Finite Group Databasebeta

← #30649200

Subgroups — #30649200

Order 16256.

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

normal subgroups only

This group has 564 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×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C2×#16861231C211yesmaximal, characteristic, complemented
4#22550450C211yesmaximal, characteristic, complemented
5#22550443C211yesmaximal, characteristic, complemented
6#22550267C212yesmaximal, complemented
7#22550262C212yesmaximal, complemented
8#22550257C212yesmaximal, complemented
9#22550247C212yesmaximal, complemented
10#22549831C211yesmaximal, characteristic, complemented
11C22×SD16×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×#14523248C2211yescharacteristic
14#16861240—24no—
15#16861235—24no—
16#16861231—24no—
17C2×Dic508C2211yescharacteristic, complemented
18C2×#14523238C2212yescomplemented
19C2×#14523237C2211yescharacteristic, complemented
20C2×#14523237C2211yescharacteristic, complemented
21C2×#14523236C2212yes—
22C2×#14523235C2212yescomplemented
23C2×D1016C2211yescharacteristic, complemented
24C2×#14523233C2212yes—
25C2×#14523232C2211yescharacteristic
26#16861207C2212yescomplemented
27#16861205C2212yes—
28#16861198C2212yescomplemented
29#16861196C2212yes—
30#16861192—22no—
31#16861185—22no—
32#16861180—22no—
33#16861178—22no—
34#16861168C2211yescharacteristic
35#16861119C2212yescomplemented
36C8×Dic127C2212yescomplemented
37C22×Q8×C127C2211yescharacteristic
38C22×D4×C127C2211yescharacteristic
39C2×SD16×C127—24no—
40C2×SD16×C127C2212yes—
41C2×SD16×C127C2212yes—
42C22×C8×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45#14523248—22no—
46#14523248—22no—
47C2×Dic254C2311yescharacteristic
48C2×Dic254C2311yescharacteristic
49C2×D508C2311yescharacteristic
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—24no—
52C2×C22⋊D127—21no—
53C2×C22⋊D127—21no—
54C2×C4×D127—21no—
55C2×C4×D127—21no—
56Dic508—22no—
57#14523238—24no—
58#14523237—22no—
59#14523237—22no—
60#14523236—24no—
61#14523235—24no—
62D1016—22no—
63#14523233—24no—
64#14523232D414yescomplemented
65C2×C127⋊C8C2312yes—
66C4⋊Dic127D412yes—
67Dic127⋊C4—24no—
68C22⋊Dic127—24no—
69C22⋊Dic127D412yes—
70(C2×D127)⋊C4—24no—
71C4×Dic127C2312yes—
72C24×C127—21no—
73C2×Q8×C127—21no—
74C2×Q8×C127—22no—
75C2×Q8×C127—22no—
76C2×Q8×C127C2312yes—
77C2×D4×C127—22no—
78C2×D4×C127—22no—
79C2×D4×C127—21no—
80C2×D4×C127C2312yes—
81C22×C4×C127—21no—
82C22×C4×C127C2311yescharacteristic
83SD16×C127—24no—
84SD16×C127—24no—
85C2×C8×C127D414yescomplemented
86C2×C8×C127C2311yescharacteristic
87C2×C8×C127C2311yescharacteristic
88C22×D127—21no—
89C22×D127—21no—
90Dic254—21no—
91Dic254—21no—
92Dic254—21no—
93Dic254—21no—
94D508—21no—
95D508—21no—
96D508—21no—
97D508—21no—
98C22⋊D127—42no—
99C22⋊D127—42no—
100C22⋊D127—84no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127C2×D414yes—
104C4×D127—42no—
105C4×D127—42no—
106C127⋊C8C2×D414yes—
107C23×C127—22no—
108C23×C127—42no—
109C23×C127—82no—
110C23×C127C2×D411yescharacteristic
111Q8×C127—22no—
112Q8×C127—22no—
113Q8×C127—42no—
114D4×C127—22no—
115D4×C127—22no—
116D4×C127—42no—
117C2×C4×C127—22no—
118C2×C4×C127—42no—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2411yescharacteristic
123C8×C127C2×D412yes—
124C8×C127C2×D412yes—
125C22×C127—22no—
126C22×C127—42no—
127C22×C127—44no—
128C22×C127—84no—
129C22×C127#12712yes—
130C22×C127C22×D411yescharacteristic
131C2×D127—41no—
132C2×D127—41no—
133C2×D127—42no—
134C2×D127—42no—
135C4×C127—42no—
136C4×C127#12712yes—
137C4×C127C22×D411yescharacteristic
138C4×C127C22×D411yescharacteristic
139Dic127—24no—
140Dic127—41no—
141Dic127—41no—
142C2×C127—22no—
143C2×C127—82no—
144C2×C127C2×#12711yescharacteristic
145C2×C127#57712yes—
146D127—81no—
147D127—81no—
148#3154—1271nomaximal, Sylow
149C127#315411yescharacteristic, complemented, Sylow
150C2×#137—1271no—
151C2×AGL(1,ℤ/8)—1271no—
152C2×#135—1271no—
153C22×SD16D12711yescharacteristic, complemented
154#536—1271no—
155#535—1271no—
156#497—1272no—
157#496—1272no—
158#495—1272no—
159#492—1272no—
160#405—1271no—
161C2×#41—1271no—
162C2×#41—1271no—
163C22×Q8C2×D12711yescharacteristic
164C22×D4C2×D12711yescharacteristic
165#137—2544no—
166AGL(1,ℤ/8)—2544no—
167#135—2544no—
168C2×Q16—1272no—
169C2×Q16—1271no—
170C2×SD16—24no—
171C2×SD16—1272no—
172C2×SD16—1272no—
173C2×SD16—1271no—
174C2×SD16—1271no—
175C2×SD16C2×D12712yes—
176C2×SD16C2×D12712yes—
177C2×D8—1272no—
178C2×D8—1271no—
179C2×M16—1271no—
180C22×C8C2×D12711yescharacteristic
181C4⋊Q8—1272no—
182#127—1272no—
183#124—1272no—
184#124—1272no—
185#123—2542no—
186C22⋊Q8—2542no—
187#121—2542no—
188#120—2542no—
189#117—1271no—
190#97—1272no—
191C4×C8—1272no—
192C24—21no—
193#41—2542no—
194#41—2542no—
195#41—2542no—
196#41—2542no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1271no—
201C2×Q8—1271no—
202C2×Q8C22×D12712yes—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—1271no—
207C2×D4—1271no—
208C2×D4—2544no—
209C2×D4—2541no—
210C2×D4—2541no—
211C2×D4C22×D12712yes—
212C22×C4—21no—
213C22×C4—2541no—
214C22×C4—2541no—
215C22×C4C22×D12711yescharacteristic
216Q16—2544no—
217Q16—2542no—
218SD16—24no—
219SD16—24no—
220SD16—2544no—
221SD16—2544no—
222SD16—2542no—
223SD16—2542no—
224D8—2544no—
225D8—2542no—
226M16—1274no—
227C2×C8—1272no—
228C2×C8C22⋊D12714yescomplemented
229C2×C8C22×D12711yescharacteristic
230C2×C8C22×D12711yescharacteristic
231C4⋊C4—1272no—
232C4⋊C4—2544no—
233C22⋊C4—1272no—
234C22⋊C4—2544no—
235C22⋊C4—2544no—
236C42—1272no—
237C23—22no—
238C23—42no—
239C23—82no—
240C23—2541no—
241C23—2541no—
242C23D4×D12711yescharacteristic
243Q8—22no—
244Q8—22no—
245Q8—42no—
246Q8—2541no—
247Q8—2541no—
248Q8—2541no—
249Q8—2541no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—10164no—
254D4—5082no—
255D4—5082no—
256D4—2541no—
257D4—2541no—
258D4—2541no—
259D4—2541no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—1274no—
263C2×C4—5082no—
264C2×C4—5082no—
265C2×C4—2541no—
266C2×C4—2541no—
267C2×C4D4×D12711yescharacteristic
268C2×C4D4×D12712yes—
269C2×C4C2×C22⋊D12712yes—
270C2×C4C23×D12711yescharacteristic
271C8—1274no—
272C8C2×C22⋊D12712yes—
273C8C2×C22⋊D12712yes—
274C22—22no—
275C22—42no—
276C22—44no—
277C22—84no—
278C22—5081no—
279C22—5081no—
280C22—5082no—
281C22—5082no—
282C22#1686120512yes—
283C22C2×D4×D12711yescharacteristic
284C4—42no—
285C4—5081no—
286C4—5081no—
287C4—2544no—
288C4#1686120512yes—
289C4C2×D4×D12711yescharacteristic
290C4C22×C22⋊D12711yescharacteristic
291C2—22no—
292C2—82no—
293C2—10161no—
294C2—10161no—
295C2C2×#1686120511yescharacteristic
296C2#2255068212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC22×C127C22×D4
hypercenterC22×SD16D127
hypocenterC127#3154
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C2×#16861235C2semidirect
3. C2×#16861231C2semidirect
4. #22550450C2semidirect
5. #22550443C2semidirect
6. #22550267C2semidirect
7. #22550262C2semidirect
8. #22550257C2semidirect
9. #22550247C2semidirect
10. #22549831C2semidirect
11. C22×SD16×C127C2semidirect
17. C2×Dic508C22semidirect
18. C2×#14523238C22semidirect
19. C2×#14523237C22semidirect
20. C2×#14523237C22semidirect
22. C2×#14523235C22semidirect
23. C2×D1016C22semidirect
26. #16861207C22semidirect
28. #16861198C22semidirect
35. #16861119C22semidirect
36. C8×Dic127C22semidirect
64. #14523232D4semidirect
85. C2×C8×C127D4semidirect
149. C127#3154semidirect
153. C22×SD16D127semidirect
228. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612352
C2×#168612312
#225504502
#225504432
#225502672
#225502622
#225502572
#225502472
#225498312
C22×SD16×C1272
#3154127