Finite Group Databasebeta

← #30650516

Subgroups — #30650516

Order 16256.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#22550684C212yesmaximal, complemented
2#22550678C211yesmaximal, characteristic, complemented
3#22550261C211yesmaximal, characteristic, complemented
4#22550148C211yesmaximal, characteristic, complemented
5#22550146C211yesmaximal, characteristic, complemented
6#22550141C212yesmaximal, complemented
7#22550052C212yesmaximal, complemented
8#22550049C212yesmaximal, complemented
9#22549993C211yesmaximal, characteristic, complemented
10#22549991C211yesmaximal, characteristic, complemented
11#578×C127C211yesmaximal, characteristic, complemented
12#16861255C2212yes—
13#16861240—22no—
14#16861240C2212yes—
15#16861236—22no—
16#16861234—22no—
17#16861234C2212yes—
18#16861234C2212yescomplemented
19#16861232—21no—
20#16861229—22no—
21#16861229C2211yescharacteristic
22C2×#14523235—22no—
23C2×#14523235C2212yescomplemented
24C2×D1016C2211yescharacteristic
25D8×D127—21no—
26#16861216C2212yescomplemented
27#16861215C2211yescharacteristic
28#16861198C2212yescomplemented
29#16861145C2211yescharacteristic, complemented
30#16861144C2211yescharacteristic, complemented
31#16861136C2211yescharacteristic, complemented
32#16861136C2212yescomplemented
33#16861135C2211yescharacteristic
34#16861128C2212yescomplemented
35#16861125C2212yescomplemented
36C8×Dic127C2211yescharacteristic, complemented
372−1+4×C127C2212yescomplemented
38#137×C127—22no—
39#137×C127C2212yescomplemented
40#135×C127—21no—
41#135×C127C2211yescharacteristic
42C2×Q16×C127—21no—
43C2×Q16×C127C2211yescharacteristic, complemented
44#131×C127C2211yescharacteristic, complemented
45#14523249—22no—
46#14523248—22no—
47#14523248C2311yescharacteristic
48C2×D508—22no—
49C2×D508C2312yes—
50D4×D127—22no—
51D4×D127—42no—
52#14523238—22no—
53#14523238D411yescharacteristic
54#14523236D411yescharacteristic
55#14523235—22no—
56#14523235—22no—
57#14523235—22no—
58#14523235—22no—
59#14523235—22no—
60#14523235D411yescharacteristic
61D1016—21no—
62D1016—21no—
63#14523233—22no—
64#14523233—22no—
65#14523233D411yescharacteristic
66#14523232—22no—
67#14523232C2311yescharacteristic
68#14523232C2312yes—
69#14523231—21no—
70C2×C127⋊C8—22no—
71C2×C127⋊C8C2311yescharacteristic
72C8×D127—21no—
73(C2×D127)⋊C4—42no—
74C4×Dic127C2311yescharacteristic
75#41×C127—22no—
76#41×C127—22no—
77#41×C127—42no—
78#41×C127C2312yes—
79#41×C127C2311yescharacteristic
80C2×Q8×C127—22no—
81C2×Q8×C127—22no—
82C2×Q8×C127C2312yes—
83Q16×C127—22no—
84Q16×C127—21no—
85Q16×C127—21no—
86Q16×C127D411yescharacteristic
87SD16×C127—22no—
88SD16×C127D412yes—
89D8×C127D411yescharacteristic
90M16×C127—21no—
91M16×C127C2311yescharacteristic
92C2×C8×C127—21no—
93C2×C8×C127C2311yescharacteristic
94C22×D127—22no—
95C22×D127—42no—
96Dic254C2×D411yescharacteristic
97D508—22no—
98D508—22no—
99D508—42no—
100D508C2×D411yescharacteristic
101C22⋊D127—21no—
102C22⋊D127—42no—
103C2×Dic127—21no—
104C4×D127—21no—
105C4×D127—42no—
106C127⋊C8—21no—
107C127⋊C8—22no—
108C127⋊C8C2×D411yescharacteristic
109C127⋊C8C2×D411yescharacteristic
110Q8×C127—22no—
111Q8×C127—22no—
112Q8×C127—42no—
113Q8×C127C2×D411yescharacteristic
114Q8×C127C2×D412yes—
115D4×C127—21no—
116D4×C127—22no—
117D4×C127—42no—
118D4×C127C2×D411yescharacteristic
119D4×C127C2×D412yes—
120C2×C4×C127—21no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127C2411yescharacteristic
126C8×C127—21no—
127C8×C127C2×D411yescharacteristic
128C8×C127C2×D411yescharacteristic
129C22×C127—21no—
130C22×C127—22no—
131C22×C127H(5,𝔽2)11yescharacteristic
132C2×D127—21no—
133C2×D127—42no—
134C2×D127—82no—
135C2×D127—162no—
136C4×C127—21no—
137C4×C127—22no—
138C4×C127—42no—
139C4×C127C22×D411yescharacteristic
140C4×C127C22×D411yescharacteristic
141Dic127—21no—
142Dic127—41no—
143C2×C127—21no—
144C2×C127—42no—
145C2×C127—41no—
146C2×C127D4211yescharacteristic
147D127—41no—
148D127—82no—
149#3290—1271nomaximal, Sylow
150C127#329011yescharacteristic, complemented, Sylow
151#578D12711yescharacteristic, complemented
152#577—1272no—
153#576—1271no—
154#495—1271no—
155#473—1271no—
156#472—1271no—
157#471—1272no—
158#455—1272no—
159#454—1272no—
160#443—1271no—
161#443—1271no—
1622−1+4C2×D12712yescomplemented
163H(5,𝔽2)—1272no—
164#137—22no—
165#137—1272no—
166#137—2542no—
167#137C2×D12712yescomplemented
168AGL(1,ℤ/8)—1272no—
169AGL(1,ℤ/8)—1272no—
170AGL(1,ℤ/8)—2542no—
171AGL(1,ℤ/8)—2542no—
172#135—21no—
173#135—1271no—
174#135—2542no—
175#135—2541no—
176#135C2×D12711yescharacteristic
177C2×Q16—21no—
178C2×Q16C2×D12711yescharacteristic, complemented
179C2×SD16—1272no—
180C2×SD16—2542no—
181C2×D8—1271no—
182C2×D8—2541no—
183#131—1272no—
184#131—1271no—
185#131C2×D12711yescharacteristic, complemented
186#124—1272no—
187#108—1271no—
188#108—1271no—
189#104—1271no—
190#104—1272no—
191#104—1271no—
192#101—1272no—
193#100—1272no—
194C4×C8—1271no—
195#41—22no—
196#41—22no—
197#41—42no—
198#41—1271no—
199#41—2542no—
200#41—2542no—
201#41C22×D12711yescharacteristic
202#41C22×D12712yes—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C22×D12712yes—
206C2×D4—1272no—
207C2×D4—5082no—
208C2×D4—2542no—
209C2×D4—2542no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—1271no—
214Q16—2542no—
215Q16C22⋊D12711yescharacteristic
216SD16—22no—
217SD16—1271no—
218SD16—1271no—
219SD16—2542no—
220SD16—2542no—
221SD16—2542no—
222SD16—2542no—
223SD16—2542no—
224SD16C22⋊D12712yes—
225D8—1271no—
226D8—2542no—
227D8—2542no—
228D8—2541no—
229D8—2541no—
230D8C22⋊D12711yescharacteristic
231M16—21no—
232M16—1272no—
233M16—1271no—
234M16—2542no—
235M16—2541no—
236M16C22×D12711yescharacteristic
237C2×C8—21no—
238C2×C8—1271no—
239C2×C8—2542no—
240C2×C8—2541no—
241C2×C8C22×D12711yescharacteristic
242C22⋊C4—5082no—
243C42—1271no—
244C23—5082no—
245C23—2542no—
246Q8—22no—
247Q8—22no—
248Q8—42no—
249Q8—1271no—
250Q8D4×D12711yescharacteristic
251Q8C2×C22⋊D12712yes—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—1271no—
256D4—5082no—
257D4—5082no—
258D4—2541no—
259D4—2542no—
260D4—2542no—
261D4D4×D12711yescharacteristic
262D4C2×C22⋊D12712yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—5082no—
269C2×C4—2541no—
270C2×C4—2541no—
271C2×C4C23×D12711yescharacteristic
272C8—21no—
273C8—1271no—
274C8—1271no—
275C8—2542no—
276C8—2541no—
277C8C2×C22⋊D12711yescharacteristic
278C8C2×C22⋊D12711yescharacteristic
279C22—21no—
280C22—22no—
281C22—20322no—
282C22—10162no—
283C22—5082no—
284C22—2541no—
285C22#1686125411yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—5081no—
290C4—2541no—
291C4C2×D4×D12711yescharacteristic
292C4C22×C22⋊D12711yescharacteristic
293C2—21no—
294C2—42no—
295C2—41no—
296C2—10162no—
297C2—5081no—
298C2D4×C22⋊D12711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#578×C127C2
socleC2×C127D42
hypercenter#578D127
hypocenterC127#3290
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #22550684C2semidirect
2. #22550678C2semidirect
3. #22550261C2semidirect
4. #22550148C2semidirect
5. #22550146C2semidirect
6. #22550141C2semidirect
7. #22550052C2semidirect
8. #22550049C2semidirect
9. #22549993C2semidirect
10. #22549991C2semidirect
11. #578×C127C2semidirect
18. #16861234C22semidirect
23. C2×#14523235C22semidirect
26. #16861216C22semidirect
28. #16861198C22semidirect
29. #16861145C22semidirect
30. #16861144C22semidirect
31. #16861136C22semidirect
32. #16861136C22semidirect
34. #16861128C22semidirect
35. #16861125C22semidirect
36. C8×Dic127C22semidirect
37. 2−1+4×C127C22semidirect
39. #137×C127C22semidirect
43. C2×Q16×C127C22semidirect
44. #131×C127C22semidirect
150. C127#3290semidirect
151. #578D127semidirect
162. 2−1+4C2×D127semidirect
167. #137C2×D127semidirect
178. C2×Q16C2×D127semidirect
185. #131C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506842
#225506782
#225502612
#225501482
#225501462
#225501412
#225500522
#225500492
#225499932
#225499912
#578×C1272
#3290127