Finite Group Databasebeta

← #30651870

Subgroups — #30651870

Order 16256.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#22550678C211yesmaximal, characteristic, complemented
2#22550677C211yesmaximal, characteristic, complemented
3#22550314C211yesmaximal, characteristic, complemented
4#22550313C212yesmaximal, complemented
5#22550310C212yesmaximal, complemented
6#22550307C212yesmaximal, complemented
7#22550306C211yesmaximal, characteristic, complemented
8C2×#16861150C211yesmaximal, characteristic, complemented
9D16×D127C212yesmaximal, complemented
10#22550294C211yesmaximal, characteristic, complemented
11#509×C127C211yesmaximal, characteristic, complemented
12#16861255—21no—
13#16861254—21no—
14#16861236—22no—
15#16861235—21no—
16#16861234—21no—
17#16861233—22no—
18#16861232C2212yescomplemented
19#16861231C2211yescharacteristic, complemented
20#16861228C2212yescomplemented
21C2×D1016C2211yescharacteristic, complemented
22D8×D127C2212yescomplemented
23D8×D127C2212yescomplemented
24#16861215C2211yescharacteristic
25#16861155C2211yescharacteristic, complemented
26#16861154C2212yescomplemented
27#16861153C2211yescharacteristic, complemented
28#16861152C2211yescharacteristic, complemented
29D2032C2212yescomplemented
30#16861150C2211yescharacteristic, complemented
31#16861150C2212yescomplemented
32#16861150C2212yescomplemented
33#16861148C2212yescomplemented
34C2×C127⋊C16C2211yescharacteristic, complemented
35C16×D127C2212yescomplemented
36#135×C127C2211yescharacteristic, complemented
37C2×D8×C127C2211yescharacteristic, complemented
38SD32×C127C2212yescomplemented
39D16×C127C2212yescomplemented
40M32×C127C2211yescharacteristic, complemented
41#14523249—22no—
42#14523248—21no—
43#14523248—41no—
44#14523248D411yescharacteristic
45#14523247—22no—
46#14523247—41no—
47C2×D508—21no—
48C2×D508—41no—
49C2×C22⋊D127—42no—
50D4×D127—22no—
51D4×D127—22no—
52D4×D127—41no—
53D4×D127—42no—
54Dic508C2311yescharacteristic
55#14523237—21no—
56#14523236—22no—
57#14523235—22no—
58D1016—21no—
59D1016C2311yescharacteristic
60D1016C2312yes—
61#14523233—22no—
62#14523233—22no—
63#14523232D411yescharacteristic
64#14523231D412yes—
65C8×D127C2312yes—
66C127⋊C16C2311yescharacteristic
67C127⋊C16C2311yescharacteristic
68#41×C127—21no—
69C2×D4×C127—21no—
70Q16×C127C2311yescharacteristic
71SD16×C127—21no—
72D8×C127—21no—
73D8×C127C2311yescharacteristic
74D8×C127C2312yes—
75C2×C8×C127C2311yescharacteristic
76C16×C127C2312yes—
77C22×D127—41no—
78C22×D127—42no—
79C22×D127—42no—
80Dic254—21no—
81Dic254D811yescharacteristic
82D508—21no—
83D508—22no—
84D508—42no—
85D508—41no—
86D508D811yescharacteristic
87C22⋊D127—41no—
88C22⋊D127—42no—
89C22⋊D127—42no—
90C22⋊D127—42no—
91C22⋊D127D812yes—
92C2×Dic127—42no—
93C4×D127—41no—
94C4×D127—42no—
95C4×D127C2×D412yes—
96C127⋊C8C2×D412yes—
97C23×C127—41no—
98Q8×C127—21no—
99D4×C127—22no—
100D4×C127—21no—
101D4×C127—41no—
102D4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127C2×D411yescharacteristic
105C8×C127C2×D411yescharacteristic
106C8×C127C2411yescharacteristic
107C22×C127—41no—
108C22×C127—42no—
109C22×C127—161no—
110C22×C127C2×D811yescharacteristic
111C2×D127—41no—
112C2×D127—42no—
113C2×D127—162no—
114C2×D127—162no—
115C2×D127—161no—
116C2×D127C2×D812yes—
117C4×C127—41no—
118C4×C127C2×D811yescharacteristic
119C4×C127C22×D411yescharacteristic
120Dic127—41no—
121Dic127C2×D812yes—
122C2×C127—21no—
123C2×C127—81no—
124C2×C127—82no—
125C2×C127C22×D811yescharacteristic
126D127—22no—
127D127—81no—
128D127—82no—
129#3415—1271nomaximal, Sylow
130C127#341511yescharacteristic, complemented, Sylow
131#576—1271no—
132#576—1271no—
133#509—1272no—
134#509—1272no—
135#509—1271no—
136#509D12711yescharacteristic, complemented
137#508—1271no—
138#508—1272no—
139C2×D16—1271no—
140C2×D16—1272no—
141#504—1271no—
142H(5,𝔽2)—2541no—
143H(5,𝔽2)—2541no—
144AGL(1,ℤ/8)—2542no—
145AGL(1,ℤ/8)—2542no—
146AGL(1,ℤ/8)—2541no—
147AGL(1,ℤ/8)—2541no—
148#135—1272no—
149#135—1272no—
150#135—1271no—
151#135C2×D12711yescharacteristic, complemented
152C2×D8—1272no—
153C2×D8—1272no—
154C2×D8—1271no—
155C2×D8C2×D12711yescharacteristic, complemented
156#131—1271no—
157Q32—1271no—
158SD32—1271no—
159SD32—1271no—
160SD32—1272no—
161SD32C2×D12712yescomplemented
162D16—1272no—
163D16—1271no—
164D16—1272no—
165D16—1272no—
166D16C2×D12712yescomplemented
167M32—1272no—
168M32C2×D12711yescharacteristic, complemented
169C2×C16—1271no—
170C2×C16—1272no—
171#41—21no—
172#41—1271no—
173#41—5081no—
174#41—5081no—
175#41—2542no—
176#41—2542no—
177#41—2541no—
178C2×D4—21no—
179C2×D4—5081no—
180C2×D4—5082no—
181C2×D4—5081no—
182C2×D4—5082no—
183C2×D4—2542no—
184C2×D4—2542no—
185C2×D4—2541no—
186Q16—1271no—
187Q16C22×D12711yescharacteristic
188SD16—21no—
189SD16—2542no—
190SD16—2542no—
191SD16—2541no—
192D8—21no—
193D8—1271no—
194D8—1272no—
195D8—2542no—
196D8—2542no—
197D8—2541no—
198D8C22×D12712yes—
199D8C22×D12711yescharacteristic
200M16—1271no—
201M16—1272no—
202C2×C8—1272no—
203C2×C8C22×D12711yescharacteristic
204C16—1271no—
205C16—1271no—
206C16C22×D12712yes—
207C23—41no—
208C23—5082no—
209C23—5082no—
210C23—5081no—
211Q8—21no—
212Q8—1271no—
213Q8—2541no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—1272no—
219D4—1271no—
220D4—5082no—
221D4—5082no—
222D4—5082no—
223D4—5082no—
224D4—5081no—
225D4—5081no—
226D4—2541no—
227D4—2542no—
228C2×C4—41no—
229C2×C4—1272no—
230C2×C4—5082no—
231C2×C4—5082no—
232C2×C4—5081no—
233C2×C4D4×D12711yescharacteristic
234C8—1272no—
235C8D4×D12711yescharacteristic
236C8C23×D12711yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—1272no—
241C22—20321no—
242C22—20322no—
243C22—20322no—
244C22—5082no—
245C22—5081no—
246C22D8×D12711yescharacteristic
247C4—41no—
248C4—1272no—
249C4—5081no—
250C4D8×D12711yescharacteristic
251C4C2×D4×D12711yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—10162no—
256C2—10161no—
257C2—2542no—
258C2C2×D8×D12711yescharacteristic
subgroupG/H
centerC2C2×D8×D127
commutator, cosocleC8×C127C24
FrattiniC8C23×D127
Fitting#509×C127C2
socleC2×C127C22×D8
hypercenter#509D127
hypocenterC127#3415
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22550678C2semidirect
2. #22550677C2semidirect
3. #22550314C2semidirect
4. #22550313C2semidirect
5. #22550310C2semidirect
6. #22550307C2semidirect
7. #22550306C2semidirect
8. C2×#16861150C2semidirect
9. D16×D127C2semidirect
10. #22550294C2semidirect
11. #509×C127C2semidirect
18. #16861232C22semidirect
19. #16861231C22semidirect
20. #16861228C22semidirect
21. C2×D1016C22semidirect
22. D8×D127C22semidirect
23. D8×D127C22semidirect
25. #16861155C22semidirect
26. #16861154C22semidirect
27. #16861153C22semidirect
28. #16861152C22semidirect
29. D2032C22semidirect
30. #16861150C22semidirect
31. #16861150C22semidirect
32. #16861150C22semidirect
33. #16861148C22semidirect
34. C2×C127⋊C16C22semidirect
35. C16×D127C22semidirect
36. #135×C127C22semidirect
37. C2×D8×C127C22semidirect
38. SD32×C127C22semidirect
39. D16×C127C22semidirect
40. M32×C127C22semidirect
130. C127#3415semidirect
136. #509D127semidirect
151. #135C2×D127semidirect
155. C2×D8C2×D127semidirect
161. SD32C2×D127semidirect
166. D16C2×D127semidirect
168. M32C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506782
#225506772
#225503142
#225503132
#225503102
#225503072
#225503062
C2×#168611502
D16×D1272
#225502942
#509×C1272
#3415127