Finite Group Databasebeta

← #30649215

Subgroups — #30649215

Order 16256.

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

normal subgroups only

This group has 596 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×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3C22×#14523233C211yesmaximal, characteristic, complemented
4#22550443C211yesmaximal, characteristic, complemented
5#22550440C211yesmaximal, characteristic, complemented
6#22550264C212yesmaximal, complemented
7#22550262C212yesmaximal, complemented
8#22550257C212yesmaximal, complemented
9#22550251C212yesmaximal, complemented
10#22549834C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14#16861237—24no—
15#16861229—24no—
16C2×#14523238C2211yescharacteristic, complemented
17C2×#14523237C2212yescomplemented
18C2×#14523236C2211yescharacteristic, complemented
19C2×#14523235C2211yescharacteristic, complemented
20C2×D1016C2212yescomplemented
21C2×#14523233—24no—
22C2×#14523233C2212yes—
23C2×#14523233C2212yes—
24C2×#14523233C2211yescharacteristic, complemented
25C22×C127⋊C8C2211yescharacteristic
26#16861205C2212yescomplemented
27#16861205C2212yescomplemented
28#16861198C2212yescomplemented
29#16861196C2212yescomplemented
30#16861183—22no—
31#16861180—22no—
32#16861178—22no—
33#16861177—22no—
34#16861168C2211yescharacteristic
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×D508C2312yes—
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—24no—
53C2×C22⋊D127—24no—
54C2×C22⋊D127—21no—
55C2×C4×D127—21no—
56#14523238—22no—
57#14523237—24no—
58#14523236—22no—
59#14523235—22no—
60D1016—24no—
61#14523233—24no—
62#14523233—24no—
63#14523233—22no—
64C2×C127⋊C8D414yescomplemented
65C2×C127⋊C8C2311yescharacteristic
66C2×C127⋊C8C2311yescharacteristic
67C4⋊Dic127D412yes—
68C22⋊Dic127—24no—
69C22⋊Dic127D412yes—
70(C2×D127)⋊C4—24no—
71C4×Dic127C2312yes—
72C24×C127—21no—
73#41×C127—22no—
74#41×C127—22no—
75C2×Q8×C127C2311yescharacteristic
76C2×D4×C127—21no—
77C2×D4×C127—22no—
78C2×D4×C127—22no—
79C2×D4×C127—21no—
80C2×D4×C127C2312yes—
81C2×D4×C127C2311yescharacteristic
82C22×C4×C127—21no—
83C22×C4×C127C2311yescharacteristic
84SD16×C127—24no—
85D8×C127—24no—
86M16×C127D414yescomplemented
87C2×C8×C127C2312yes—
88C22×D127—22no—
89C22×D127—21no—
90C22×D127—42no—
91C22×D127—82no—
92Dic254—21no—
93Dic254—21no—
94D508—22no—
95D508—21no—
96D508—21no—
97D508—22no—
98D508—42no—
99C22⋊D127—42no—
100C22⋊D127—84no—
101C22⋊D127—84no—
102C2×Dic127—21no—
103C2×Dic127C2×D414yes—
104C4×D127—42no—
105C127⋊C8C2×D412yes—
106C127⋊C8C2×D412yes—
107C23×C127—21no—
108C23×C127—22no—
109C23×C127—42no—
110C23×C127—82no—
111C23×C127C2×D411yescharacteristic
112Q8×C127—21no—
113Q8×C127—21no—
114D4×C127—21no—
115D4×C127—22no—
116D4×C127—22no—
117D4×C127—21no—
118D4×C127—42no—
119D4×C127—42no—
120C2×C4×C127—21no—
121C2×C4×C127—42no—
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127C2×D412yes—
124C2×C4×C127C2×D412yes—
125C2×C4×C127C2411yescharacteristic
126C8×C127C2×D414yes—
127C22×C127—22no—
128C22×C127—41no—
129C22×C127—42no—
130C22×C127—44no—
131C22×C127—42no—
132C22×C127—84no—
133C22×C127#12712yes—
134C22×C127C22×D411yescharacteristic
135C2×D127—41no—
136C2×D127—42no—
137C2×D127—42no—
138C2×D127—44no—
139C2×D127—84no—
140C4×C127—41no—
141C4×C127#12712yes—
142C4×C127C22×D411yescharacteristic
143C4×C127C22×D411yescharacteristic
144Dic127—24no—
145Dic127—41no—
146C2×C127—22no—
147C2×C127—81no—
148C2×C127—82no—
149C2×C127C2×#12711yescharacteristic
150C2×C127#57612yes—
151D127—81no—
152D127—82no—
153#3155—1271nomaximal, Sylow
154C127#315511yescharacteristic, complemented, Sylow
155C2×AGL(1,ℤ/8)—1271no—
156C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
157C2×#135—1271no—
158C22×D8—1271no—
159#535—1271no—
160#535—1271no—
161#496—1272no—
162#496—1272no—
163#495—1272no—
164#493—1272no—
165#405—1271no—
166C2×#41—1271no—
167C2×#41C2×D12711yescharacteristic
168C22×D4—1271no—
169C22×D4C2×D12711yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—2544no—
172#135—2544no—
173C2×Q16—1271no—
174C2×SD16—1271no—
175C2×SD16—1271no—
176C2×SD16—1272no—
177C2×SD16C2×D12712yescomplemented
178C2×D8—1272no—
179C2×D8—1272no—
180C2×D8—1271no—
181C2×D8—1272no—
182C2×D8—2544no—
183C2×D8C2×D12712yescomplemented
184C2×M16C2×D12711yescharacteristic
185C22×C8—1271no—
186#127—1272no—
187#127—1272no—
188#124—1272no—
189#124—1272no—
190#121—2542no—
191#121—2542no—
192#120—2542no—
193#120—2542no—
194#117—1271no—
195#97—1272no—
196C4×C8—1272no—
197C24—21no—
198C24—2541no—
199#41—22no—
200#41—22no—
201#41—2542no—
202#41—2542no—
203C2×Q8—1271no—
204C2×Q8C22×D12711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1272no—
210C2×D4—1271no—
211C2×D4—2544no—
212C2×D4—2544no—
213C2×D4—2541no—
214C2×D4—2541no—
215C2×D4—2542no—
216C2×D4—2542no—
217C2×D4C22×D12712yes—
218C2×D4C22×D12711yescharacteristic
219C22×C4—21no—
220C22×C4—2541no—
221C22×C4C22×D12711yescharacteristic
222Q16—2542no—
223SD16—24no—
224SD16—2542no—
225SD16—2542no—
226SD16—2544no—
227D8—24no—
228D8—2544no—
229D8—2544no—
230D8—2542no—
231D8—2544no—
232M16C22⋊D12714yescomplemented
233C2×C8—1274no—
234C2×C8—1271no—
235C2×C8—1271no—
236C2×C8C22×D12712yes—
237C4⋊C4—1272no—
238C22⋊C4—1272no—
239C22⋊C4—2544no—
240C22⋊C4—2544no—
241C42—1272no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—10162no—
247C23—5082no—
248C23—2542no—
249C23—2541no—
250C23D4×D12711yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2541no—
254Q8—2541no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—10164no—
262D4—10164no—
263D4—5082no—
264D4—5082no—
265D4—2542no—
266D4—2541no—
267D4—2541no—
268D4—2542no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—1274no—
272C2×C4—5082no—
273C2×C4—2541no—
274C2×C4D4×D12711yescharacteristic
275C2×C4D4×D12712yes—
276C2×C4C2×C22⋊D12712yes—
277C2×C4C23×D12711yescharacteristic
278C8—1272no—
279C8—1272no—
280C8C2×C22⋊D12714yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—10164no—
288C22—5081no—
289C22—5082no—
290C22—5082no—
291C22—5084no—
292C22#1686120512yes—
293C22C2×D4×D12711yescharacteristic
294C4—41no—
295C4—5081no—
296C4—2544no—
297C4#1686120512yes—
298C4C2×D4×D12711yescharacteristic
299C4C22×C22⋊D12711yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—10162no—
304C2—10161no—
305C2C2×#1686120511yescharacteristic
306C2#2255068012yes—
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#3155
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×#16861229C2semidirect
3. C22×#14523233C2semidirect
4. #22550443C2semidirect
5. #22550440C2semidirect
6. #22550264C2semidirect
7. #22550262C2semidirect
8. #22550257C2semidirect
9. #22550251C2semidirect
10. #22549834C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
16. C2×#14523238C22semidirect
17. C2×#14523237C22semidirect
18. C2×#14523236C22semidirect
19. C2×#14523235C22semidirect
20. C2×D1016C22semidirect
24. C2×#14523233C22semidirect
26. #16861205C22semidirect
27. #16861205C22semidirect
28. #16861198C22semidirect
29. #16861196C22semidirect
35. #16861119C22semidirect
36. C8×Dic127C22semidirect
40. C2×SD16×C127C22semidirect
41. C2×D8×C127C22semidirect
64. C2×C127⋊C8D4semidirect
86. M16×C127D4semidirect
154. C127#3155semidirect
156. C2×AGL(1,ℤ/8)D127semidirect
177. C2×SD16C2×D127semidirect
183. C2×D8C2×D127semidirect
232. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612292
C22×#145232332
#225504432
#225504402
#225502642
#225502622
#225502572
#225502512
#225498342
C2×AGL(1,ℤ/8)×C1272
#3155127