Finite Group Databasebeta

← #30649210

Subgroups — #30649210

Order 16256.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16861233C212yesmaximal, complemented
2C2×#16861228C211yesmaximal, characteristic, complemented
3C2×D8×D127C211yesmaximal, characteristic, complemented
4#22550439C212yesmaximal, complemented
5#22550264C212yesmaximal, complemented
6#22550263C212yesmaximal, complemented
7#22550260C211yesmaximal, characteristic, complemented
8#22550259C211yesmaximal, characteristic, complemented
9#22550251C211yesmaximal, characteristic, complemented
10#22549830C211yesmaximal, characteristic, complemented
11#493×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yes—
13C2×D4×D127C2212yes—
14#16861233—28no—
15#16861228—24no—
16C2×Dic508C2211yescharacteristic, complemented
17C2×#14523237C2212yescomplemented
18C2×#14523236C2212yescomplemented
19C2×#14523236C2212yescomplemented
20C2×D1016C2211yescharacteristic
21C2×#14523233C2212yes—
22C2×#14523233C2212yescomplemented
23D8×D127—24no—
24C2×#14523231C2211yescharacteristic
25C2×C8×D127C2211yescharacteristic
26#16861205C2212yescomplemented
27#16861197C2212yescomplemented
28#16861196C2212yescomplemented
29#16861181—24no—
30#16861177—24no—
31#16861166C2211yescharacteristic
32#16861120C2211yescharacteristic, complemented
33#16861119C2211yescharacteristic, complemented
34C8×Dic127C2211yescharacteristic, complemented
35C2×D8×C127C2211yescharacteristic
36C2×D8×C127C2211yescharacteristic, complemented
37C2×D8×C127C2212yescomplemented
38#127×C127C2212yescomplemented
39C4×C8×C127C2211yescharacteristic, complemented
40C23×D127—22no—
41#14523247—24no—
42#14523247—24no—
43C2×Dic254C2312yes—
44C2×D508C2312yes—
45C2×C22⋊D127—22no—
46C2×C22⋊D127—22no—
47C2×C22⋊D127—24no—
48D4×D127—24no—
49D4×D127—24no—
50C22×Dic127—22no—
51C2×C4×D127C2311yescharacteristic
52Dic508—22no—
53#14523237—24no—
54#14523236—24no—
55#14523236—24no—
56D1016—22no—
57#14523233—24no—
58#14523233—24no—
59#14523231D414yescomplemented
60C2×C127⋊C8C2311yescharacteristic
61C2×C127⋊C8C2311yescharacteristic
62C8×D127D414yescomplemented
63Dic127⋊C4D412yes—
64C22⋊Dic127—24no—
65(C2×D127)⋊C4—24no—
66(C2×D127)⋊C4D412yes—
67C4×Dic127C2311yescharacteristic
68C2×D4×C127—24no—
69C2×D4×C127C2312yes—
70C2×D4×C127C2312yes—
71D8×C127—22no—
72D8×C127—22no—
73D8×C127—24no—
74C2×C8×C127C2311yescharacteristic
75C2×C8×C127C2311yescharacteristic
76C42×C127C2311yescharacteristic
77C22×D127—22no—
78C22×D127—44no—
79C22×D127—84no—
80C22×D127C2×D411yescharacteristic
81Dic254—22no—
82Dic254—22no—
83D508—22no—
84D508—22no—
85C22⋊D127—44no—
86C22⋊D127—44no—
87C22⋊D127—84no—
88C2×Dic127—22no—
89C2×Dic127—44no—
90C2×Dic127C2×D412yes—
91C2×Dic127C2×D411yescharacteristic
92C4×D127C2×D412yes—
93C4×D127C2×D412yes—
94C127⋊C8C2×D412yes—
95C127⋊C8C2×D412yes—
96C23×C127—22no—
97C23×C127—22no—
98D4×C127—22no—
99D4×C127—22no—
100D4×C127—22no—
101D4×C127—22no—
102D4×C127—84no—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2411yescharacteristic
105C8×C127C2×D412yes—
106C8×C127C2×D412yes—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—44no—
110C22×C127—44no—
111C22×C127C22×D411yescharacteristic
112C2×D127—22no—
113C2×D127—42no—
114C2×D127—44no—
115C2×D127—88no—
116C2×D127#12712yes—
117C4×C127—22no—
118C4×C127C22×D411yescharacteristic
119C4×C127C22×D411yescharacteristic
120Dic127—22no—
121Dic127—42no—
122Dic127#12712yes—
123C2×C127—82no—
124C2×C127—82no—
125C2×C127C2×#12711yescharacteristic
126C2×C127#57612yes—
127D127—22no—
128D127—82no—
129#3155—1271nomaximal, Sylow
130C127#315511yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—1272no—
132C2×#135—1271no—
133C22×D8—1271no—
134#535—1272no—
135#496—1272no—
136#496—1272no—
137#495—1271no—
138#495—1271no—
139#493—1271no—
140#493D12711yescharacteristic, complemented
141#405—1271no—
142C2×#41—1272no—
143C22×D4—1272no—
144AGL(1,ℤ/8)—2548no—
145#135—2544no—
146C2×Q16—1271no—
147C2×SD16—1272no—
148C2×SD16—1272no—
149C2×SD16—1272no—
150C2×D8—1272no—
151C2×D8—1272no—
152C2×D8—1271no—
153C2×D8—2544no—
154C2×D8C2×D12711yescharacteristic
155C2×D8C2×D12711yescharacteristic, complemented
156C2×D8C2×D12712yescomplemented
157C2×M16—1271no—
158C22×C8—1271no—
159#127—1272no—
160#127C2×D12712yescomplemented
161#124—1272no—
162#124—1272no—
163#121—2544no—
164#120—2544no—
165#117—1271no—
166#97—1271no—
167#97—1271no—
168C4×C8—1271no—
169C4×C8C2×D12711yescharacteristic, complemented
170C24—2542no—
171#41—2544no—
172#41—2544no—
173C2×Q8—1272no—
174C2×D4—24no—
175C2×D4—1272no—
176C2×D4—2544no—
177C2×D4—2544no—
178C2×D4—2544no—
179C2×D4—2542no—
180C2×D4—2542no—
181C2×D4C22×D12712yes—
182C2×D4C22×D12712yes—
183C22×C4—1271no—
184C22×C4—2542no—
185Q16—2542no—
186SD16—2544no—
187SD16—2544no—
188SD16—2544no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—2544no—
193D8—2544no—
194D8—2542no—
195M16—1274no—
196C2×C8—1271no—
197C2×C8—1271no—
198C2×C8—1274no—
199C2×C8C22×D12711yescharacteristic
200C2×C8C22×D12711yescharacteristic
201C4⋊C4—1272no—
202C22⋊C4—1272no—
203C22⋊C4—2544no—
204C22⋊C4—2544no—
205C42—1271no—
206C42C22×D12711yescharacteristic
207C23—22no—
208C23—22no—
209C23—1271no—
210C23—10164no—
211C23—5084no—
212C23—2542no—
213Q8—2542no—
214Q8—2542no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—10164no—
221D4—5084no—
222D4—5084no—
223D4—2542no—
224D4—2542no—
225C2×C4—1271no—
226C2×C4—1272no—
227C2×C4—1272no—
228C2×C4—1272no—
229C2×C4—5084no—
230C2×C4—2542no—
231C2×C4D4×D12712yes—
232C2×C4C23×D12711yescharacteristic
233C8—1272no—
234C8—1272no—
235C8D4×D12712yes—
236C8D4×D12712yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—1272no—
242C22—10168no—
243C22—5082no—
244C22—5084no—
245C22—2542no—
246C22C2×D4×D12711yescharacteristic
247C4—22no—
248C4—1272no—
249C4—5082no—
250C4—2542no—
251C4C2×D4×D12711yescharacteristic
252C4C2×D4×D12711yescharacteristic
253C2—82no—
254C2—82no—
255C2—10162no—
256C2—2542no—
257C2#127×D12711yescharacteristic
258C2#2255067712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#493×C127C2
socleC22×C127C22×D4
hypercenter#493D127
hypocenterC127#3155
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×#16861228C2semidirect
3. C2×D8×D127C2semidirect
4. #22550439C2semidirect
5. #22550264C2semidirect
6. #22550263C2semidirect
7. #22550260C2semidirect
8. #22550259C2semidirect
9. #22550251C2semidirect
10. #22549830C2semidirect
11. #493×C127C2semidirect
16. C2×Dic508C22semidirect
17. C2×#14523237C22semidirect
18. C2×#14523236C22semidirect
19. C2×#14523236C22semidirect
22. C2×#14523233C22semidirect
26. #16861205C22semidirect
27. #16861197C22semidirect
28. #16861196C22semidirect
32. #16861120C22semidirect
33. #16861119C22semidirect
34. C8×Dic127C22semidirect
36. C2×D8×C127C22semidirect
37. C2×D8×C127C22semidirect
38. #127×C127C22semidirect
39. C4×C8×C127C22semidirect
59. #14523231D4semidirect
62. C8×D127D4semidirect
130. C127#3155semidirect
140. #493D127semidirect
155. C2×D8C2×D127semidirect
156. C2×D8C2×D127semidirect
160. #127C2×D127semidirect
169. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
C2×#168612282
C2×D8×D1272
#225504392
#225502642
#225502632
#225502602
#225502592
#225502512
#225498302
#493×C1272
#3155127