Finite Group Databasebeta

← #30647994

Subgroups — #30647994

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×#16861254C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3#22550443C211yesmaximal, characteristic, complemented
4#22550018C214yesmaximal, complemented
5#22550008C214yesmaximal, complemented
6#22549881C211yesmaximal, characteristic, complemented
7#22549873C211yesmaximal, characteristic, complemented
8#22549842C211yesmaximal, characteristic, complemented
9C22×D8×C127C211yesmaximal, characteristic, complemented
10#16861254—24no—
11#16861254—24no—
12C2×#14523248C2211yescharacteristic
13C2×#14523248C2211yescharacteristic
14C2×#14523247—21no—
15C2×#14523247C2212yescomplemented
16C22×C22⋊D127—22no—
17C2×D4×D127—21no—
18C2×D4×D127C2212yescomplemented
19#16861235—24no—
20C2×#14523236C2212yescomplemented
21C2×#14523233C2212yescomplemented
22C2×#14523232C2211yescharacteristic, complemented
23#16861205—21no—
24#16861196—21no—
25#16861180C2214yescomplemented
26#16861178—22no—
27#16861168C2211yescharacteristic, complemented
28#16861134C2212yescomplemented
29#16861131C2212yescomplemented
30#16861130C2214yescomplemented
31(C2×D127)⋊C8C2214yescomplemented
32C22×D4×C127C2211yescharacteristic
33C22×D4×C127C2211yescharacteristic, complemented
34C2×D8×C127—24no—
35C2×D8×C127C2214yescomplemented
36C22×C8×C127C2211yescharacteristic, complemented
37C23×D127—22no—
38#14523248—22no—
39#14523248—22no—
40#14523248—22no—
41#14523248D414yescomplemented
42#14523247—28no—
43#14523247—24no—
44#14523247—24no—
45C2×Dic254D411yescharacteristic
46C2×Dic254C2311yescharacteristic
47C2×D508D411yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—216no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—24no—
52C2×C22⋊D127—22no—
53C2×C22⋊D127—21no—
54C2×C22⋊D127D412yes—
55D4×D127—28no—
56D4×D127—24no—
57D4×D127—24no—
58C22×Dic127—22no—
59C2×C4×D127—21no—
60C2×C4×D127C2312yes—
61#14523236—24no—
62#14523233—24no—
63#14523232—22no—
64C2×C127⋊C8C2312yes—
65C4⋊Dic127C2312yes—
66C22⋊Dic127—24no—
67C22⋊Dic127—21no—
68C4×Dic127—21no—
69C24×C127—21no—
70C24×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127—22no—
74C2×D4×C127—22no—
75C2×D4×C127—21no—
76C2×D4×C127D414yescomplemented
77C2×D4×C127C2312yes—
78C2×D4×C127C2312yes—
79C22×C4×C127C2311yescharacteristic
80D8×C127—28no—
81C2×C8×C127—22no—
82C2×C8×C127C2312yes—
83C22×D127—21no—
84C22×D127—28no—
85C22×D127—84no—
86C22×D127C2×D412yes—
87Dic254—22no—
88Dic254—21no—
89Dic254C2×D412yes—
90D508—22no—
91D508—21no—
92D508C2×D412yes—
93C22⋊D127—28no—
94C22⋊D127—28no—
95C22⋊D127—24no—
96C22⋊D127—24no—
97C22⋊D127—84no—
98C2×Dic127—28no—
99C2×Dic127—21no—
100C2×Dic127—22no—
101C2×Dic127C2×D412yes—
102C4×D127—24no—
103C4×D127—22no—
104C4×D127—22no—
105C127⋊C8—22no—
106C23×C127—22no—
107C23×C127—24no—
108C23×C127—22no—
109C23×C127—42no—
110C23×C127—82no—
111C23×C127—82no—
112C23×C127C2×D411yescharacteristic
113D4×C127—24no—
114D4×C127—22no—
115D4×C127—22no—
116D4×C127—22no—
117D4×C127—42no—
118D4×C127C2×D414yes—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2411yescharacteristic
123C8×C127—22no—
124C22×C127—22no—
125C22×C127—24no—
126C22×C127—44no—
127C22×C127—42no—
128C22×C127—44no—
129C22×C127—84no—
130C22×C127—84no—
131C22×C127#12012yes—
132C22×C127C22×D411yescharacteristic
133C2×D127—22no—
134C2×D127—22no—
135C2×D127—24no—
136C2×D127—42no—
137C2×D127—88no—
138C4×C127#12012yes—
139C4×C127C22×D411yescharacteristic
140C4×C127C22×D411yescharacteristic
141Dic127—22no—
142Dic127—22no—
143Dic127—42no—
144C2×C127—22no—
145C2×C127—44no—
146C2×C127—82no—
147C2×C127C2×#12011yescharacteristic
148C2×C127#57612yes—
149D127—42no—
150D127—42no—
151#3008—1271nomaximal, Sylow
152C127#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—1271no—
154C2×AGL(1,ℤ/8)—1271no—
155C22×D8D12711yescharacteristic, complemented
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—1271no—
167C2×#41—2541no—
168C22×D4—1272no—
169C22×D4—2541no—
170C22×D4—2542no—
171C22×D4C2×D12711yescharacteristic
172C22×D4C2×D12711yescharacteristic, complemented
173AGL(1,ℤ/8)—2544no—
174C2×SD16—1272no—
175C2×D8—24no—
176C2×D8—1272no—
177C2×D8C2×D12714yescomplemented
178C2×M16—1271no—
179C22×C8C2×D12711yescharacteristic, complemented
180#127—2541no—
181#124—2541no—
182#121—1274no—
183#120—2542no—
184#117—1271no—
185#103—1272no—
186#102—1274no—
187#102—1272no—
188C22⋊C8—1274no—
189C24—21no—
190C24—21no—
191C24—2542no—
192#41—1274no—
193#41—2544no—
194#41—2548no—
195#41—2544no—
196#41—2542no—
197#41—2542no—
198#41—2542no—
199C2×Q8—1271no—
200C2×Q8—1271no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—1272no—
207C2×D4—1271no—
208C2×D4—1271no—
209C2×D4—2542no—
210C2×D4—2544no—
211C2×D4—2548no—
212C2×D4—2544no—
213C2×D4—2542no—
214C2×D4—25416no—
215C2×D4—2541no—
216C2×D4—2544no—
217C2×D4C22⋊D12714yescomplemented
218C2×D4C22×D12712yes—
219C2×D4C22×D12712yes—
220C22×C4—1272no—
221C22×C4—2542no—
222C22×C4—2541no—
223C22×C4C22×D12711yescharacteristic
224SD16—2544no—
225D8—28no—
226D8—2544no—
227M16—2542no—
228C2×C8—22no—
229C2×C8—1272no—
230C2×C8C22×D12712yes—
231C4⋊C4—1272no—
232C22⋊C4—2544no—
233C22⋊C4—2541no—
234C42—2541no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—42no—
239C23—82no—
240C23—82no—
241C23—1272no—
242C23—10164no—
243C23—2548no—
244C23—2541no—
245C23D4×D12711yescharacteristic
246Q8—1272no—
247Q8—2541no—
248Q8—2542no—
249D4—22no—
250D4—22no—
251D4—22no—
252D4—24no—
253D4—42no—
254D4—1272no—
255D4—10164no—
256D4—2548no—
257D4—2548no—
258D4—2544no—
259D4—2544no—
260D4—2541no—
261D4—2542no—
262D4C2×C22⋊D12714yes—
263C2×C4—1272no—
264C2×C4—2542no—
265C2×C4—2548no—
266C2×C4—2541no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4—2544no—
270C2×C4D4×D12712yes—
271C2×C4D4×D12711yescharacteristic
272C2×C4C2×C22⋊D12712yes—
273C2×C4C23×D12711yescharacteristic
274C8—22no—
275C8—2542no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—84no—
283C22—10168no—
284C22—5082no—
285C22—2542no—
286C22—2542no—
287C22—2544no—
288C22#1686117712yes—
289C22C2×D4×D12711yescharacteristic
290C4—5082no—
291C4—2542no—
292C4—2542no—
293C4#1686117712yes—
294C4C2×D4×D12711yescharacteristic
295C4C22×C22⋊D12711yescharacteristic
296C2—22no—
297C2—44no—
298C2—82no—
299C2—5082no—
300C2—5082no—
301C2C2×#1686117711yescharacteristic
302C2#2255067712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC22×C127C22×D4
hypercenterC22×D8D127
hypocenterC127#3008
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861254C2semidirect
2. C2×#16861235C2semidirect
3. #22550443C2semidirect
4. #22550018C2semidirect
5. #22550008C2semidirect
6. #22549881C2semidirect
7. #22549873C2semidirect
8. #22549842C2semidirect
9. C22×D8×C127C2semidirect
15. C2×#14523247C22semidirect
18. C2×D4×D127C22semidirect
20. C2×#14523236C22semidirect
21. C2×#14523233C22semidirect
22. C2×#14523232C22semidirect
25. #16861180C22semidirect
27. #16861168C22semidirect
28. #16861134C22semidirect
29. #16861131C22semidirect
30. #16861130C22semidirect
31. (C2×D127)⋊C8C22semidirect
33. C22×D4×C127C22semidirect
35. C2×D8×C127C22semidirect
36. C22×C8×C127C22semidirect
41. #14523248D4semidirect
76. C2×D4×C127D4semidirect
152. C127#3008semidirect
155. C22×D8D127semidirect
172. C22×D4C2×D127semidirect
177. C2×D8C2×D127semidirect
179. C22×C8C2×D127semidirect
217. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168612352
#225504432
#225500182
#225500082
#225498812
#225498732
#225498422
C22×D8×C1272
#3008127