Finite Group Databasebeta

← #30648015

Subgroups — #30648015

Order 16256.

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

normal subgroups only

This group has 732 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
1C2×#16861256C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3#22550443C211yesmaximal, characteristic, complemented
4#22550018C214yesmaximal, complemented
5#22550013C214yesmaximal, complemented
6#22549886C211yesmaximal, characteristic, complemented
7#22549872C211yesmaximal, characteristic, complemented
8#22549844C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
10#16861256—24no—
11#16861256—24no—
12C2×#14523248—21no—
13C2×#14523248C2211yescharacteristic
14C2×#14523247—22no—
15C2×#14523247C2212yescomplemented
16C22×Dic254—21no—
17C22×Dic254C2211yescharacteristic
18C2×Q8×D127C2212yescomplemented
19C2×#14523236—24no—
20C2×#14523236C2214yes—
21C22×C127⋊C8C2211yescharacteristic
22#16861205—21no—
23#16861196—21no—
24#16861180C2214yes—
25#16861178—22no—
26#16861168C2211yescharacteristic
27#16861134C2214yescomplemented
28#16861133C2212yescomplemented
29#16861130C2212yescomplemented
30(C2×D127)⋊C8C2214yescomplemented
31C2×#41×C127C2211yescharacteristic
32C22×D4×C127C2211yescharacteristic
33AGL(1,ℤ/8)×C127—24no—
34C2×SD16×C127C2212yes—
35C2×D8×C127C2212yes—
36C2×M16×C127C2211yescharacteristic
37#14523248—28no—
38#14523248—22no—
39#14523248—22no—
40#14523247—216no—
41#14523247—24no—
42#14523247—24no—
43C2×Dic254—28no—
44C2×Dic254—22no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254D411yescharacteristic
48C2×Dic254D414yescomplemented
49C2×Dic254C2312yes—
50Q8×D127—24no—
51Q8×D127—24no—
52C2×D508D411yescharacteristic
53C2×C22⋊D127—22no—
54C2×C22⋊D127—24no—
55C2×C22⋊D127D412yes—
56C22×Dic127—22no—
57C22×Dic127—21no—
58C2×C4×D127—22no—
59C2×C4×D127C2312yes—
60#14523236—28no—
61C2×C127⋊C8—22no—
62C2×C127⋊C8C2312yes—
63C4⋊Dic127C2312yes—
64C22⋊Dic127—24no—
65C22⋊Dic127—21no—
66C4×Dic127—21no—
67C24×C127—21no—
68#41×C127—22no—
69#41×C127D414yescomplemented
70C2×Q8×C127C2311yescharacteristic
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127—22no—
74C2×D4×C127—21no—
75C2×D4×C127C2312yes—
76C2×D4×C127C2311yescharacteristic
77C22×C4×C127—21no—
78C22×C4×C127C2311yescharacteristic
79SD16×C127—24no—
80D8×C127—24no—
81M16×C127—22no—
82C2×C8×C127C2312yes—
83C22×D127C2×D412yes—
84Dic254—28no—
85Dic254—22no—
86Dic254—24no—
87Dic254—22no—
88Dic254C2×D414yes—
89D508—22no—
90C22⋊D127—28no—
91C22⋊D127—24no—
92C22⋊D127—84no—
93C2×Dic127—28no—
94C2×Dic127—24no—
95C2×Dic127—22no—
96C2×Dic127—22no—
97C2×Dic127C2×D412yes—
98C4×D127—28no—
99C4×D127—22no—
100C4×D127—22no—
101C127⋊C8—22no—
102C23×C127—21no—
103C23×C127—22no—
104C23×C127—42no—
105C23×C127—82no—
106C23×C127C2×D411yescharacteristic
107Q8×C127—21no—
108Q8×C127C2×D412yes—
109D4×C127—24no—
110D4×C127—21no—
111D4×C127—22no—
112D4×C127—22no—
113D4×C127—42no—
114D4×C127C2×D412yes—
115C2×C4×C127—21no—
116C2×C4×C127—24no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2411yescharacteristic
121C8×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—42no—
125C22×C127—42no—
126C22×C127—44no—
127C22×C127—84no—
128C22×C127#12012yes—
129C22×C127C22×D411yescharacteristic
130C2×D127—22no—
131C2×D127—24no—
132C4×C127—22no—
133C4×C127#12012yes—
134C4×C127C22×D411yescharacteristic
135C4×C127C22×D411yescharacteristic
136Dic127—24no—
137Dic127—22no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—42no—
141C2×C127—82no—
142C2×C127C2×#12011yescharacteristic
143C2×C127#57712yes—
144D127—42no—
145#3010—1271nomaximal, Sylow
146C127#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1271no—
148C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
149C22×SD16—1271no—
150#535—1271no—
151#449—1274no—
152#448—1274no—
153#419—1271no—
154#416—1271no—
155#408—1271no—
1562−1+4—2544no—
1572−1+4—2544no—
158C2×#41—1271no—
159C2×#41—1272no—
160C2×#41—2541no—
161C2×#41—2542no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8—1272no—
164C22×Q8—1271no—
165C22×Q8—2541no—
166C22×D4C2×D12711yescharacteristic
167AGL(1,ℤ/8)—24no—
168C2×SD16—1274no—
169C2×SD16—2544no—
170C2×SD16C2×D12712yes—
171C2×D8C2×D12712yes—
172C2×M16C2×D12711yescharacteristic
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—21no—
184#41—22no—
185#41—2548no—
186#41—2544no—
187#41—25416no—
188#41—2544no—
189#41—2542no—
190#41—2542no—
191#41C22⋊D12714yescomplemented
192C2×Q8—1271no—
193C2×Q8—1274no—
194C2×Q8—1272no—
195C2×Q8—2542no—
196C2×Q8—2548no—
197C2×Q8—2544no—
198C2×Q8—2544no—
199C2×Q8—2541no—
200C2×Q8—2542no—
201C2×Q8C22×D12711yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—1272no—
207C2×D4—1271no—
208C2×D4—2544no—
209C2×D4—2542no—
210C2×D4C22×D12712yes—
211C2×D4C22×D12711yescharacteristic
212C22×C4—21no—
213C22×C4—1272no—
214C22×C4—2542no—
215C22×C4—2542no—
216C22×C4—2541no—
217C22×C4C22×D12711yescharacteristic
218SD16—24no—
219SD16—2548no—
220D8—24no—
221M16—22no—
222C2×C8—1272no—
223C2×C8—2542no—
224C2×C8C22×D12712yes—
225C4⋊C4—1272no—
226C22⋊C4—2544no—
227C22⋊C4—2541no—
228C42—2541no—
229C23—21no—
230C23—22no—
231C23—42no—
232C23—82no—
233C23—1272no—
234C23D4×D12711yescharacteristic
235Q8—21no—
236Q8—1274no—
237Q8—2548no—
238Q8—2542no—
239Q8—2542no—
240Q8—2544no—
241Q8C2×C22⋊D12712yes—
242D4—24no—
243D4—21no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—10164no—
248D4—2548no—
249D4—2544no—
250D4—2542no—
251D4C2×C22⋊D12712yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—1272no—
255C2×C4—2548no—
256C2×C4—2542no—
257C2×C4—2548no—
258C2×C4—2542no—
259C2×C4—2544no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4D4×D12712yes—
263C2×C4D4×D12711yescharacteristic
264C2×C4C2×C22⋊D12712yes—
265C2×C4C23×D12711yescharacteristic
266C8—22no—
267C8—2542no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—42no—
272C22—44no—
273C22—84no—
274C22—2542no—
275C22—2544no—
276C22#1686117712yes—
277C22C2×D4×D12711yescharacteristic
278C4—22no—
279C4—5082no—
280C4—2542no—
281C4—2544no—
282C4#1686117712yes—
283C4C2×D4×D12711yescharacteristic
284C4C22×C22⋊D12711yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—5082no—
289C2C2×#1686117711yescharacteristic
290C2#2255068612yes—
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#3010
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. C2×#16861256C2semidirect
2. C22×#14523236C2semidirect
3. #22550443C2semidirect
4. #22550018C2semidirect
5. #22550013C2semidirect
6. #22549886C2semidirect
7. #22549872C2semidirect
8. #22549844C2semidirect
9. C2×AGL(1,ℤ/8)×C127C2semidirect
15. C2×#14523247C22semidirect
18. C2×Q8×D127C22semidirect
27. #16861134C22semidirect
28. #16861133C22semidirect
29. #16861130C22semidirect
30. (C2×D127)⋊C8C22semidirect
48. C2×Dic254D4semidirect
69. #41×C127D4semidirect
146. C127#3010semidirect
148. C2×AGL(1,ℤ/8)D127semidirect
191. #41C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C22×#145232362
#225504432
#225500182
#225500132
#225498862
#225498722
#225498442
C2×AGL(1,ℤ/8)×C1272
#3010127