Finite Group Databasebeta

← #30648017

Subgroups — #30648017

Order 16256.

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

normal subgroups only

This group has 732 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×#16861257C211yesmaximal, characteristic, complemented
2C2×#16861237C211yesmaximal, characteristic, complemented
3C22×#14523235C211yesmaximal, characteristic, complemented
4#22550445C211yesmaximal, characteristic, complemented
5#22550024C214yesmaximal, complemented
6#22550016C214yesmaximal, complemented
7#22549870C211yesmaximal, characteristic, complemented
8#22549844C211yesmaximal, characteristic, complemented
9#419×C127C211yesmaximal, characteristic, complemented
10#16861257—24no—
11#16861257—24no—
12C2×#14523249—21no—
13C2×#14523249C2212yescomplemented
14C2×#14523248—22no—
15C2×#14523248C2211yescharacteristic
16C2×#14523248C2211yescharacteristic
17C2×Q8×D127—21no—
18C2×Q8×D127C2212yescomplemented
19C22×D508C2211yescharacteristic
20#16861237—24no—
21C2×#14523237C2212yes—
22C2×#14523235—24no—
23C2×#14523235C2214yes—
24C2×D1016C2212yes—
25C22×C127⋊C8C2211yescharacteristic
26#16861206—21no—
27#16861197—21no—
28#16861184C2214yes—
29#16861179—22no—
30#16861132C2212yescomplemented
31#16861129C2212yescomplemented
32(C2×D127)⋊C8C2214yescomplemented
33C22×Q8×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic
35#117×C127C2211yescharacteristic
36#103×C127C2214yescomplemented
37C23×D127—21no—
38#14523249—28no—
39#14523249—24no—
40#14523249—24no—
41#14523248—216no—
42#14523248—22no—
43#14523248—22no—
44#14523248—22no—
45#14523248D414yescomplemented
46C2×Dic254—22no—
47C2×Dic254D411yescharacteristic
48C2×Dic254C2311yescharacteristic
49Q8×D127—28no—
50Q8×D127—24no—
51Q8×D127—24no—
52C2×D508—22no—
53C2×D508—24no—
54C2×D508—22no—
55C2×D508—22no—
56C2×D508—21no—
57C2×D508D411yescharacteristic
58C2×D508C2312yes—
59C2×D508C2311yescharacteristic
60C2×C22⋊D127—21no—
61C2×C22⋊D127D412yes—
62C2×C4×D127—22no—
63C2×C4×D127—22no—
64C2×C4×D127—21no—
65C2×C4×D127C2312yes—
66#14523237—24no—
67#14523235—28no—
68D1016—24no—
69C2×C127⋊C8—22no—
70C2×C127⋊C8C2312yes—
71(C2×D127)⋊C4—24no—
72C2×Q8×C127—21no—
73C2×Q8×C127—22no—
74C2×Q8×C127D414yescomplemented
75C2×Q8×C127C2312yes—
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78M16×C127—22no—
79C2×C8×C127C2312yes—
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C42×C127—21no—
83C22×D127—21no—
84C22×D127—22no—
85C22×D127—42no—
86C22×D127—82no—
87C22×D127C2×D412yes—
88Dic254—28no—
89Dic254—22no—
90Dic254—21no—
91Dic254C2×D412yes—
92D508—28no—
93D508—22no—
94D508—21no—
95D508—22no—
96D508—22no—
97D508—42no—
98D508—84no—
99D508C2×D412yes—
100C22⋊D127—24no—
101C22⋊D127—24no—
102C2×Dic127—21no—
103C2×Dic127C2×D412yes—
104C4×D127—28no—
105C4×D127—28no—
106C4×D127—24no—
107C4×D127—22no—
108C4×D127—22no—
109C127⋊C8—22no—
110C23×C127C2×D411yescharacteristic
111Q8×C127—24no—
112Q8×C127—22no—
113Q8×C127C2×D414yes—
114C2×C4×C127—24no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2411yescharacteristic
121C8×C127—22no—
122C22×C127—22no—
123C22×C127#12012yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—22no—
127C2×D127—24no—
128C2×D127—42no—
129C2×D127—42no—
130C2×D127—44no—
131C2×D127—84no—
132C4×C127—24no—
133C4×C127—42no—
134C4×C127#12012yes—
135C4×C127C22×D411yescharacteristic
136C4×C127C22×D411yescharacteristic
137Dic127—22no—
138Dic127—22no—
139C2×C127—22no—
140C2×C127C2×#12011yescharacteristic
141C2×C127#57712yes—
142D127—42no—
143D127—42no—
144D127—82no—
145#3010—1271nomaximal, Sylow
146C127#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1271no—
148C2×AGL(1,ℤ/8)—1271no—
149C22×SD16—1271no—
150#535—1271no—
151#449—1274no—
152#448—1274no—
153#419D12711yescharacteristic, complemented
154#416—1271no—
155#408—1271no—
1562−1+4—2544no—
1572−1+4—2544no—
158C2×#41—1271no—
159C2×#41—1272no—
160C2×#41—1271no—
161C2×#41—2542no—
162C2×#41—2541no—
163C22×Q8—1272no—
164C22×Q8—2541no—
165C22×Q8C2×D12711yescharacteristic
166C22×D4—1271no—
167AGL(1,ℤ/8)—2544no—
168C2×SD16—1274no—
169C2×SD16—1272no—
170C2×SD16—2544no—
171C2×D8—1272no—
172C2×M16C2×D12711yescharacteristic
173C22×C8—1271no—
174#127—2541no—
175#124—2541no—
176#121—1274no—
177#120—2542no—
178#117C2×D12711yescharacteristic
179#103—1272no—
180#103C2×D12714yescomplemented
181#102—1272no—
182C22⋊C8—1274no—
183C24—2541no—
184#41—1274no—
185#41—25416no—
186#41—2544no—
187#41—2548no—
188#41—2544no—
189#41—2542no—
190#41—2542no—
191#41—2542no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—1271no—
195C2×Q8—1271no—
196C2×Q8—2542no—
197C2×Q8—2544no—
198C2×Q8—2548no—
199C2×Q8—2544no—
200C2×Q8D50814yescomplemented
201C2×Q8C22×D12712yes—
202C2×D4—1272no—
203C2×D4—1271no—
204C2×D4—1271no—
205C2×D4—1272no—
206C2×D4—2544no—
207C2×D4—2542no—
208C2×D4—2541no—
209C2×D4—2541no—
210C2×D4—2542no—
211C2×D4—2542no—
212C22×C4—21no—
213C22×C4—1272no—
214C22×C4—2542no—
215C22×C4—2542no—
216C22×C4—2541no—
217C22×C4C22×D12711yescharacteristic
218SD16—2544no—
219SD16—2548no—
220D8—2544no—
221M16—22no—
222C2×C8—1272no—
223C2×C8—2542no—
224C2×C8C22×D12712yes—
225C4⋊C4C22×D12712yes—
226C22⋊C4—21no—
227C22⋊C4—2544no—
228C42—21no—
229C23—1272no—
230C23—10162no—
231C23—5082no—
232C23—2542no—
233C23—2541no—
234C23D4×D12711yescharacteristic
235Q8—22no—
236Q8—24no—
237Q8—1272no—
238Q8—2548no—
239Q8—2541no—
240Q8—2542no—
241Q8C2×D50814yes—
242D4—1272no—
243D4—10164no—
244D4—5082no—
245D4—2548no—
246D4—2544no—
247D4—2544no—
248D4—2542no—
249D4—2542no—
250D4—2541no—
251D4—2542no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—1272no—
256C2×C4—2548no—
257C2×C4—2548no—
258C2×C4—2541no—
259C2×C4—2542no—
260C2×C4—2542no—
261C2×C4—2544no—
262C2×C4D4×D12712yes—
263C2×C4D4×D12711yescharacteristic
264C2×C4C2×D50812yes—
265C2×C4C23×D12711yescharacteristic
266C8—22no—
267C8—2542no—
268C22—22no—
269C22—10164no—
270C22—5082no—
271C22—5084no—
272C22—5082no—
273C22—2542no—
274C22—2542no—
275C22—2544no—
276C22#1686117912yes—
277C22C2×D4×D12711yescharacteristic
278C4—24no—
279C4—42no—
280C4—2542no—
281C4—2542no—
282C4#1686117912yes—
283C4C2×D4×D12711yescharacteristic
284C4C22×D50811yescharacteristic
285C2—22no—
286C2—10162no—
287C2—5082no—
288C2—5082no—
289C2C2×#1686117911yescharacteristic
290C2#2255068512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
hypercenter#419D127
hypocenterC127#3010
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. C2×#16861237C2semidirect
3. C22×#14523235C2semidirect
4. #22550445C2semidirect
5. #22550024C2semidirect
6. #22550016C2semidirect
7. #22549870C2semidirect
8. #22549844C2semidirect
9. #419×C127C2semidirect
13. C2×#14523249C22semidirect
18. C2×Q8×D127C22semidirect
30. #16861132C22semidirect
31. #16861129C22semidirect
32. (C2×D127)⋊C8C22semidirect
36. #103×C127C22semidirect
45. #14523248D4semidirect
74. C2×Q8×C127D4semidirect
146. C127#3010semidirect
153. #419D127semidirect
180. #103C2×D127semidirect
200. C2×Q8D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
C2×#168612372
C22×#145232352
#225504452
#225500242
#225500162
#225498702
#225498442
#419×C1272
#3010127