Finite Group Databasebeta

← #30648010

Subgroups — #30648010

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
2C2×#16861239C211yesmaximal, characteristic, complemented
3C22×#14523237C211yesmaximal, characteristic, complemented
4#22550452C211yesmaximal, characteristic, complemented
5#22550041C214yesmaximal, complemented
6#22550028C214yesmaximal, complemented
7#22549880C211yesmaximal, characteristic, complemented
8#22549842C211yesmaximal, characteristic, complemented
9#416×C127C211yesmaximal, characteristic, complemented
10#16861255—24no—
11#16861255—24no—
12C2×#14523249C2212yes—
13C2×#14523248—21no—
14C2×#14523248C2211yescharacteristic
15C22×Dic254C2211yescharacteristic, complemented
16C22×D508—21no—
17C22×D508C2211yescharacteristic
18C2×D4×D127—22no—
19C2×D4×D127C2212yes—
20#16861239—24no—
21C2×#14523238C2212yescomplemented
22C2×#14523237—24no—
23C2×#14523237C2214yescomplemented
24C2×#14523236C2212yescomplemented
25C2×#14523232C2211yescharacteristic, complemented
26#16861209—21no—
27#16861197—21no—
28#16861193—22no—
29#16861189C2214yescomplemented
30#16861129C2214yescomplemented
31(C2×D127)⋊C8C2214yescomplemented
32C2×#41×C127C2211yescharacteristic
33C22×C8×C127C2211yescharacteristic, complemented
34#117×C127C2211yescharacteristic, complemented
35#103×C127C2212yescomplemented
36#102×C127C2212yescomplemented
37C23×D127—22no—
38C23×D127—21no—
39#14523249—24no—
40#14523249—24no—
41#14523248—28no—
42#14523248—22no—
43#14523248—22no—
44C2×Dic254—21no—
45C2×Dic254—22no—
46C2×Dic254—22no—
47C2×Dic254D411yescharacteristic
48C2×Dic254C2312yes—
49C2×D508—28no—
50C2×D508—22no—
51C2×D508—21no—
52C2×D508—22no—
53C2×D508D411yescharacteristic
54C2×D508D414yes—
55C2×D508C2312yes—
56C2×C22⋊D127—22no—
57C2×C22⋊D127D412yes—
58D4×D127—216no—
59D4×D127—24no—
60D4×D127—24no—
61C22×Dic127—21no—
62C2×C4×D127—22no—
63C2×C4×D127C2312yes—
64#14523238—24no—
65#14523237—28no—
66#14523236—24no—
67#14523232—22no—
68C2×C127⋊C8C2312yes—
69C4⋊Dic127—24no—
70(C2×D127)⋊C4—24no—
71#41×C127—22no—
72#41×C127D414yes—
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78C2×C8×C127—22no—
79C2×C8×C127C2312yes—
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C42×C127—21no—
83C22×D127—22no—
84C22×D127—28no—
85C22×D127—24no—
86C22×D127—84no—
87C22×D127—82no—
88C22×D127C2×D412yes—
89Dic254—22no—
90Dic254—22no—
91Dic254—22no—
92Dic254—42no—
93D508—28no—
94D508—22no—
95D508—24no—
96D508—22no—
97D508C2×D414yes—
98C22⋊D127—28no—
99C22⋊D127—24no—
100C2×Dic127—22no—
101C2×Dic127—42no—
102C2×Dic127C2×D412yes—
103C4×D127—28no—
104C4×D127—22no—
105C4×D127—22no—
106C127⋊C8—22no—
107C23×C127—21no—
108C23×C127C2×D411yescharacteristic
109Q8×C127—21no—
110Q8×C127C2×D412yes—
111D4×C127—24no—
112D4×C127—21no—
113D4×C127C2×D412yes—
114C2×C4×C127—21no—
115C2×C4×C127—24no—
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—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×#137—1271no—
150C22×SD16—1271no—
151#536—1271no—
152#452—1274no—
153#450—1274no—
154#418—1271no—
155#416D12711yescharacteristic, complemented
156#408—1271no—
157H(5,𝔽2)—2544no—
158H(5,𝔽2)—2544no—
159C2×#41—1271no—
160C2×#41—1272no—
161C2×#41—2541no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8—1271no—
164C22×D4—1272no—
165C22×D4—1271no—
166C22×D4—2541no—
167C22×D4—2542no—
168#137—2544no—
169C2×Q16—1272no—
170C2×SD16—1272no—
171C2×SD16—1274no—
172C2×SD16—2544no—
173C2×M16—1271no—
174C22×C8C2×D12711yescharacteristic, complemented
175C4⋊Q8—2541no—
176#124—2541no—
177#123—2542no—
178C22⋊Q8—1274no—
179#117C2×D12711yescharacteristic, complemented
180#103C2×D12712yescomplemented
181#102—1274no—
182#102C2×D12712yescomplemented
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#41D50814yes—
193C2×Q8—1271no—
194C2×Q8—1272no—
195C2×Q8—2542no—
196C2×Q8—2542no—
197C2×Q8—2541no—
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—1272no—
215C22×C4—2542no—
216C22×C4—2541no—
217C22×C4C22×D12711yescharacteristic
218Q16—2544no—
219SD16—2548no—
220SD16—2544no—
221M16—2542no—
222C2×C8—22no—
223C2×C8—1272no—
224C2×C8C22×D12712yes—
225C4⋊C4—2544no—
226C4⋊C4C22×D12712yes—
227C22⋊C4—21no—
228C22⋊C4—2544no—
229C42—21no—
230C23—21no—
231C23—1272no—
232C23—10162no—
233C23—10164no—
234C23—2548no—
235C23—2544no—
236C23—2542no—
237C23D4×D12711yescharacteristic
238Q8—21no—
239Q8—5082no—
240Q8—2542no—
241Q8—2542no—
242Q8—2542no—
243Q8C2×D50812yes—
244D4—24no—
245D4—21no—
246D4—1274no—
247D4—2548no—
248D4—2548no—
249D4—2544no—
250D4—2542no—
251D4—2544no—
252D4—2542no—
253D4C2×D50812yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—1272no—
258C2×C4—5082no—
259C2×C4—2548no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4—2542no—
263C2×C4D4×D12712yes—
264C2×C4D4×D12711yescharacteristic
265C2×C4C2×D50812yes—
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#1686117912yes—
279C22C2×D4×D12711yescharacteristic
280C4—22no—
281C4—42no—
282C4—5082no—
283C4—2542no—
284C4#1686117912yes—
285C4C2×D4×D12711yescharacteristic
286C4C22×D50811yescharacteristic
287C2—22no—
288C2—42no—
289C2—5084no—
290C2—5082no—
291C2C2×#1686117911yescharacteristic
292C2#2255068312yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861255C2
hypercenter#416D127
hypocenterC127#3009
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#16861239C2semidirect
3. C22×#14523237C2semidirect
4. #22550452C2semidirect
5. #22550041C2semidirect
6. #22550028C2semidirect
7. #22549880C2semidirect
8. #22549842C2semidirect
9. #416×C127C2semidirect
15. C22×Dic254C22semidirect
21. C2×#14523238C22semidirect
23. C2×#14523237C22semidirect
24. C2×#14523236C22semidirect
25. C2×#14523232C22semidirect
29. #16861189C22semidirect
30. #16861129C22semidirect
31. (C2×D127)⋊C8C22semidirect
33. C22×C8×C127C22semidirect
34. #117×C127C22semidirect
35. #103×C127C22semidirect
36. #102×C127C22semidirect
147. C127#3009semidirect
155. #416D127semidirect
174. C22×C8C2×D127semidirect
179. #117C2×D127semidirect
180. #103C2×D127semidirect
182. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612392
C22×#145232372
#225504522
#225500412
#225500282
#225498802
#225498422
#416×C1272
#3009127