescharacteristic, complemented
18C2×#14523238C2212yescomplemented
19C2×#14523236—24no—
20C2×#14523236C2212yes—
21C2×#14523236C2212yes—
22C2×#14523236C2212yes—
23C2×#14523236C2211yescharacteristic, complemented
24C2×#14523236C2212yescomplemented
25C2×#14523235C2211yescharacteristic, complemented
26C2×#14523233C2212yes—
27C2×#14523233C2211yescharacteristic, complemented
28C2×#14523232C2211yescharacteristic
29C22×C127⋊C8C2211yescharacteristic
30#16861209C2212yescomplemented
31#16861197C2212yescomplemented
32#16861193—22no—
33#16861189—22no—
34#16861120C2212yescomplemented
35C4×C127⋊C8C2212yescomplemented
36C2×#41×C127C2211yescharacteristic
37C22×D4×C127C2211yescharacteristic
38#127×C127C2212yes—
39#124×C127C2212yes—
40#121×C127—22no—
41#120×C127—22no—
42#117×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254—22no—
48C2×Dic254C2312yes—
49C2×Dic254C2311yescharacteristic
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—21no—
52C22×Dic127—21no—
53C2×C4×D127—21no—
54#14523238—22no—
55#14523238—24no—
56#14523236—24no—
57#14523236—24no—
58#14523236—24no—
59#14523236—22no—
60#14523236—24no—
61#14523235—22no—
62#14523233—24no—
63#14523233—22no—
64#14523232D414yescomplemented
65C2×C127⋊C8D414yescomplemented
66C2×C127⋊C8C2311yescharacteristic
67C2×C127⋊C8C2311yescharacteristic
68C2×C127⋊C8C2312yes—
69C4⋊Dic127—24no—
70(C2×D127)⋊C4—24no—
71C24×C127—21no—
72#41×C127—22no—
73#41×C127—22no—
74C2×Q8×C127C2311yescharacteristic
75C2×D4×C127—21no—
76C2×D4×C127—22no—
77C2×D4×C127—22no—
78C2×D4×C127—21no—
79C2×D4×C127—24no—
80C2×D4×C127C2312yes—
81C2×D4×C127C2311yescharacteristic
82C22×C4×C127—21no—
83C22×C4×C127C2311yescharacteristic
84C4⋊C4×C127D412yes—
85C22⋊C4×C127—24no—
86C22⋊C4×C127D412yes—
87C42×C127C2312yes—
88C22×D127—21no—
89Dic254—22no—
90Dic254—21no—
91Dic254—22no—
92Dic254—21no—
93Dic254—42no—
94D508—21no—
95D508—21no—
96C22⋊D127—42no—
97C2×Dic127—22no—
98C2×Dic127—21no—
99C2×Dic127—42no—
100C4×D127—42no—
101C127⋊C8C2×D412yes—
102C127⋊C8C2×D412yes—
103C127⋊C8C2×D414yes—
104C23×C127—22no—
105C23×C127—21no—
106C23×C127—42no—
107C23×C127—82no—
108C23×C127C2×D411yescharacteristic
109Q8×C127—21no—
110Q8×C127—21no—
111D4×C127—22no—
112D4×C127—22no—
113D4×C127—21no—
114D4×C127—21no—
115D4×C127—42no—
116D4×C127—42no—
117D4×C127—84no—
118C2×C4×C127—21no—
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—42no—
129C22×C127—44no—
130C22×C127—84no—
131C22×C127#12712yes—
132C22×C127C22×D411yescharacteristic
133C2×D127—41no—
134C2×D127—42no—
135C4×C127—24no—
136C4×C127—41no—
137C4×C127#12712yes—
138C4×C127C22×D411yescharacteristic
139C4×C127C22×D411yescharacteristic
140Dic127—41no—
141Dic127—42no—
142C2×C127—22no—
143C2×C127—81no—
144C2×C127—82no—
145C2×C127C2×#12711yescharacteristic
146C2×C127#57712yes—
147D127—81no—
148#3154—1271nomaximal, Sylow
149C127#315411yescharacteristic, complemented, Sylow
150C2×#137—1271no—
151C2×AGL(1,ℤ/8)—1271no—
152C2×#135—1271no—
153C22×SD16—1271no—
154#536—1271no—
155#535D12711yescharacteristic, complemented
156#497—1272no—
157#496—1272no—
158#495—1272no—
159#492—1272no—
160#405—1271no—
161C2×#41—1271no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8—1271no—
164C22×D4C2×D12711yescharacteristic
165#137—2544no—
166AGL(1,ℤ/8)—2544no—
167#135—2544no—
168C2×Q16—1271no—
169C2×Q16—1272no—
170C2×SD16—1271no—
171C2×SD16—1272no—
172C2×SD16—1272no—
173C2×SD16—1272no—
174C2×SD16—1271no—
175C2×SD16—1272no—
176C2×SD16—2544no—
177C2×D8—1272no—
178C2×D8—1271no—
179C2×M16—1271no—
180C22×C8—1271no—
181C4⋊Q8—1272no—
182#127C2×D12712yes—
183#124—1272no—
184#124C2×D12712yes—
185#123—2542no—
186C22⋊Q8—2542no—
187#121—22no—
188#120—22no—
189#117C2×D12711yescharacteristic
190#97—1272no—
191C4×C8—1272no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—2542no—
196#41—2542no—
197C2×Q8—1272no—
198C2×Q8—1271no—
199C2×Q8—2542no—
200C2×Q8—2542no—
201C2×Q8—2541no—
202C2×Q8C22×D12711yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—24no—
208C2×D4—1271no—
209C2×D4—2541no—
210C2×D4C22×D12712yes—
211C2×D4C22×D12711yescharacteristic
212C22×C4—21no—
213C22×C4—2541no—
214C22×C4—2541no—
215C22×C4C22×D12711yescharacteristic
216Q16—2542no—
217Q16—2544no—
218SD16—2544no—
219SD16—2544no—
220SD16—2542no—
221SD16—2544no—
222SD16—2544no—
223SD16—2542no—
224D8—2542no—
225D8—2544no—
226M16—1274no—
227C2×C8—1274no—
228C2×C8—1271no—
229C2×C8—1272no—
230C2×C8—1271no—
231C4⋊C4—2544no—
232C4⋊C4C22⋊D12712yes—
233C22⋊C4—24no—
234C22⋊C4—2544no—
235C22⋊C4C22⋊D12712yes—
236C42C22×D12712yes—
237C23—22no—
238C23—21no—
239C23—42no—
240C23—82no—
241C23—2541no—
242C23C2×C22⋊D12711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—5082no—
246Q8—2542no—
247Q8—2541no—
248Q8—2542no—
249Q8—2541no—
250D4—22no—
251D4—21no—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—84no—
257D4—5082no—
258D4—2541no—
259D4—2541no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—5082no—
263C2×C4—5082no—
264C2×C4—2542no—
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—42no—
278C22—44no—
279C22—84no—
280C22—5081no—
281C22—5082no—
282C22#1686120512yes—
283C22C22×C22⋊D12711yescharacteristic
284C4—24no—
285C4—41no—
286C4—5081no—
287C4—5082no—
288C4#1686120512yes—
289C4C2×D4×D12711yescharacteristic
290C4C2×D4×D12711yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—10161no—
295C2C2×#1686120511yescharacteristic
296C2#2255068712yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#535×C127C2
socleC22×C127C22×D4
hypercenter#535D127
hypocenterC127#3154
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861235C2semidirect
3. C2×#16861229C2semidirect
4. C22×#14523236C2semidirect
5. #22550452C2semidirect
6. #22550271C2semidirect
7. #22550263C2semidirect
8. #22550258C2semidirect
9. #22550248C2semidirect
10. #22549832C2semidirect
11. #535×C127C2semidirect
17. C2×#14523238C22semidirect
18. C2×#14523238C22semidirect
23. C2×#14523236C22semidirect
24. C2×#14523236C22semidirect
25. C2×#14523235C22semidirect
27. C2×#14523233C22semidirect
30. #16861209C22semidirect
31. #16861197C22semidirect
34. #16861120C22semidirect
35. C4×C127⋊C8C22semidirect
64. #14523232D4semidirect
65. C2×C127⋊C8D4semidirect
149. C127#3154semidirect
155. #535D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612352
C2×#168612292
C22×#145232362
#225504522
#225502712
#225502632
#225502582
#225502482
#225498322
#535×C1272
#3154127