Finite Group Databasebeta

← #30649178

Subgroups — #30649178

Order 16256.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C2×#16861234C211yesmaximal, characteristic, complemented
4#22550422C211yesmaximal, characteristic, complemented
5C2×#16861205C211yesmaximal, characteristic, complemented
6#22550264C212yesmaximal, complemented
7#22550262C212yesmaximal, complemented
8#22550251C212yesmaximal, complemented
9#22550247C212yesmaximal, complemented
10M16×Dic127C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14C22×C22⋊D127—22no—
15#16861237—24no—
16#16861235—24no—
17#16861234—24no—
18C2×#14523237C2212yescomplemented
19C2×#14523236C2212yescomplemented
20C2×#14523235C2212yescomplemented
21C2×D1016C2212yescomplemented
22C2×#14523233C2212yescomplemented
23C2×#14523233C2212yescomplemented
24C2×#14523232C2211yescharacteristic
25#16861207C2211yescharacteristic, complemented
26#16861205—22no—
27#16861205C2212yes—
28#16861205C2212yes—
29#16861205C2211yescharacteristic, complemented
30#16861198C2211yescharacteristic, complemented
31#16861196C2211yescharacteristic, complemented
32#16861181—22no—
33C4×C22⋊D127—22no—
34C2×C4×Dic127C2211yescharacteristic
35#16861119C2212yescomplemented
36C8×Dic127C2212yescomplemented
37C2×#41×C127C2211yescharacteristic
38C22×D4×C127C2211yescharacteristic
39AGL(1,ℤ/8)×C127—24no—
40C2×SD16×C127C2212yescomplemented
41C2×D8×C127C2212yescomplemented
42C2×M16×C127C2211yescharacteristic
43C23×D127—21no—
44#14523248—22no—
45#14523248—22no—
46C2×Dic254C2311yescharacteristic
47C2×D508—22no—
48C2×D508—22no—
49C2×D508—21no—
50C2×D508C2311yescharacteristic
51C2×D508C2312yes—
52C2×C22⋊D127—24no—
53C2×C22⋊D127—24no—
54C2×C22⋊D127—22no—
55C2×C22⋊D127—21no—
56C2×C22⋊D127—48no—
57C22×Dic127D412yes—
58C2×C4×D127—21no—
59#14523237—24no—
60#14523236—24no—
61#14523235—24no—
62D1016—24no—
63#14523233—24no—
64#14523233—24no—
65#14523232D414yescomplemented
66C2×C127⋊C8C2312yes—
67Dic127⋊C4—22no—
68C22⋊Dic127—22no—
69(C2×D127)⋊C4—22no—
70C4×Dic127D412yes—
71C4×Dic127C2311yescharacteristic
72C4×Dic127C2311yescharacteristic
73C24×C127—21no—
74#41×C127—22no—
75#41×C127—22no—
76C2×Q8×C127C2311yescharacteristic
77C2×D4×C127—21no—
78C2×D4×C127—22no—
79C2×D4×C127—22no—
80C2×D4×C127—21no—
81C2×D4×C127C2311yescharacteristic
82C2×D4×C127C2312yes—
83C22×C4×C127—21no—
84C22×C4×C127C2311yescharacteristic
85SD16×C127—24no—
86D8×C127—24no—
87M16×C127D414yescomplemented
88C2×C8×C127C2312yes—
89C22×D127—21no—
90C22×D127—22no—
91C22×D127—44no—
92C22×D127—42no—
93Dic254—21no—
94Dic254—21no—
95D508—21no—
96D508—22no—
97D508—21no—
98D508—22no—
99D508—42no—
100C22⋊D127—48no—
101C22⋊D127—42no—
102C22⋊D127—44no—
103C22⋊D127—48no—
104C2×Dic127—24no—
105C2×Dic127—21no—
106C2×Dic127C2×D412yes—
107C2×Dic127C2×D412yes—
108C4×D127—42no—
109C127⋊C8C2×D414yescomplemented
110C23×C127—21no—
111C23×C127—22no—
112C23×C127—44no—
113C23×C127—42no—
114C23×C127C2×D411yescharacteristic
115Q8×C127—21no—
116Q8×C127—21no—
117D4×C127—21no—
118D4×C127—22no—
119D4×C127—22no—
120D4×C127—21no—
121D4×C127—42no—
122D4×C127—42no—
123C2×C4×C127—21no—
124C2×C4×C127—42no—
125C2×C4×C127C2×D411yescharacteristic
126C2×C4×C127C2×D412yes—
127C2×C4×C127C2×D412yes—
128C2×C4×C127C2411yescharacteristic
129C8×C127C2×D414yescomplemented
130C22×C127—22no—
131C22×C127—41no—
132C22×C127—42no—
133C22×C127—44no—
134C22×C127—42no—
135C22×C127—84no—
136C22×C127#12712yes—
137C22×C127C22×D411yescharacteristic
138C2×D127—41no—
139C2×D127—42no—
140C2×D127—44no—
141C2×D127—42no—
142C2×D127—84no—
143C4×C127—41no—
144C4×C127#12712yes—
145C4×C127C22×D411yescharacteristic
146C4×C127C22×D411yescharacteristic
147Dic127—22no—
148Dic127—41no—
149Dic127AGL(1,ℤ/8)14yescomplemented
150C2×C127—22no—
151C2×C127—81no—
152C2×C127—82no—
153C2×C127C2×#12711yescharacteristic
154C2×C127C2×AGL(1,ℤ/8)12yes—
155D127—81no—
156D127—82no—
157#3151—1271nomaximal, Sylow
158C127#315111yescharacteristic, complemented, Sylow
159C2×AGL(1,ℤ/8)—1271no—
160C2×AGL(1,ℤ/8)—1271no—
161C2×AGL(1,ℤ/8)—1271no—
162C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
163#532—1271no—
164C2×#127—1271no—
165#496—1272no—
166#496—1272no—
167#493—1272no—
168#492—1272no—
169C4×M16—1271no—
170C2×#41—1271no—
171C2×#41C2×D12711yescharacteristic
172C22×D4—1271no—
173C22×D4—2542no—
174C22×D4C2×D12711yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2544no—
177AGL(1,ℤ/8)—2544no—
178AGL(1,ℤ/8)—2544no—
179C2×SD16—1272no—
180C2×SD16—1272no—
181C2×SD16—1272no—
182C2×SD16C2×D12712yescomplemented
183C2×D8—1272no—
184C2×D8—1272no—
185C2×D8—1272no—
186C2×D8C2×D12712yescomplemented
187C2×M16—1271no—
188C2×M16C2×D12711yescharacteristic
189C4⋊Q8—1271no—
190#127—1272no—
191#127—1272no—
192#127—1271no—
193#127—2542no—
194#124—1271no—
195#124—1271no—
196#121—2542no—
197C4×D4—2542no—
198C2×C42—1271no—
199#97—1272no—
200C4×C8—1272no—
201C24—21no—
202C24—2541no—
203#41—22no—
204#41—22no—
205#41—2542no—
206#41—2542no—
207C2×Q8—1271no—
208C2×Q8C22×D12711yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—1271no—
214C2×D4—1272no—
215C2×D4—5088no—
216C2×D4—2542no—
217C2×D4—2544no—
218C2×D4—2544no—
219C2×D4—2541no—
220C2×D4—2541no—
221C2×D4—2542no—
222C2×D4—2542no—
223C2×D4C22×D12711yescharacteristic
224C2×D4C22×D12712yes—
225C22×C4—21no—
226C22×C4—1272no—
227C22×C4—2541no—
228C22×C4C22×D12711yescharacteristic
229SD16—24no—
230SD16—2544no—
231SD16—2544no—
232SD16—2544no—
233D8—24no—
234D8—2544no—
235D8—2544no—
236D8—2544no—
237M16—1274no—
238M16C22⋊D12714yescomplemented
239C2×C8—1272no—
240C2×C8C22×D12712yes—
241C4⋊C4—2542no—
242C22⋊C4—2542no—
243C22⋊C4—2542no—
244C42—1272no—
245C42—1271no—
246C42—1271no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—5084no—
252C23—5082no—
253C23—2541no—
254C23—2542no—
255C23D4×D12711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2541no—
259Q8—2541no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—5088no—
267D4—5084no—
268D4—5082no—
269D4—5088no—
270D4—5082no—
271D4—2541no—
272D4—2542no—
273D4—2541no—
274D4—2542no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—1272no—
278C2×C4—1272no—
279C2×C4—5082no—
280C2×C4—2544no—
281C2×C4—2541no—
282C2×C4D4×D12711yescharacteristic
283C2×C4D4×D12712yes—
284C2×C4C2×C22⋊D12712yes—
285C2×C4C23×D12711yescharacteristic
286C8—1274no—
287C8C2×C22⋊D12714yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—10164no—
295C22—5084no—
296C22—5082no—
297C22—5081no—
298C22—5082no—
299C22#1686120512yes—
300C22C2×D4×D12711yescharacteristic
301C4—41no—
302C4—1274no—
303C4—5081no—
304C4—2542no—
305C4#1686120512yes—
306C4C2×D4×D12711yescharacteristic
307C4C22×C22⋊D12711yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—10162no—
312C2—10161no—
313C2C2×#1686120511yescharacteristic
314C2AGL(1,ℤ/8)×D12712yes—
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#3151
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×#16861237C2semidirect
2. C2×#16861235C2semidirect
3. C2×#16861234C2semidirect
4. #22550422C2semidirect
5. C2×#16861205C2semidirect
6. #22550264C2semidirect
7. #22550262C2semidirect
8. #22550251C2semidirect
9. #22550247C2semidirect
10. M16×Dic127C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
18. C2×#14523237C22semidirect
19. C2×#14523236C22semidirect
20. C2×#14523235C22semidirect
21. C2×D1016C22semidirect
22. C2×#14523233C22semidirect
23. C2×#14523233C22semidirect
25. #16861207C22semidirect
29. #16861205C22semidirect
30. #16861198C22semidirect
31. #16861196C22semidirect
35. #16861119C22semidirect
36. C8×Dic127C22semidirect
40. C2×SD16×C127C22semidirect
41. C2×D8×C127C22semidirect
65. #14523232D4semidirect
87. M16×C127D4semidirect
109. C127⋊C8C2×D4semidirect
129. C8×C127C2×D4semidirect
149. Dic127AGL(1,ℤ/8)semidirect
158. C127#3151semidirect
162. C2×AGL(1,ℤ/8)D127semidirect
182. C2×SD16C2×D127semidirect
186. C2×D8C2×D127semidirect
238. M16C22⋊D127semidirect
287. C8C2×C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612352
C2×#168612342
#225504222
C2×#168612052
#225502642
#225502622
#225502512
#225502472
M16×Dic1272
C2×AGL(1,ℤ/8)×C1272
#3151127