Finite Group Databasebeta

← #30647189

Subgroups — #30647189

Order 16256.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16861255C211yesmaximal, characteristic
2#22550350C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic, complemented
4#22549849C212yesmaximal, complemented
5#99×D127C218yesmaximal, complemented
6#410×C127C211yesmaximal, characteristic, complemented
7#16861255—24no—
8#16861255—24no—
9C2×#14523249C412yescomplemented
10C2×#14523248C412yescomplemented
11C2×#14523248C2211yescharacteristic
12C22×D508C412yescomplemented
13C22×D508C2211yescharacteristic
14C2×D4×D127C412yescomplemented
15C2×D4×D127C2214yes—
16C4×D508—24no—
17#16861171—24no—
18#16861168C2212yescomplemented
19C22⋊C4×D127C2218yescomplemented
20#16861124C2214yescomplemented
21#16861123C2218yescomplemented
22C2×#41×C127C2211yescharacteristic
23#117×C127C2212yescomplemented
24#99×C127C2214yescomplemented
25C23×D127C2×C414yescomplemented
26C23×D127C2312yes—
27#14523249—28no—
28#14523248—24no—
29#14523248—24no—
30#14523248—22no—
31#14523248—22no—
32C2×Dic254C2×C411yescharacteristic, complemented
33C2×D508—24no—
34C2×D508—24no—
35C2×D508C2×C412yes—
36C2×D508C2×C412yes—
37C2×D508C2×C414yescomplemented
38C2×D508C2×C411yescharacteristic, complemented
39C2×D508D414yes—
40C2×D508D414yes—
41C2×C22⋊D127C2×C414yescomplemented
42C2×C22⋊D127C2312yes—
43D4×D127—216no—
44D4×D127—28no—
45C2×C4×D127—28no—
46C2×C4×D127C2×C412yescomplemented
47C2×C4×D127C2×C414yescomplemented
48C4⋊Dic127—22no—
49C22⋊Dic127C2314yes—
50(C2×D127)⋊C4—28no—
51C4×Dic127—22no—
52#41×C127—22no—
53#41×C127—22no—
54C2×Q8×C127C2×C411yescharacteristic, complemented
55C2×D4×C127C2×C411yescharacteristic, complemented
56C2×D4×C127C2312yes—
57C22×C4×C127C2×C412yescomplemented
58C22×C4×C127C2311yescharacteristic
59C4⋊C4×C127—22no—
60C22⋊C4×C127C2314yes—
61C42×C127—22no—
62C22×D127—28no—
63C22×D127—82no—
64C22×D127—84no—
65C22×D127C22×C412yes—
66C22×D127C22×C414yes—
67C22×D127C2×D418yes—
68Dic254—21no—
69Dic254—21no—
70D508—28no—
71D508—21no—
72D508—21no—
73D508C22⋊C418yes—
74D508C22⋊C414yes—
75D508C22⋊C414yes—
76C22⋊D127—24no—
77C22⋊D127—28no—
78C2×Dic127—24no—
79C2×Dic127C22×C412yes—
80C4×D127—28no—
81C4×D127—24no—
82C4×D127—88no—
83C23×C127C22×C412yes—
84C23×C127C2411yescharacteristic
85Q8×C127—22no—
86D4×C127—24no—
87D4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—24no—
91C2×C4×C127C22×C412yes—
92C2×C4×C127C22×C411yescharacteristic
93C2×C4×C127C22×C411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127C2×D412yes—
96C22×C127—22no—
97C22×C127—41no—
98C22×C127—42no—
99C22×C127C23×C411yescharacteristic
100C22×C127C22×D412yes—
101C2×D127—22no—
102C2×D127—48no—
103C2×D127—44no—
104C2×D127—42no—
105C2×D127—88no—
106C2×D127C2×C22⋊C418yes—
107C4×C127—22no—
108C4×C127—44no—
109C4×C127C2×C22⋊C411yescharacteristic
110C4×C127C2×C22⋊C411yescharacteristic
111C4×C127C2×C22⋊C412yes—
112Dic127—22no—
113Dic127—44no—
114C2×C127—22no—
115C2×C127—21no—
116C2×C127—42no—
117C2×C127C22×C22⋊C411yescharacteristic
118D127—28no—
119D127—42no—
120#2883—1271nomaximal, Sylow
121C127#288311yescharacteristic, complemented, Sylow
122C2×H(5,𝔽2)—1271no—
123#518—1272no—
124#410—1271no—
125#410—1272no—
126#410D12711yescharacteristic, complemented
127C2×#99—1278no—
128H(5,𝔽2)—2544no—
129H(5,𝔽2)—2544no—
130C2×#41—1272no—
131C2×#41—1272no—
132C2×#41—1271no—
133C2×#41C2×D12711yescharacteristic
134C22×D4—1272no—
135C22×D4—1274no—
136C22×D4—1272no—
137C22×D4—1271no—
138C4×D4—2544no—
139C4×D4—2544no—
140#117—1272no—
141#117C2×D12712yescomplemented
142C2×C22⋊C4—1278no—
143#99—1274no—
144#99—1278no—
145#99C2×D12714yescomplemented
146C24—1272no—
147C24—1274no—
148#41—22no—
149#41—22no—
150#41—2544no—
151#41—2544no—
152#41—2548no—
153#41—2542no—
154#41—2542no—
155C2×Q8—1271no—
156C2×Q8C4×D12711yescharacteristic, complemented
157C2×D4—1274no—
158C2×D4—1272no—
159C2×D4—1272no—
160C2×D4—1272no—
161C2×D4—1274no—
162C2×D4—1271no—
163C2×D4—1274no—
164C2×D4—1274no—
165C2×D4—2544no—
166C2×D4—2544no—
167C2×D4—2548no—
168C2×D4—25416no—
169C2×D4C4×D12711yescharacteristic, complemented
170C2×D4C22×D12712yes—
171C22×C4—1274no—
172C22×C4—1272no—
173C22×C4—2548no—
174C22×C4C4×D12712yescomplemented
175C22×C4C22×D12711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—2542no—
178C22⋊C4—1274no—
179C22⋊C4—2548no—
180C22⋊C4C22×D12714yes—
181C42—22no—
182C42—2542no—
183C23—1274no—
184C23—1278no—
185C23—1272no—
186C23—10164no—
187C23—10162no—
188C23—2548no—
189C23C2×C4×D12712yes—
190C23C23×D12711yescharacteristic
191Q8—22no—
192Q8—2541no—
193Q8—2541no—
194D4—24no—
195D4—22no—
196D4—1274no—
197D4—1274no—
198D4—1278no—
199D4—2548no—
200D4—2548no—
201D4—2544no—
202D4—2541no—
203D4—2541no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—1272no—
208C2×C4—10168no—
209C2×C4—2548no—
210C2×C4—2544no—
211C2×C4—2544no—
212C2×C4C2×C4×D12712yes—
213C2×C4C2×C4×D12711yescharacteristic
214C2×C4C2×C4×D12711yescharacteristic
215C2×C4D4×D12712yes—
216C2×C4D4×D12712yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—1278no—
221C22—10168no—
222C22—5082no—
223C22—5088no—
224C22—5084no—
225C22—2542no—
226C22C22×C4×D12711yescharacteristic
227C22C2×D4×D12712yes—
228C4—22no—
229C4—44no—
230C4—5084no—
231C4—2542no—
232C4C22⋊C4×D12711yescharacteristic
233C4C22⋊C4×D12711yescharacteristic
234C4C22⋊C4×D12712yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—5082no—
239C2—2548no—
240C2C2×C22⋊C4×D12711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D127
commutatorC22×C127C23×C4
FrattiniC23C23×D127
Fitting#410×C127C2
socleC2×C127C22×C22⋊C4
prime radicalC2×#16861255C2
cosocleC23×C127C24
hypercenter#410D127
hypocenterC127#2883
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22550350C2semidirect
3. #22549850C2semidirect
4. #22549849C2semidirect
5. #99×D127C2semidirect
6. #410×C127C2semidirect
9. C2×#14523249C4semidirect
10. C2×#14523248C4semidirect
12. C22×D508C4semidirect
14. C2×D4×D127C4semidirect
18. #16861168C22semidirect
19. C22⋊C4×D127C22semidirect
20. #16861124C22semidirect
21. #16861123C22semidirect
23. #117×C127C22semidirect
24. #99×C127C22semidirect
25. C23×D127C2×C4semidirect
32. C2×Dic254C2×C4semidirect
37. C2×D508C2×C4semidirect
38. C2×D508C2×C4semidirect
41. C2×C22⋊D127C2×C4semidirect
46. C2×C4×D127C2×C4semidirect
47. C2×C4×D127C2×C4semidirect
54. C2×Q8×C127C2×C4semidirect
55. C2×D4×C127C2×C4semidirect
57. C22×C4×C127C2×C4semidirect
121. C127#2883semidirect
126. #410D127semidirect
141. #117C2×D127semidirect
145. #99C2×D127semidirect
156. C2×Q8C4×D127semidirect
169. C2×D4C4×D127semidirect
174. C22×C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
#225503502
#225498502
#225498492
#99×D1272
#410×C1272
#2883127