Finite Group Databasebeta

← #30648006

Subgroups — #30648006

Order 16256.

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

normal subgroups only

This group has 752 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
2C22×#14523235C211yesmaximal, characteristic, complemented
3#22550450C211yesmaximal, characteristic, complemented
4#22550037C214yesmaximal, complemented
5#22550025C214yesmaximal, complemented
6#22549882C211yesmaximal, characteristic, complemented
7#22549872C211yesmaximal, characteristic, complemented
8#22549844C211yesmaximal, characteristic, complemented
9C2×#137×C127C211yesmaximal, characteristic, complemented
10#16861255—24no—
11#16861255—24no—
12C2×#14523249C2212yes—
13C2×#14523248—21no—
14C2×#14523248C2211yescharacteristic
15C22×D508—21no—
16C22×D508C2211yescharacteristic
17C2×D4×D127—22no—
18C2×D4×D127C2212yes—
19C2×#14523235—24no—
20C2×#14523235C2214yescomplemented
21C22×C127⋊C8C2211yescharacteristic, complemented
22#16861207—21no—
23#16861198—21no—
24#16861192—22no—
25#16861185C2214yescomplemented
26#16861168C2211yescharacteristic, complemented
27#16861133C2212yescomplemented
28#16861131C2214yescomplemented
29#16861130C2212yescomplemented
30(C2×D127)⋊C8C2214yescomplemented
31C2×#41×C127C2211yescharacteristic
32C22×Q8×C127C2211yescharacteristic, complemented
33#137×C127—24no—
34C2×Q16×C127C2212yescomplemented
35C2×SD16×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×D508D414yes—
51C2×D508C2312yes—
52C2×C22⋊D127—22no—
53C2×C22⋊D127D412yes—
54D4×D127—216no—
55D4×D127—24no—
56D4×D127—24no—
57C2×C4×D127—22no—
58C2×C4×D127C2312yes—
59#14523235—28no—
60C2×C127⋊C8—22no—
61C2×C127⋊C8C2312yes—
62C4⋊Dic127C2312yes—
63Dic127⋊C4—24no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—24no—
66C4×Dic127—21no—
67#41×C127—22no—
68#41×C127D414yes—
69C2×Q8×C127—21no—
70C2×Q8×C127—22no—
71C2×Q8×C127—22no—
72C2×Q8×C127C2312yes—
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127C2311yescharacteristic
79Q16×C127—24no—
80SD16×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—
97C2×Dic127—22no—
98C2×Dic127C2×D412yes—
99C4×D127—28no—
100C4×D127—22no—
101C4×D127—22no—
102C127⋊C8—22no—
103C23×C127—21no—
104C23×C127C2×D411yescharacteristic
105Q8×C127—21no—
106Q8×C127—22no—
107Q8×C127—22no—
108Q8×C127—42no—
109Q8×C127C2×D412yes—
110D4×C127—24no—
111D4×C127—21no—
112D4×C127C2×D412yes—
113C2×C4×C127—21no—
114C2×C4×C127—24no—
115C2×C4×C127—22no—
116C2×C4×C127—42no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2411yescharacteristic
121C8×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—42no—
125C22×C127#12012yes—
126C22×C127C22×D411yescharacteristic
127C2×D127—22no—
128C2×D127—24no—
129C2×D127—24no—
130C2×D127—44no—
131C2×D127—84no—
132C2×D127—88no—
133C4×C127—22no—
134C4×C127—42no—
135C4×C127#12012yes—
136C4×C127C22×D411yescharacteristic
137C4×C127C22×D411yescharacteristic
138Dic127—22no—
139Dic127—42no—
140C2×C127—22no—
141C2×C127—42no—
142C2×C127C2×#12011yescharacteristic
143C2×C127#57712yes—
144D127—42no—
145D127—44no—
146#3009—1271nomaximal, Sylow
147C127#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—1271no—
149C2×#137D12711yescharacteristic, complemented
150C22×SD16—1271no—
151#536—1271no—
152#452—1274no—
153#450—1274no—
154#418—1271no—
155#416—1271no—
156#408—1271no—
157H(5,𝔽2)—2544no—
158H(5,𝔽2)—2544no—
159C2×#41—1271no—
160C2×#41—1272no—
161C2×#41—2541no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8C2×D12711yescharacteristic, complemented
164C22×D4—1272no—
165C22×D4—1271no—
166C22×D4—2541no—
167C22×D4—2542no—
168#137—24no—
169C2×Q16C2×D12712yescomplemented
170C2×SD16—1274no—
171C2×SD16—2544no—
172C2×SD16C2×D12712yescomplemented
173C2×M16C2×D12711yescharacteristic, complemented
174C22×C8—1271no—
175C4⋊Q8—2541no—
176#124—2541no—
177#123—2542no—
178C22⋊Q8—1274no—
179#117—1271no—
180#103—1272no—
181#102—1272no—
182#102—1274no—
183C22⋊C8—1274no—
184C24—2542no—
185C24—2541no—
186#41—22no—
187#41—2548no—
188#41—2544no—
189#41—2544no—
190#41—2542no—
191#41—2542no—
192#41C22⋊D12714yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1271no—
197C2×Q8C22×D12712yes—
198C2×Q8C22×D12711yescharacteristic
199C2×D4—21no—
200C2×D4—1272no—
201C2×D4—1274no—
202C2×D4—1271no—
203C2×D4—1272no—
204C2×D4—2542no—
205C2×D4—2548no—
206C2×D4—2542no—
207C2×D4—2544no—
208C2×D4—25416no—
209C2×D4—2544no—
210C2×D4—2541no—
211C2×D4—2542no—
212C2×D4C22×D12711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—1272no—
216C22×C4—2542no—
217C22×C4C22×D12711yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—2548no—
221M16—22no—
222C2×C8—1272no—
223C2×C8—2542no—
224C2×C8C22×D12712yes—
225C4⋊C4—1272no—
226C4⋊C4—2544no—
227C22⋊C4—2544no—
228C22⋊C4—2541no—
229C42—2541no—
230C23—21no—
231C23—1272no—
232C23—10162no—
233C23—10164no—
234C23—2548no—
235C23—2544no—
236C23—2542no—
237C23D4×D12711yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2542no—
243Q8C2×C22⋊D12712yes—
244D4—24no—
245D4—21no—
246D4—1274no—
247D4—2548no—
248D4—2548no—
249D4—2544no—
250D4—2542no—
251D4—2544no—
252D4—2542no—
253D4C2×C22⋊D12712yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—1272no—
259C2×C4—2548no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4—2542no—
263C2×C4D4×D12712yes—
264C2×C4D4×D12711yescharacteristic
265C2×C4C2×C22⋊D12712yes—
266C2×C4C23×D12711yescharacteristic
267C8—22no—
268C8—2542no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—10164no—
273C22—10168no—
274C22—5084no—
275C22—2544no—
276C22—2542no—
277C22—2544no—
278C22#1686117712yes—
279C22C2×D4×D12711yescharacteristic
280C4—22no—
281C4—42no—
282C4—5082no—
283C4—2542no—
284C4#1686117712yes—
285C4C2×D4×D12711yescharacteristic
286C4C22×C22⋊D12711yescharacteristic
287C2—22no—
288C2—42no—
289C2—5084no—
290C2—5082no—
291C2C2×#1686117711yescharacteristic
292C2#2255068512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861255C2
hypercenterC2×#137D127
hypocenterC127#3009
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14523235C2semidirect
3. #22550450C2semidirect
4. #22550037C2semidirect
5. #22550025C2semidirect
6. #22549882C2semidirect
7. #22549872C2semidirect
8. #22549844C2semidirect
9. C2×#137×C127C2semidirect
20. C2×#14523235C22semidirect
21. C22×C127⋊C8C22semidirect
25. #16861185C22semidirect
26. #16861168C22semidirect
27. #16861133C22semidirect
28. #16861131C22semidirect
29. #16861130C22semidirect
30. (C2×D127)⋊C8C22semidirect
32. C22×Q8×C127C22semidirect
34. C2×Q16×C127C22semidirect
35. C2×SD16×C127C22semidirect
36. C2×M16×C127C22semidirect
147. C127#3009semidirect
149. C2×#137D127semidirect
163. C22×Q8C2×D127semidirect
169. C2×Q16C2×D127semidirect
172. C2×SD16C2×D127semidirect
173. C2×M16C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C22×#145232352
#225504502
#225500372
#225500252
#225498822
#225498722
#225498442
C2×#137×C1272
#3009127