Finite Group Databasebeta

← #30647971

Subgroups — #30647971

Order 16256.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#14523249C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic, complemented
3#22550383C211yesmaximal, characteristic, complemented
4#22550037C214yesmaximal, complemented
5#22550020C214yesmaximal, complemented
6#22549885C211yesmaximal, characteristic, complemented
7#22549882C211yesmaximal, characteristic, complemented
8#22549839C211yesmaximal, characteristic, complemented
9C2×#137×C127C211yesmaximal, characteristic, complemented
10C2×#14523249—24no—
11C2×#14523249—22no—
12C2×#14523249—24no—
13C2×#14523249C2214yes—
14C22×D508—21no—
15C22×D508C2211yescharacteristic
16C22×C4×D127—21no—
17C22×C4×D127C2211yescharacteristic
18#16861234—24no—
19C2×#14523235C2212yescomplemented
20C2×#14523233C2212yescomplemented
21C2×#14523232C2211yescharacteristic
22#16861190—21no—
23#16861185C2212yescomplemented
24#16861180C2212yescomplemented
25#16861177—21no—
26C4×C22⋊D127—22no—
27#16861168C2211yescharacteristic
28#16861133C2214yescomplemented
29#16861131C2214yescomplemented
30(C2×D127)⋊C8C2214yescomplemented
31C2×#41×C127C2211yescharacteristic
32C22×Q8×C127C2211yescharacteristic
33#137×C127—24no—
34C2×Q16×C127C2212yescomplemented
35C2×SD16×C127C2212yescomplemented
36C2×M16×C127C2211yescharacteristic
37C23×D127—21no—
38C23×D127D411yescharacteristic
39#14523249—216no—
40#14523249—28no—
41#14523249—28no—
42C2×D508—28no—
43C2×D508—24no—
44C2×D508—21no—
45C2×D508—22no—
46C2×D508D414yescomplemented
47C2×D508C2312yes—
48C2×C22⋊D127—22no—
49C22×Dic127D411yescharacteristic
50C2×C4×D127—28no—
51C2×C4×D127—24no—
52C2×C4×D127—22no—
53C2×C4×D127—22no—
54C2×C4×D127D412yes—
55C2×C4×D127C2312yes—
56#14523235—24no—
57#14523233—24no—
58#14523232—22no—
59C2×C127⋊C8C2312yes—
60C4⋊Dic127C2312yes—
61Dic127⋊C4—22no—
62C22⋊Dic127—22no—
63C22⋊Dic127—21no—
64(C2×D127)⋊C4—22no—
65C4×Dic127—21no—
66#41×C127—22no—
67#41×C127—22no—
68C2×Q8×C127—22no—
69C2×Q8×C127—21no—
70C2×Q8×C127D414yescomplemented
71C2×Q8×C127C2311yescharacteristic
72C2×Q8×C127C2312yes—
73C2×D4×C127—21no—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78Q16×C127—24no—
79SD16×C127—24no—
80M16×C127—22no—
81C2×C8×C127C2312yes—
82C22×D127—22no—
83C22×D127—24no—
84C22×D127—22no—
85C22×D127—42no—
86C22×D127—44no—
87C22×D127C2×D412yes—
88D508—216no—
89D508—24no—
90D508—22no—
91D508C2×D414yes—
92C22⋊D127—44no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C2×Dic127C2×D412yes—
96C4×D127—216no—
97C4×D127—24no—
98C4×D127—22no—
99C4×D127—22no—
100C127⋊C8—22no—
101C23×C127—21no—
102C23×C127C2×D411yescharacteristic
103Q8×C127—24no—
104Q8×C127—22no—
105Q8×C127—21no—
106Q8×C127—21no—
107Q8×C127C2×D414yes—
108D4×C127—21no—
109D4×C127—21no—
110D4×C127—42no—
111C2×C4×C127—24no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—42no—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2411yescharacteristic
119C8×C127—22no—
120C22×C127—22no—
121C22×C127—41no—
122C22×C127—42no—
123C22×C127#12012yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—24no—
127C2×D127—22no—
128C2×D127—22no—
129C2×D127—44no—
130C2×D127—48no—
131C2×D127—44no—
132C4×C127—24no—
133C4×C127—41no—
134C4×C127#12012yes—
135C4×C127C22×D411yescharacteristic
136C4×C127C22×D411yescharacteristic
137Dic127—22no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—81no—
141C2×C127C2×#12011yescharacteristic
142C2×C127#57512yes—
143D127—42no—
144D127—44no—
145#3005—1271nomaximal, Sylow
146C127#300511yescharacteristic, complemented, Sylow
147C22×#41—1271no—
148C2×#137D12711yescharacteristic, complemented
149C2×AGL(1,ℤ/8)—1271no—
150#525—1271no—
151#452—1274no—
152#449—1274no—
153#419—1271no—
154#418—1271no—
155#407—1271no—
156C2×#41—1274no—
157C2×#41—2542no—
158C2×#41—2544no—
159C2×#41—2544no—
160C2×#41C2×D12711yescharacteristic
161C22×Q8C2×D12711yescharacteristic
162C22×D4—1271no—
163C22×D4—2541no—
164C23×C4—1271no—
165C23×C4—2541no—
166#137—24no—
167AGL(1,ℤ/8)—2544no—
168C2×Q16C2×D12712yescomplemented
169C2×SD16—1272no—
170C2×SD16C2×D12712yescomplemented
171C2×D8—1272no—
172C2×M16—1271no—
173C2×M16C2×D12711yescharacteristic
174#123—2541no—
175C22⋊Q8—1272no—
176#121—1272no—
177#120—2541no—
178C4×D4—2542no—
179#117—1271no—
180#103—1274no—
181#102—1274no—
182C22⋊C8—1274no—
183C24—1271no—
184C24—2541no—
185#41—22no—
186#41—22no—
187#41—2548no—
188#41—25416no—
189#41—2548no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8C22⋊D12714yescomplemented
193C2×Q8C22×D12711yescharacteristic
194C2×Q8C22×D12712yes—
195C2×D4—21no—
196C2×D4—1274no—
197C2×D4—1272no—
198C2×D4—2544no—
199C2×D4—2548no—
200C2×D4—2542no—
201C2×D4—2541no—
202C2×D4—2542no—
203C2×D4C22×D12711yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1271no—
207C22×C4—1272no—
208C22×C4—1272no—
209C22×C4—2544no—
210C22×C4—2548no—
211C22×C4—2542no—
212C22×C4—2542no—
213C22×C4C22×D12711yescharacteristic
214Q16—24no—
215SD16—24no—
216SD16—2544no—
217D8—2544no—
218M16—22no—
219M16—2542no—
220C2×C8—1272no—
221C2×C8C22×D12712yes—
222C4⋊C4—1272no—
223C4⋊C4—2542no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C22⋊C4—2541no—
227C42—2541no—
228C23—21no—
229C23—1272no—
230C23—5084no—
231C23—5082no—
232C23—2542no—
233C23—2544no—
234C23—2542no—
235C23D4×D12711yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—21no—
239Q8—21no—
240Q8C2×C22⋊D12714yes—
241D4—21no—
242D4—21no—
243D4—42no—
244D4—1274no—
245D4—5084no—
246D4—25416no—
247D4—2542no—
248D4—2544no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—1272no—
254C2×C4—25416no—
255C2×C4—2542no—
256C2×C4—2542no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4—2544no—
260C2×C4D4×D12712yes—
261C2×C4D4×D12711yescharacteristic
262C2×C4C2×C22⋊D12712yes—
263C2×C4C23×D12711yescharacteristic
264C8—22no—
265C8—2542no—
266C22—22no—
267C22—41no—
268C22—42no—
269C22—5088no—
270C22—5084no—
271C22—5084no—
272C22—2544no—
273C22—2542no—
274C22—2542no—
275C22—2542no—
276C22#1686117712yes—
277C22C2×D4×D12711yescharacteristic
278C4—24no—
279C4—41no—
280C4—5082no—
281C4—2542no—
282C4#1686117712yes—
283C4C2×D4×D12711yescharacteristic
284C4C22×C22⋊D12711yescharacteristic
285C2—22no—
286C2—81no—
287C2—5084no—
288C2—5082no—
289C2C2×#1686117711yescharacteristic
290C2#2255067512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
hypercenterC2×#137D127
hypocenterC127#3005
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#14523249C2semidirect
2. C2×#16861234C2semidirect
3. #22550383C2semidirect
4. #22550037C2semidirect
5. #22550020C2semidirect
6. #22549885C2semidirect
7. #22549882C2semidirect
8. #22549839C2semidirect
9. C2×#137×C127C2semidirect
19. C2×#14523235C22semidirect
20. C2×#14523233C22semidirect
23. #16861185C22semidirect
24. #16861180C22semidirect
28. #16861133C22semidirect
29. #16861131C22semidirect
30. (C2×D127)⋊C8C22semidirect
34. C2×Q16×C127C22semidirect
35. C2×SD16×C127C22semidirect
46. C2×D508D4semidirect
70. C2×Q8×C127D4semidirect
146. C127#3005semidirect
148. C2×#137D127semidirect
168. C2×Q16C2×D127semidirect
170. C2×SD16C2×D127semidirect
192. C2×Q8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232492
C2×#168612342
#225503832
#225500372
#225500202
#225498852
#225498822
#225498392
C2×#137×C1272
#3005127