Finite Group Databasebeta

← #30648016

Subgroups — #30648016

Order 16256.

732 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16861234C211yesmaximal, characteristic, complemented
2C22×#14523235C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550021C214yesmaximal, complemented
5#22550015C214yesmaximal, complemented
6#22549885C211yesmaximal, characteristic, complemented
7#22549872C211yesmaximal, characteristic, complemented
8#22549843C211yesmaximal, characteristic, complemented
9C2×2−1+4×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11#16861234—24no—
12C2×#14523235—24no—
13C2×#14523235C2214yes—
14C2×#14523235C2212yes—
15C2×#14523233C2212yes—
16C2×#14523232C2211yescharacteristic
17C22×C127⋊C8C2211yescharacteristic
18#16861205—21no—
19#16861198—21no—
20#16861183C2214yes—
21#16861177—22no—
22#16861168C2211yescharacteristic
23#16861133C2214yescomplemented
24#16861133C2212yescomplemented
25#16861130C2212yescomplemented
26C22⋊(C127⋊C8)C2214yescomplemented
272−1+4×C127—24no—
282−1+4×C127—24no—
29C2×#41×C127—22no—
30C2×#41×C127—21no—
31C2×#41×C127C2211yescharacteristic
32C2×#41×C127C2211yescharacteristic
33C2×#41×C127C2212yescomplemented
34C22×Q8×C127—21no—
35C22×Q8×C127C2211yescharacteristic
36C22×Q8×C127C2212yescomplemented
37C23×D127—21no—
38C2×D508—22no—
39C2×D508—22no—
40C2×D508—21no—
41C2×D508C2312yes—
42C2×C22⋊D127—24no—
43#14523235—28no—
44#14523235—24no—
45#14523233—24no—
46#14523232—22no—
47C2×C127⋊C8—22no—
48C2×C127⋊C8C2312yes—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127C2312yes—
51C22⋊Dic127—21no—
52(C2×D127)⋊C4—24no—
53C4×Dic127—21no—
54#41×C127—216no—
55#41×C127—28no—
56#41×C127—24no—
57#41×C127—24no—
58#41×C127—22no—
59#41×C127—22no—
60#41×C127—22no—
61#41×C127D414yescomplemented
62C2×Q8×C127—28no—
63C2×Q8×C127—24no—
64C2×Q8×C127—24no—
65C2×Q8×C127—21no—
66C2×Q8×C127—22no—
67C2×Q8×C127—22no—
68C2×Q8×C127D411yescharacteristic
69C2×Q8×C127D414yescomplemented
70C2×Q8×C127C2312yes—
71C2×Q8×C127C2311yescharacteristic
72C2×D4×C127—21no—
73C2×D4×C127—22no—
74C2×D4×C127D411yescharacteristic
75C2×D4×C127D412yes—
76C2×D4×C127C2311yescharacteristic
77C22×C4×C127—21no—
78C22×C4×C127—22no—
79C22×C4×C127—21no—
80C22×C4×C127—22no—
81C22×C4×C127C2312yes—
82C22×C4×C127C2311yescharacteristic
83C22×D127—22no—
84C22×D127—42no—
85C22×D127—82no—
86D508—22no—
87D508—22no—
88D508—42no—
89C22⋊D127—84no—
90C2×Dic127—22no—
91C127⋊C8—22no—
92C127⋊C8—22no—
93C23×C127—21no—
94C23×C127C2×D412yes—
95C23×C127C2×D411yescharacteristic
96Q8×C127—28no—
97Q8×C127—22no—
98Q8×C127—24no—
99Q8×C127—21no—
100Q8×C127—22no—
101Q8×C127C2×D412yes—
102Q8×C127C2×D414yes—
103D4×C127—28no—
104D4×C127—24no—
105D4×C127—24no—
106D4×C127—22no—
107D4×C127—21no—
108D4×C127C2×D412yes—
109C2×C4×C127—28no—
110C2×C4×C127—28no—
111C2×C4×C127—24no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—24no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2411yescharacteristic
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—24no—
126C22×C127—42no—
127C22×C127#12012yes—
128C22×C127C22×D411yescharacteristic
129C2×D127—42no—
130C2×D127—44no—
131C2×D127—84no—
132C4×C127—24no—
133C4×C127—22no—
134C4×C127—22no—
135C4×C127#12012yes—
136C4×C127C22×D411yescharacteristic
137C4×C127C22×D411yescharacteristic
138Dic127—42no—
139C2×C127—22no—
140C2×C127—42no—
141C2×C127—42no—
142C2×C127C2×#12011yescharacteristic
143C2×C127#57712yes—
144D127—82no—
145#3010—1271nomaximal, Sylow
146C127#301011yescharacteristic, complemented, Sylow
147C2×2−1+4D12711yescharacteristic, complemented
148C2×AGL(1,ℤ/8)—1271no—
149C22×SD16—1271no—
150#535—1271no—
151#449—1274no—
152#448—1274no—
153#419—1271no—
154#416—1271no—
155#408—1271no—
1562−1+4—24no—
1572−1+4—24no—
158C2×#41—22no—
159C2×#41—21no—
160C2×#41C2×D12711yescharacteristic
161C2×#41C2×D12711yescharacteristic
162C2×#41C2×D12712yescomplemented
163C22×Q8—21no—
164C22×Q8C2×D12711yescharacteristic
165C22×Q8C2×D12712yescomplemented
166C22×D4—1271no—
167AGL(1,ℤ/8)—2544no—
168C2×SD16—1274no—
169C2×SD16—1272no—
170C2×SD16—2544no—
171C2×D8—1272no—
172C2×M16—1271no—
173C22×C8—1271no—
174#127—2541no—
175#124—2541no—
176#121—1274no—
177#120—2542no—
178#117—1271no—
179#103—1272no—
180#103—1274no—
181#102—1272no—
182C22⋊C8—1274no—
183C24—2541no—
184#41—216no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41C22⋊D12714yescomplemented
192C2×Q8—28no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8C22⋊D12711yescharacteristic
199C2×Q8C22⋊D12714yescomplemented
200C2×Q8C22×D12712yes—
201C2×Q8C22×D12711yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—1272no—
205C2×D4—2544no—
206C2×D4—2541no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4C22⋊D12712yes—
210C2×D4C22⋊D12711yescharacteristic
211C2×D4C22×D12711yescharacteristic
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4C22×D12712yes—
217C22×C4C22×D12711yescharacteristic
218SD16—2548no—
219SD16—2544no—
220D8—2544no—
221M16—2542no—
222C2×C8—1272no—
223C2×C8—1272no—
224C2×C8—2542no—
225C4⋊C4—1272no—
226C22⋊C4—2541no—
227C22⋊C4—2544no—
228C42—2541no—
229C23—21no—
230C23—10162no—
231C23—5082no—
232C23—2542no—
233C23C2×C22⋊D12712yes—
234C23C2×C22⋊D12711yescharacteristic
235Q8—28no—
236Q8—21no—
237Q8—22no—
238Q8—22no—
239Q8—24no—
240Q8C2×C22⋊D12712yes—
241Q8C2×C22⋊D12714yes—
242D4—28no—
243D4—24no—
244D4—24no—
245D4—21no—
246D4—22no—
247D4—10164no—
248D4—5082no—
249D4—2542no—
250D4—2542no—
251D4C2×C22⋊D12712yes—
252C2×C4—28no—
253C2×C4—28no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—2542no—
261C2×C4C2×C22⋊D12712yes—
262C2×C4C2×C22⋊D12712yes—
263C2×C4C2×C22⋊D12712yes—
264C2×C4C2×C22⋊D12711yescharacteristic
265C2×C4C23×D12711yescharacteristic
266C8—2542no—
267C8—2542no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—42no—
273C22—10164no—
274C22—5082no—
275C22—5084no—
276C22#1686117812yes—
277C22C22×C22⋊D12711yescharacteristic
278C4—24no—
279C4—22no—
280C4—22no—
281C4—5082no—
282C4#1686117812yes—
283C4C22×C22⋊D12711yescharacteristic
284C4C22×C22⋊D12711yescharacteristic
285C2—22no—
286C2—42no—
287C2—42no—
288C2—10162no—
289C2C2×#1686117811yescharacteristic
290C2#2255068412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×2−1+4×C127C2
socleC22×C127C22×D4
hypercenterC2×2−1+4D127
hypocenterC127#3010
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C22×#14523235C2semidirect
3. #22550440C2semidirect
4. #22550021C2semidirect
5. #22550015C2semidirect
6. #22549885C2semidirect
7. #22549872C2semidirect
8. #22549843C2semidirect
9. C2×2−1+4×C127C2semidirect
23. #16861133C22semidirect
24. #16861133C22semidirect
25. #16861130C22semidirect
26. C22⋊(C127⋊C8)C22semidirect
33. C2×#41×C127C22semidirect
36. C22×Q8×C127C22semidirect
61. #41×C127D4semidirect
69. C2×Q8×C127D4semidirect
146. C127#3010semidirect
147. C2×2−1+4D127semidirect
162. C2×#41C2×D127semidirect
165. C22×Q8C2×D127semidirect
191. #41C22⋊D127semidirect
199. C2×Q8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×#145232352
#225504402
#225500212
#225500152
#225498852
#225498722
#225498432
C2×2−1+4×C1272
#3010127