Finite Group Databasebeta

← #30650329

Subgroups — #30650329

Order 16256.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16861236C212yesmaximal, complemented
2#22550529C211yesmaximal, characteristic, complemented
3#22550511C211yesmaximal, characteristic, complemented
4#22550119C212yesmaximal, complemented
5#22550078C211yesmaximal, characteristic, complemented
6#22550068C211yesmaximal, characteristic, complemented
7#22550025C212yesmaximal, complemented
8#22550020C212yesmaximal, complemented
9C4×D1016C211yesmaximal, characteristic, complemented
10#22549955C211yesmaximal, characteristic, complemented
11#492×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yescomplemented
13C2×D4×D127—21no—
14C2×D4×D127C2212yes—
15#16861236—28no—
16C2×#14523235C2212yes—
17C2×D1016C2211yescharacteristic
18C2×#14523233C2212yescomplemented
19C2×#14523231C2212yescomplemented
20#16861194—22no—
21#16861185C2212yescomplemented
22#16861184—21no—
23#16861180—21no—
24#16861180C2212yes—
25#16861177—22no—
26C4×D508C2211yescharacteristic
27C4×D508C2211yescharacteristic
28C4×D508C2211yescharacteristic, complemented
29C4⋊C4×D127—21no—
30#16861143C2211yescharacteristic, complemented
31C4⋊(C127⋊C8)C2211yescharacteristic, complemented
32#16861133C2212yescomplemented
33#16861131C2211yescharacteristic
34#16861131C2211yescharacteristic, complemented
35#16861130C2212yescomplemented
36(C2×D127)⋊C8C2212yescomplemented
37C2×SD16×C127C2212yescomplemented
38C2×SD16×C127C2212yescomplemented
39C4⋊Q8×C127C2211yescharacteristic, complemented
40#127×C127C2211yescharacteristic
41C4×C8×C127C2211yescharacteristic, complemented
42C23×D127—22no—
43#14523249—24no—
44#14523249—42no—
45C2×D508—22no—
46C2×D508D411yescharacteristic
47C2×D508C2311yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51D4×D127—24no—
52D4×D127—42no—
53D4×D127—44no—
54C2×C4×D127—22no—
55C2×C4×D127—21no—
56C2×C4×D127—21no—
57C2×C4×D127C2312yes—
58#14523235—24no—
59D1016D414yescomplemented
60#14523233—24no—
61#14523231—24no—
62C2×C127⋊C8C2312yes—
63C4⋊Dic127D411yescharacteristic
64C4⋊Dic127C2311yescharacteristic
65C4⋊Dic127C2311yescharacteristic
66Dic127⋊C4—22no—
67C22⋊Dic127—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—21no—
70(C2×D127)⋊C4—21no—
71(C2×D127)⋊C4D412yes—
72C2×Q8×C127C2312yes—
73C2×D4×C127—21no—
74C2×D4×C127—21no—
75C2×D4×C127C2312yes—
76SD16×C127—24no—
77SD16×C127—42no—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C4⋊C4×C127—21no—
81C4⋊C4×C127—21no—
82C42×C127C2311yescharacteristic
83C22×D127—21no—
84C22×D127—21no—
85C22×D127—44no—
86C22×D127—44no—
87C22×D127—82no—
88C22×D127C2×D412yes—
89D508—22no—
90D508—21no—
91D508—21no—
92D508—44no—
93D508C2×D412yes—
94D508C2×D412yescomplemented
95C22⋊D127—44no—
96C22⋊D127—44no—
97C2×Dic127—21no—
98C2×Dic127—21no—
99C2×Dic127C2×D412yes—
100C4×D127—22no—
101C4×D127—22no—
102C4×D127—44no—
103C4×D127—42no—
104C4×D127—42no—
105C127⋊C8—22no—
106C23×C127—22no—
107Q8×C127—22no—
108Q8×C127—22no—
109D4×C127—22no—
110D4×C127—22no—
111D4×C127—42no—
112D4×C127—81no—
113C2×C4×C127—22no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2411yescharacteristic
117C8×C127—21no—
118C8×C127C2×D412yescomplemented
119C22×C127—42no—
120C22×C127—42no—
121C22×C127—42no—
122C22×C127C22×D411yescharacteristic
123C2×D127—22no—
124C2×D127—22no—
125C2×D127—22no—
126C2×D127—22no—
127C2×D127—22no—
128C2×D127—41no—
129C2×D127—41no—
130C2×D127—41no—
131C2×D127—41no—
132C2×D127—88no—
133C4×C127—21no—
134C4×C127—42no—
135C4×C127AGL(1,ℤ/8)12yescomplemented
136C4×C127C22×D411yescharacteristic
137C4×C127H(5,𝔽2)11yescharacteristic
138Dic127—22no—
139Dic127—41no—
140Dic127—41no—
141C2×C127—82no—
142C2×C127D4211yescharacteristic
143C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C127#57711yescharacteristic
145D127—42no—
146D127—42no—
147D127—42no—
148#3272—1271nomaximal, Sylow
149C127#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1272no—
151#547—1271no—
152D42—1271no—
153#492D12711yescharacteristic, complemented
154#468—1272no—
155#461—1271no—
156#459—1271no—
157#450—1272no—
158#449—1272no—
159C4×D8—1271no—
160#436—1271no—
161C2×#41—1272no—
162C22×D4—1272no—
163C22×D4—2541no—
164AGL(1,ℤ/8)—2548no—
165C2×SD16—1272no—
166C2×SD16C2×D12712yescomplemented
167C2×SD16C2×D12712yescomplemented
168C2×D8—1272no—
169C2×D8—1271no—
170C2×M16—1272no—
171C4⋊Q8C2×D12711yescharacteristic, complemented
172#127C2×D12711yescharacteristic
173#123—2542no—
174C22⋊Q8—1272no—
175#121—1272no—
176#121—2541no—
177#121—2541no—
178#120—2542no—
179C4×D4—1271no—
180C4×D4—1271no—
181C4×D4—1271no—
182C2×C4⋊C4—2541no—
183#107—1271no—
184C4⋊C8—1271no—
185#103—1272no—
186#102—1272no—
187#102—1271no—
188#102—1271no—
189C22⋊C8—1272no—
190C4×C8C2×D12711yescharacteristic, complemented
191C24—2542no—
192#41—5082no—
193#41—2544no—
194C2×Q8C22×D12712yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—1271no—
198C2×D4—1271no—
199C2×D4—1271no—
200C2×D4—5084no—
201C2×D4—5082no—
202C2×D4—2542no—
203C2×D4—2542no—
204C2×D4—2544no—
205C2×D4—2542no—
206C2×D4C22×D12712yes—
207C22×C4—1272no—
208C22×C4—2542no—
209C22×C4—2541no—
210C22×C4—2541no—
211SD16—24no—
212SD16—42no—
213SD16—2544no—
214D8—1274no—
215D8—2544no—
216M16—2544no—
217C2×C8—1272no—
218C2×C8C22×D12711yescharacteristic
219C2×C8C22×D12711yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—1271no—
223C4⋊C4—1271no—
224C4⋊C4—1271no—
225C4⋊C4—2542no—
226C22⋊C4—1272no—
227C22⋊C4—2542no—
228C22⋊C4—2542no—
229C22⋊C4—2541no—
230C22⋊C4—2541no—
231C42C22×D12711yescharacteristic
232C23—22no—
233C23—1272no—
234C23—10162no—
235C23—5084no—
236C23—5084no—
237C23—2541no—
238C23—2541no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—1272no—
246D4—1272no—
247D4—5084no—
248D4—5084no—
249D4—5084no—
250D4—2542no—
251D4—2541no—
252D4—2541no—
253C2×C4—22no—
254C2×C4—1272no—
255C2×C4—5084no—
256C2×C4—5082no—
257C2×C4—5082no—
258C2×C4—2541no—
259C2×C4—2541no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4D4×D12711yescharacteristic
263C2×C4D4×D12711yescharacteristic
264C2×C4C23×D12711yescharacteristic
265C8—21no—
266C8—2542no—
267C8D4×D12712yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—10168no—
272C22—5081no—
273C22—5081no—
274C22—5081no—
275C22—5081no—
276C22—2542no—
277C22—2542no—
278C22—2542no—
279C22—2542no—
280C22—2542no—
281C22C2×D4×D12711yescharacteristic
282C4—21no—
283C4—42no—
284C4—5081no—
285C4—5081no—
286C4—2542no—
287C4#1686123612yescomplemented
288C4C2×D4×D12711yescharacteristic
289C4#1686125411yescharacteristic
290C2—82no—
291C2—5082no—
292C2—5082no—
293C2—5082no—
294C2#2255051111yescharacteristic
295C2C2×#1686123611yescharacteristic
296C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#492×C127C2
socleC22×C127C22×D4
hypercenter#492D127
hypocenterC127#3272
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. #22550529C2semidirect
3. #22550511C2semidirect
4. #22550119C2semidirect
5. #22550078C2semidirect
6. #22550068C2semidirect
7. #22550025C2semidirect
8. #22550020C2semidirect
9. C4×D1016C2semidirect
10. #22549955C2semidirect
11. #492×C127C2semidirect
12. C2×#14523249C22semidirect
18. C2×#14523233C22semidirect
19. C2×#14523231C22semidirect
21. #16861185C22semidirect
28. C4×D508C22semidirect
30. #16861143C22semidirect
31. C4⋊(C127⋊C8)C22semidirect
32. #16861133C22semidirect
34. #16861131C22semidirect
35. #16861130C22semidirect
36. (C2×D127)⋊C8C22semidirect
37. C2×SD16×C127C22semidirect
38. C2×SD16×C127C22semidirect
39. C4⋊Q8×C127C22semidirect
41. C4×C8×C127C22semidirect
59. D1016D4semidirect
94. D508C2×D4semidirect
118. C8×C127C2×D4semidirect
135. C4×C127AGL(1,ℤ/8)semidirect
149. C127#3272semidirect
153. #492D127semidirect
166. C2×SD16C2×D127semidirect
167. C2×SD16C2×D127semidirect
171. C4⋊Q8C2×D127semidirect
190. C4×C8C2×D127semidirect
267. C8D4×D127semidirect
287. C4#16861236semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
#225505292
#225505112
#225501192
#225500782
#225500682
#225500252
#225500202
C4×D10162
#225499552
#492×C1272
#3272127