Finite Group Databasebeta

← #30647959

Subgroups — #30647959

Order 16256.

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

normal subgroups only

This group has 994 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
1C22×D4×D127C211yesmaximal, characteristic, complemented
2C2×#16861237C211yesmaximal, characteristic, complemented
3C2×#16861235C211yesmaximal, characteristic, complemented
4#22550388C211yesmaximal, characteristic, complemented
5#22550028C214yesmaximal, complemented
6#22550011C214yesmaximal, complemented
7#22549880C211yesmaximal, characteristic, complemented
8#22549839C211yesmaximal, characteristic, complemented
9#418×C127C211yesmaximal, characteristic, complemented
10C24×D127—21no—
11C2×#14523248C2211yescharacteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C22⋊D127—21no—
14C2×D4×D127—24no—
15C2×D4×D127—22no—
16C2×D4×D127—24no—
17C2×D4×D127C2214yes—
18C22×C4×D127C2211yescharacteristic
19#16861237—24no—
20#16861235—24no—
21C2×#14523237C2212yescomplemented
22C2×#14523236C2212yescomplemented
23C2×D1016C2212yescomplemented
24C2×#14523233C2212yescomplemented
25C2×#14523232C2211yescharacteristic, complemented
26#16861193—21no—
27#16861189C2212yescomplemented
28#16861184C2212yescomplemented
29#16861179—21no—
30C4×D508—22no—
31#16861129C2214yescomplemented
32(C2×D127)⋊C8C2214yescomplemented
33C22×D4×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic, complemented
35#117×C127C2211yescharacteristic, complemented
36#102×C127C2214yescomplemented
37C23×D127—28no—
38C23×D127—24no—
39C23×D127—21no—
40C23×D127—48no—
41C23×D127D411yescharacteristic
42#14523248—22no—
43#14523248—22no—
44C2×Dic254C2311yescharacteristic, complemented
45C2×D508—22no—
46C2×D508—21no—
47C2×D508—22no—
48C2×D508D414yescomplemented
49C2×D508C2312yes—
50C2×D508C2311yescharacteristic, complemented
51C2×C22⋊D127—28no—
52C2×C22⋊D127—24no—
53C2×C22⋊D127—21no—
54D4×D127—216no—
55D4×D127—28no—
56D4×D127—28no—
57C22×Dic127D411yescharacteristic
58C2×C4×D127—22no—
59C2×C4×D127—22no—
60C2×C4×D127—21no—
61C2×C4×D127D412yes—
62C2×C4×D127C2312yes—
63#14523237—24no—
64#14523236—24no—
65D1016—24no—
66#14523233—24no—
67#14523232—22no—
68C2×C127⋊C8C2312yescomplemented
69C4⋊Dic127—22no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72C24×C127—21no—
73C2×D4×C127—21no—
74C2×D4×C127—22no—
75C2×D4×C127D414yescomplemented
76C2×D4×C127C2312yes—
77C22×C4×C127C2311yescharacteristic
78M16×C127—22no—
79C2×C8×C127C2312yescomplemented
80C4⋊C4×C127C2312yescomplemented
81C22⋊C4×C127—21no—
82C42×C127—21no—
83C22×D127—22no—
84C22×D127—24no—
85C22×D127—216no—
86C22×D127—24no—
87C22×D127—22no—
88C22×D127—21no—
89C22×D127—432no—
90C22×D127—48no—
91C22×D127—44no—
92C22×D127—48no—
93C22×D127C2×D412yes—
94Dic254—21no—
95Dic254—21no—
96D508—24no—
97D508—22no—
98D508—21no—
99D508—21no—
100D508—44no—
101D508C2×D414yes—
102C22⋊D127—216no—
103C22⋊D127—42no—
104C2×Dic127—22no—
105C2×Dic127—21no—
106C2×Dic127C2×D412yes—
107C4×D127—24no—
108C4×D127—22no—
109C4×D127—22no—
110C4×D127—42no—
111C127⋊C8—22no—
112C23×C127—24no—
113C23×C127—22no—
114C23×C127—44no—
115C23×C127C2×D411yescharacteristic
116D4×C127—24no—
117D4×C127—22no—
118D4×C127C2×D414yes—
119C2×C4×C127—22no—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D412yes—
123C2×C4×C127C2411yescharacteristic
124C8×C127—22no—
125C22×C127—22no—
126C22×C127—24no—
127C22×C127—48no—
128C22×C127—44no—
129C22×C127#12012yes—
130C22×C127C22×D411yescharacteristic
131C2×D127—28no—
132C2×D127—22no—
133C2×D127—22no—
134C2×D127—24no—
135C2×D127—432no—
136C2×D127—48no—
137C2×D127—44no—
138C2×D127—41no—
139C2×D127—42no—
140C2×D127AGL(1,ℤ/8)14yescomplemented
141C4×C127—42no—
142C4×C127#12012yes—
143C4×C127C22×D411yescharacteristic
144C4×C127C22×D411yescharacteristic
145Dic127—22no—
146Dic127—41no—
147C2×C127—22no—
148C2×C127—44no—
149C2×C127C2×#12011yescharacteristic
150C2×C127C2×AGL(1,ℤ/8)12yes—
151D127—24no—
152D127—44no—
153D127—81no—
154#3003—1271nomaximal, Sylow
155C127#300311yescharacteristic, complemented, Sylow
156C23×D4—1271no—
157C2×AGL(1,ℤ/8)—1271no—
158C2×AGL(1,ℤ/8)—1271no—
159#525—1271no—
160#450—1274no—
161#447—1274no—
162#418—1271no—
163#418D12711yescharacteristic, complemented
164#407—1271no—
165C25—2541no—
166C2×#41—1271no—
167C22×D4—1274no—
168C22×D4—1271no—
169C22×D4—2542no—
170C22×D4—2541no—
171C22×D4—2544no—
172C22×D4—2544no—
173C22×D4C2×D12711yescharacteristic
174C23×C4—1271no—
175AGL(1,ℤ/8)—2544no—
176AGL(1,ℤ/8)—2544no—
177C2×SD16—1272no—
178C2×SD16—1272no—
179C2×D8—1272no—
180C2×D8—1272no—
181C2×M16—1271no—
182C2×M16C2×D12711yescharacteristic, complemented
183#123—2541no—
184C22⋊Q8—1272no—
185#121—1272no—
186#120—2541no—
187C4×D4—2542no—
188#117C2×D12711yescharacteristic, complemented
189#102—1274no—
190#102C2×D12714yescomplemented
191C22⋊C8—1274no—
192C24—21no—
193C24—1271no—
194C24—5088no—
195C24—2544no—
196C24—2548no—
197C24—2541no—
198#41—2542no—
199#41—2542no—
200C2×Q8—1271no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—1274no—
204C2×D4—1272no—
205C2×D4—1271no—
206C2×D4—2542no—
207C2×D4—2544no—
208C2×D4—2548no—
209C2×D4—25416no—
210C2×D4—2541no—
211C2×D4—2548no—
212C2×D4—2548no—
213C2×D4—2541no—
214C2×D4—2542no—
215C2×D4D50814yescomplemented
216C2×D4C22×D12712yes—
217C22×C4—1271no—
218C22×C4—1272no—
219C22×C4—1272no—
220C22×C4—2541no—
221C22×C4—2542no—
222C22×C4—2542no—
223C22×C4C22×D12711yescharacteristic
224SD16—2544no—
225SD16—2544no—
226D8—2544no—
227D8—2544no—
228M16—22no—
229M16—2542no—
230C2×C8—1272no—
231C2×C8C22×D12712yescomplemented
232C4⋊C4—2542no—
233C4⋊C4C22×D12712yescomplemented
234C22⋊C4—21no—
235C22⋊C4—2542no—
236C22⋊C4—2542no—
237C42—21no—
238C23—22no—
239C23—24no—
240C23—44no—
241C23—1272no—
242C23—5084no—
243C23—5088no—
244C23—50832no—
245C23—5088no—
246C23—2544no—
247C23—25416no—
248C23—2541no—
249C23—2542no—
250C23—2544no—
251C23—2542no—
252C23D4×D12711yescharacteristic
253Q8—2541no—
254Q8—2541no—
255D4—22no—
256D4—24no—
257D4—1274no—
258D4—5084no—
259D4—5082no—
260D4—25416no—
261D4—2541no—
262D4—2541no—
263D4—2542no—
264D4—2544no—
265D4C2×D50814yes—
266C2×C4—22no—
267C2×C4—1272no—
268C2×C4—5082no—
269C2×C4—2541no—
270C2×C4—2542no—
271C2×C4—2542no—
272C2×C4—2542no—
273C2×C4—2544no—
274C2×C4D4×D12712yes—
275C2×C4D4×D12711yescharacteristic
276C2×C4C2×D50812yes—
277C2×C4C23×D12711yescharacteristic
278C8—22no—
279C8—2542no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—1274no—
285C22—5088no—
286C22—50832no—
287C22—5081no—
288C22—5082no—
289C22—5084no—
290C22—2548no—
291C22—2544no—
292C22—2542no—
293C22—2542no—
294C22#1686117912yes—
295C22C2×D4×D12711yescharacteristic
296C4—42no—
297C4—5081no—
298C4—2542no—
299C4#1686117912yes—
300C4C2×D4×D12711yescharacteristic
301C4C22×D50811yescharacteristic
302C2—22no—
303C2—44no—
304C2—10161no—
305C2—5084no—
306C2—2544no—
307C2C2×#1686117911yescharacteristic
308C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
hypercenter#418D127
hypocenterC127#3003
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D127C2semidirect
2. C2×#16861237C2semidirect
3. C2×#16861235C2semidirect
4. #22550388C2semidirect
5. #22550028C2semidirect
6. #22550011C2semidirect
7. #22549880C2semidirect
8. #22549839C2semidirect
9. #418×C127C2semidirect
11. C2×#14523248C22semidirect
21. C2×#14523237C22semidirect
22. C2×#14523236C22semidirect
23. C2×D1016C22semidirect
24. C2×#14523233C22semidirect
25. C2×#14523232C22semidirect
27. #16861189C22semidirect
28. #16861184C22semidirect
31. #16861129C22semidirect
32. (C2×D127)⋊C8C22semidirect
34. C2×M16×C127C22semidirect
35. #117×C127C22semidirect
36. #102×C127C22semidirect
44. C2×Dic254C23semidirect
48. C2×D508D4semidirect
50. C2×D508C23semidirect
68. C2×C127⋊C8C23semidirect
75. C2×D4×C127D4semidirect
79. C2×C8×C127C23semidirect
80. C4⋊C4×C127C23semidirect
140. C2×D127AGL(1,ℤ/8)semidirect
155. C127#3003semidirect
163. #418D127semidirect
182. C2×M16C2×D127semidirect
188. #117C2×D127semidirect
190. #102C2×D127semidirect
215. C2×D4D508semidirect
231. C2×C8C22×D127semidirect
233. C4⋊C4C22×D127semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×#168612372
C2×#168612352
#225503882
#225500282
#225500112
#225498802
#225498392
#418×C1272
#3003127