Finite Group Databasebeta

← #30647969

Subgroups — #30647969

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×#14523247C211yesmaximal, characteristic, complemented
2C2×#16861239C211yesmaximal, characteristic, complemented
3#22550383C211yesmaximal, characteristic, complemented
4#22550035C214yesmaximal, complemented
5#22550018C214yesmaximal, complemented
6#22549886C211yesmaximal, characteristic, complemented
7#22549881C211yesmaximal, characteristic, complemented
8#22549839C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
10C2×#14523247—24no—
11C2×#14523247—22no—
12C2×#14523247—24no—
13C2×#14523247C2214yes—
14C22×Dic254C2211yescharacteristic
15C22×C22⋊D127—21no—
16C23×Dic127—21no—
17C22×C4×D127C2211yescharacteristic
18#16861239—24no—
19C2×#14523238C2212yescomplemented
20C2×#14523236C2212yescomplemented
21C2×#14523232C2211yescharacteristic
22#16861190—21no—
23#16861185C2212yescomplemented
24#16861180C2212yescomplemented
25#16861177—21no—
26C4×C22⋊D127—22no—
27#16861168C2211yescharacteristic
28#16861134C2214yescomplemented
29#16861130C2214yescomplemented
30(C2×D127)⋊C8C2214yescomplemented
31C2×#41×C127C2211yescharacteristic
32C22×D4×C127C2211yescharacteristic
33AGL(1,ℤ/8)×C127—24no—
34C2×SD16×C127C2212yescomplemented
35C2×D8×C127C2212yescomplemented
36C2×M16×C127C2211yescharacteristic
37C23×D127D411yescharacteristic
38#14523247—216no—
39#14523247—28no—
40#14523247—28no—
41C2×Dic254—21no—
42C2×Dic254—22no—
43C2×Dic254D414yescomplemented
44C2×Dic254C2312yes—
45C2×C22⋊D127—28no—
46C2×C22⋊D127—24no—
47C2×C22⋊D127—22no—
48C22×Dic127—28no—
49C22×Dic127—24no—
50C22×Dic127—21no—
51C22×Dic127D411yescharacteristic
52C2×C4×D127—22no—
53C2×C4×D127—22no—
54C2×C4×D127D412yes—
55C2×C4×D127C2312yes—
56#14523238—24no—
57#14523236—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—
66C24×C127—21no—
67#41×C127—22no—
68#41×C127—22no—
69C2×Q8×C127C2311yescharacteristic
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127D414yescomplemented
74C2×D4×C127C2311yescharacteristic
75C2×D4×C127C2312yes—
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78SD16×C127—24no—
79D8×C127—24no—
80M16×C127—22no—
81C2×C8×C127C2312yes—
82C22×D127—22no—
83C22×D127—42no—
84C22×D127C2×D412yes—
85Dic254—24no—
86Dic254—22no—
87Dic254C2×D414yes—
88C22⋊D127—216no—
89C22⋊D127—44no—
90C2×Dic127—216no—
91C2×Dic127—22no—
92C2×Dic127—24no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C2×Dic127C2×D412yes—
96C4×D127—24no—
97C4×D127—22no—
98C4×D127—22no—
99C127⋊C8—22no—
100C23×C127—24no—
101C23×C127—22no—
102C23×C127—21no—
103C23×C127—44no—
104C23×C127C2×D411yescharacteristic
105Q8×C127—21no—
106Q8×C127—21no—
107D4×C127—24no—
108D4×C127—22no—
109D4×C127—21no—
110D4×C127—21no—
111D4×C127—42no—
112D4×C127C2×D414yes—
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—24no—
122C22×C127—48no—
123C22×C127—44no—
124C22×C127—41no—
125C22×C127—42no—
126C22×C127#12012yes—
127C22×C127C22×D411yescharacteristic
128C2×D127—22no—
129C2×D127—22no—
130C2×D127—22no—
131C2×D127—44no—
132C4×C127—41no—
133C4×C127#12012yes—
134C4×C127C22×D411yescharacteristic
135C4×C127C22×D411yescharacteristic
136Dic127—22no—
137Dic127—24no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—44no—
141C2×C127—81no—
142C2×C127C2×#12011yescharacteristic
143C2×C127#57512yes—
144D127—42no—
145#3005—1271nomaximal, Sylow
146C127#300511yescharacteristic, complemented, Sylow
147C22×#41—1271no—
148C2×#137—1271no—
149C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
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×Q8—1271no—
162C22×D4—2541no—
163C22×D4C2×D12711yescharacteristic
164C23×C4—1271no—
165C23×C4—2541no—
166#137—2544no—
167AGL(1,ℤ/8)—24no—
168C2×Q16—1272no—
169C2×SD16—1272no—
170C2×SD16C2×D12712yescomplemented
171C2×D8C2×D12712yescomplemented
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—21no—
184C24—1271no—
185#41—22no—
186#41—22no—
187#41—2548no—
188#41—25416no—
189#41—2548no—
190C2×Q8—1274no—
191C2×Q8—1272no—
192C2×Q8—2541no—
193C2×Q8—2542no—
194C2×Q8C22×D12711yescharacteristic
195C2×D4—21no—
196C2×D4—22no—
197C2×D4—21no—
198C2×D4—2542no—
199C2×D4—2544no—
200C2×D4—2548no—
201C2×D4C22⋊D12714yescomplemented
202C2×D4C22×D12711yescharacteristic
203C2×D4C22×D12712yes—
204C22×C4—21no—
205C22×C4—1271no—
206C22×C4—1272no—
207C22×C4—1272no—
208C22×C4—2544no—
209C22×C4—2548no—
210C22×C4—2541no—
211C22×C4—2542no—
212C22×C4—2542no—
213C22×C4C22×D12711yescharacteristic
214Q16—2544no—
215SD16—24no—
216SD16—2544no—
217D8—24no—
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—22no—
229C23—24no—
230C23—21no—
231C23—44no—
232C23—1272no—
233C23—5082no—
234C23—2542no—
235C23D4×D12711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—1274no—
239Q8—2542no—
240Q8—2544no—
241D4—22no—
242D4—21no—
243D4—21no—
244D4—24no—
245D4—42no—
246D4—5084no—
247D4—25416no—
248D4C2×C22⋊D12714yes—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—1272no—
252C2×C4—2542no—
253C2×C4—25416no—
254C2×C4—2542no—
255C2×C4—2542no—
256C2×C4—2544no—
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—24no—
268C22—48no—
269C22—44no—
270C22—41no—
271C22—42no—
272C22—5084no—
273C22—2542no—
274C22—2542no—
275C22—2542no—
276C22#1686117712yes—
277C22C2×D4×D12711yescharacteristic
278C4—41no—
279C4—5082no—
280C4—2542no—
281C4—2544no—
282C4#1686117712yes—
283C4C2×D4×D12711yescharacteristic
284C4C22×C22⋊D12711yescharacteristic
285C2—22no—
286C2—44no—
287C2—81no—
288C2—5082no—
289C2C2×#1686117711yescharacteristic
290C2#2255067212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3005
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×#14523247C2semidirect
2. C2×#16861239C2semidirect
3. #22550383C2semidirect
4. #22550035C2semidirect
5. #22550018C2semidirect
6. #22549886C2semidirect
7. #22549881C2semidirect
8. #22549839C2semidirect
9. C2×AGL(1,ℤ/8)×C127C2semidirect
19. C2×#14523238C22semidirect
20. C2×#14523236C22semidirect
23. #16861185C22semidirect
24. #16861180C22semidirect
28. #16861134C22semidirect
29. #16861130C22semidirect
30. (C2×D127)⋊C8C22semidirect
34. C2×SD16×C127C22semidirect
35. C2×D8×C127C22semidirect
43. C2×Dic254D4semidirect
73. C2×D4×C127D4semidirect
146. C127#3005semidirect
149. C2×AGL(1,ℤ/8)D127semidirect
170. C2×SD16C2×D127semidirect
171. C2×D8C2×D127semidirect
201. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168612392
#225503832
#225500352
#225500182
#225498862
#225498812
#225498392
C2×AGL(1,ℤ/8)×C1272
#3005127