Finite Group Databasebeta

← #30647997

Subgroups — #30647997

Order 16256.

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

normal subgroups only

This group has 784 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×#16861255C211yesmaximal, characteristic, complemented
2C22×#14523233C211yesmaximal, characteristic, complemented
3#22550443C211yesmaximal, characteristic, complemented
4#22550020C214yesmaximal, complemented
5#22550008C214yesmaximal, complemented
6#22549882C211yesmaximal, characteristic, complemented
7#22549872C211yesmaximal, characteristic, complemented
8#22549844C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
10#16861255—24no—
11#16861255—24no—
12C2×#14523249C2212yescomplemented
13C2×#14523248—21no—
14C2×#14523248C2211yescharacteristic
15C22×D508—21no—
16C22×D508C2211yescharacteristic
17C2×D4×D127—22no—
18C2×D4×D127C2212yescomplemented
19C2×#14523233—24no—
20C2×#14523233C2214yescomplemented
21C22×C127⋊C8C2211yescharacteristic, complemented
22#16861205—21no—
23#16861196—21no—
24#16861180C2214yescomplemented
25#16861178—22no—
26#16861168C2211yescharacteristic, complemented
27#16861133C2212yescomplemented
28#16861131C2214yescomplemented
29#16861130C2212yescomplemented
30(C2×D127)⋊C8C2214yescomplemented
31C2×#41×C127C2211yescharacteristic
32C22×D4×C127C2211yescharacteristic, complemented
33AGL(1,ℤ/8)×C127—24no—
34C2×SD16×C127C2212yescomplemented
35C2×D8×C127C2212yescomplemented
36C2×M16×C127C2211yescharacteristic, complemented
37C23×D127—22no—
38C23×D127—21no—
39#14523249—24no—
40#14523249—24no—
41#14523248—28no—
42#14523248—22no—
43#14523248—22no—
44C2×Dic254D411yescharacteristic
45C2×D508—28no—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—22no—
49C2×D508D411yescharacteristic
50C2×D508D414yescomplemented
51C2×D508C2312yes—
52C2×C22⋊D127—24no—
53C2×C22⋊D127—22no—
54C2×C22⋊D127D412yes—
55D4×D127—216no—
56D4×D127—24no—
57D4×D127—24no—
58C2×C4×D127—22no—
59C2×C4×D127C2312yes—
60#14523233—28no—
61C2×C127⋊C8—22no—
62C2×C127⋊C8C2312yes—
63C4⋊Dic127C2312yes—
64C22⋊Dic127—24no—
65C22⋊Dic127—21no—
66C4×Dic127—21no—
67C24×C127—21no—
68#41×C127—22no—
69#41×C127D414yescomplemented
70C2×Q8×C127C2311yescharacteristic
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127—22no—
74C2×D4×C127—21no—
75C2×D4×C127C2312yes—
76C2×D4×C127C2311yescharacteristic
77C22×C4×C127—21no—
78C22×C4×C127C2311yescharacteristic
79SD16×C127—24no—
80D8×C127—24no—
81M16×C127—22no—
82C2×C8×C127C2312yes—
83C22×D127—22no—
84C22×D127—28no—
85C22×D127—24no—
86C22×D127—84no—
87C22×D127—82no—
88C22×D127C2×D412yes—
89Dic254—22no—
90D508—28no—
91D508—22no—
92D508—24no—
93D508—22no—
94D508C2×D414yes—
95C22⋊D127—28no—
96C22⋊D127—24no—
97C22⋊D127—84no—
98C2×Dic127—22no—
99C2×Dic127C2×D412yes—
100C4×D127—28no—
101C4×D127—22no—
102C4×D127—22no—
103C127⋊C8—22no—
104C23×C127—21no—
105C23×C127—22no—
106C23×C127—42no—
107C23×C127—82no—
108C23×C127C2×D411yescharacteristic
109Q8×C127—21no—
110Q8×C127C2×D412yes—
111D4×C127—24no—
112D4×C127—21no—
113D4×C127—22no—
114D4×C127—22no—
115D4×C127—42no—
116D4×C127C2×D412yes—
117C2×C4×C127—21no—
118C2×C4×C127—24no—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2411yescharacteristic
123C8×C127—22no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—42no—
127C22×C127—42no—
128C22×C127—44no—
129C22×C127—84no—
130C22×C127#12012yes—
131C22×C127C22×D411yescharacteristic
132C2×D127—22no—
133C2×D127—24no—
134C2×D127—24no—
135C2×D127—44no—
136C2×D127—84no—
137C2×D127—88no—
138C4×C127—22no—
139C4×C127#12012yes—
140C4×C127C22×D411yescharacteristic
141C4×C127C22×D411yescharacteristic
142Dic127—22no—
143Dic127—42no—
144C2×C127—22no—
145C2×C127—42no—
146C2×C127—82no—
147C2×C127C2×#12011yescharacteristic
148C2×C127#57612yes—
149D127—42no—
150D127—44no—
151#3008—1271nomaximal, Sylow
152C127#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1271no—
154C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
155C22×D8—1271no—
156#535—1271no—
157#449—1274no—
158#447—1274no—
159#418—1271no—
160#416—1271no—
161#408—1271no—
162H(5,𝔽2)—2544no—
163H(5,𝔽2)—2544no—
164C2×#41—1271no—
165C2×#41—1272no—
166C2×#41—2541no—
167C2×#41C2×D12711yescharacteristic
168C22×D4—1272no—
169C22×D4—1271no—
170C22×D4—2541no—
171C22×D4—2542no—
172C22×D4C2×D12711yescharacteristic, complemented
173AGL(1,ℤ/8)—24no—
174C2×SD16C2×D12712yescomplemented
175C2×D8—1274no—
176C2×D8—2544no—
177C2×D8C2×D12712yescomplemented
178C2×M16C2×D12711yescharacteristic, complemented
179C22×C8—1271no—
180#127—2541no—
181#124—2541no—
182#121—1274no—
183#120—2542no—
184#117—1271no—
185#103—1272no—
186#102—1272no—
187#102—1274no—
188C22⋊C8—1274no—
189C24—21no—
190C24—2542no—
191C24—2541no—
192#41—22no—
193#41—2548no—
194#41—2544no—
195#41—2544no—
196#41—2542no—
197#41—2542no—
198#41C22⋊D12714yescomplemented
199C2×Q8—1271no—
200C2×Q8C22×D12711yescharacteristic
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—1272no—
206C2×D4—1274no—
207C2×D4—1271no—
208C2×D4—1272no—
209C2×D4—2542no—
210C2×D4—2548no—
211C2×D4—2542no—
212C2×D4—2544no—
213C2×D4—25416no—
214C2×D4—2544no—
215C2×D4—2544no—
216C2×D4—2541no—
217C2×D4—2542no—
218C2×D4C22×D12712yes—
219C2×D4C22×D12711yescharacteristic
220C22×C4—21no—
221C22×C4—1272no—
222C22×C4—2542no—
223C22×C4C22×D12711yescharacteristic
224SD16—24no—
225D8—24no—
226D8—2548no—
227M16—22no—
228C2×C8—1272no—
229C2×C8—2542no—
230C2×C8C22×D12712yes—
231C4⋊C4—1272no—
232C22⋊C4—2544no—
233C22⋊C4—2541no—
234C42—2541no—
235C23—21no—
236C23—22no—
237C23—42no—
238C23—82no—
239C23—1272no—
240C23—10162no—
241C23—10164no—
242C23—2548no—
243C23—2544no—
244C23—2542no—
245C23D4×D12711yescharacteristic
246Q8—21no—
247Q8—2542no—
248Q8C2×C22⋊D12712yes—
249D4—24no—
250D4—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—1274no—
255D4—10164no—
256D4—2548no—
257D4—2548no—
258D4—2544no—
259D4—2542no—
260D4—2544no—
261D4—2542no—
262D4C2×C22⋊D12712yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—1272no—
266C2×C4—2548no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4—2542no—
270C2×C4D4×D12712yes—
271C2×C4D4×D12711yescharacteristic
272C2×C4C2×C22⋊D12712yes—
273C2×C4C23×D12711yescharacteristic
274C8—22no—
275C8—2542no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—10164no—
283C22—10168no—
284C22—5084no—
285C22—2544no—
286C22—2542no—
287C22—2544no—
288C22#1686117712yes—
289C22C2×D4×D12711yescharacteristic
290C4—22no—
291C4—5082no—
292C4—2542no—
293C4#1686117712yes—
294C4C2×D4×D12711yescharacteristic
295C4C22×C22⋊D12711yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—5084no—
300C2—5082no—
301C2C2×#1686117711yescharacteristic
302C2#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#3008
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×#16861255C2semidirect
2. C22×#14523233C2semidirect
3. #22550443C2semidirect
4. #22550020C2semidirect
5. #22550008C2semidirect
6. #22549882C2semidirect
7. #22549872C2semidirect
8. #22549844C2semidirect
9. C2×AGL(1,ℤ/8)×C127C2semidirect
12. C2×#14523249C22semidirect
18. C2×D4×D127C22semidirect
20. C2×#14523233C22semidirect
21. C22×C127⋊C8C22semidirect
24. #16861180C22semidirect
26. #16861168C22semidirect
27. #16861133C22semidirect
28. #16861131C22semidirect
29. #16861130C22semidirect
30. (C2×D127)⋊C8C22semidirect
32. C22×D4×C127C22semidirect
34. C2×SD16×C127C22semidirect
35. C2×D8×C127C22semidirect
36. C2×M16×C127C22semidirect
50. C2×D508D4semidirect
69. #41×C127D4semidirect
152. C127#3008semidirect
154. C2×AGL(1,ℤ/8)D127semidirect
172. C22×D4C2×D127semidirect
174. C2×SD16C2×D127semidirect
177. C2×D8C2×D127semidirect
178. C2×M16C2×D127semidirect
198. #41C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C22×#145232332
#225504432
#225500202
#225500082
#225498822
#225498722
#225498442
C2×AGL(1,ℤ/8)×C1272
#3008127