Finite Group Databasebeta

← #30648373

Subgroups — #30648373

Order 16256.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D127C211yesmaximal, characteristic, complemented
2C2×#16861215C211yesmaximal, characteristic, complemented
3#22550442C212yesmaximal, complemented
4#22550128C212yesmaximal, complemented
5#22550127C212yesmaximal, complemented
6#22550113C211yesmaximal, characteristic, complemented
7#22550112C211yesmaximal, characteristic, complemented
8#22550109C211yesmaximal, characteristic, complemented
9#22549913C211yesmaximal, characteristic, complemented
10#22549879C212yesmaximal, complemented
11#466×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×D4×D127C2212yes—
14C2×D1016C2211yescharacteristic
15C2×#14523233C2212yes—
16D8×D127—24no—
17#16861215—24no—
18C2×#14523232C2211yescharacteristic, complemented
19C2×#14523231C2211yescharacteristic
20C2×#14523231C2211yescharacteristic, complemented
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic, complemented
23#16861205—22no—
24#16861195—22no—
25#16861184C2212yescomplemented
26#16861183C2212yescomplemented
27#16861180C2212yescomplemented
28#16861179—22no—
29#16861177—22no—
30#16861167C2212yes—
31#16861143C2211yescharacteristic, complemented
32#16861140C2212yescomplemented
33#16861131C2212yescomplemented
34#16861130C2212yescomplemented
35#16861129C2212yescomplemented
36C2×D8×C127C2211yescharacteristic
37C22×C8×C127C2211yescharacteristic, complemented
38#121×C127C2212yescomplemented
39#107×C127C2211yescharacteristic, complemented
40#102×C127C2212yescomplemented
41C23×D127—22no—
42#14523248—22no—
43#14523248—22no—
44C2×Dic254D411yescharacteristic
45C2×D508—22no—
46C2×D508D411yescharacteristic
47C2×D508C2312yes—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127D411yescharacteristic
52C2×C22⋊D127D411yescharacteristic
53D4×D127—24no—
54D4×D127—24no—
55C2×C4×D127C2311yescharacteristic
56C2×C4×D127C2311yescharacteristic
57D1016—22no—
58#14523233—24no—
59#14523232—22no—
60#14523231—22no—
61#14523231—22no—
62C2×C127⋊C8C2311yescharacteristic
63C2×C127⋊C8C2311yescharacteristic
64C8×D127—22no—
65C8×D127D414yescomplemented
66C4⋊Dic127C2312yes—
67C22⋊Dic127—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71C4×Dic127—22no—
72C2×D4×C127—22no—
73C2×D4×C127C2312yes—
74C22×C4×C127C2311yescharacteristic
75D8×C127—22no—
76C2×C8×C127—22no—
77C2×C8×C127C2311yescharacteristic
78C2×C8×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22×D127—22no—
82C22×D127—44no—
83C22×D127—82no—
84C22×D127—82no—
85C22×D127C2×D411yescharacteristic
86C22×D127C2×D411yescharacteristic
87Dic254—22no—
88D508—22no—
89D508—22no—
90D508—22no—
91D508—82no—
92C22⋊D127—22no—
93C22⋊D127—22no—
94C22⋊D127—44no—
95C22⋊D127—82no—
96C22⋊D127—82no—
97C2×Dic127—22no—
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D411yescharacteristic
100C4×D127—21no—
101C4×D127—21no—
102C4×D127C2×D412yes—
103C4×D127#4112yes—
104C127⋊C8—21no—
105C127⋊C8C2×D412yes—
106C23×C127—22no—
107C23×C127C2×D411yescharacteristic
108D4×C127—22no—
109D4×C127—22no—
110D4×C127—82no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—22no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2411yescharacteristic
116C8×C127—21no—
117C8×C127C2×D412yes—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—21no—
121C22×C127—42no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127C22×D411yescharacteristic
125C2×D127—21no—
126C2×D127—21no—
127C2×D127—21no—
128C2×D127—21no—
129C2×D127—21no—
130C2×D127—42no—
131C2×D127—42no—
132C2×D127—42no—
133C2×D127—88no—
134C2×D127#12112yes—
135C4×C127—21no—
136C4×C127—42no—
137C4×C127C22×D411yescharacteristic
138C4×C127C2×#4111yescharacteristic
139Dic127—21no—
140Dic127—42no—
141Dic127#12112yes—
142C2×C127—41no—
143C2×C127—82no—
144C2×C127C2×#12111yescharacteristic
145C2×C127#57611yescharacteristic
146C2×C127#57611yescharacteristic
147D127—22no—
148D127—41no—
149D127—82no—
150#3058—1271nomaximal, Sylow
151C127#305811yescharacteristic, complemented, Sylow
152C22×D8—1271no—
153C2×#131—1271no—
154#535—1272no—
155#469—1272no—
156#469—1272no—
157#466—1271no—
158#466—1271no—
159#466—1271no—
160#466D12711yescharacteristic, complemented
161#427—1271no—
162#418—1272no—
163C2×#41—1271no—
164C22×D4—1272no—
165C2×D8—1272no—
166C2×D8—1271no—
167C2×D8—2544no—
168C2×D8C2×D12711yescharacteristic
169#131—2544no—
170C2×M16—1271no—
171C2×M16—1271no—
172C2×M16—1271no—
173C22×C8—1271no—
174C22×C8—1271no—
175C22×C8C2×D12711yescharacteristic, complemented
176#127—2542no—
177#124—2542no—
178#121—1272no—
179#121—1272no—
180#121—1272no—
181#121C2×D12712yescomplemented
182#120—2542no—
183#120—2542no—
184#117—1272no—
185#107—1271no—
186#107C2×D12711yescharacteristic, complemented
187#106—1272no—
188#102—1272no—
189#102—1272no—
190#102—1272no—
191#102C2×D12712yescomplemented
192C24—2542no—
193#41—2542no—
194#41—2542no—
195C2×Q8—1271no—
196C2×D4—22no—
197C2×D4—1271no—
198C2×D4—1271no—
199C2×D4—1271no—
200C2×D4—1272no—
201C2×D4—2542no—
202C2×D4—2544no—
203C2×D4—2544no—
204C2×D4—2542no—
205C2×D4—2542no—
206C2×D4—2542no—
207C2×D4C22×D12712yes—
208C22×C4—1271no—
209C22×C4—1271no—
210C22×C4C22×D12711yescharacteristic
211D8—22no—
212D8—2544no—
213D8—2542no—
214M16—2542no—
215M16—2542no—
216M16—2542no—
217C2×C8—22no—
218C2×C8—1271no—
219C2×C8—1274no—
220C2×C8—1271no—
221C2×C8—2542no—
222C2×C8C22×D12711yescharacteristic
223C2×C8C22×D12711yescharacteristic
224C4⋊C4—1272no—
225C4⋊C4C22×D12712yes—
226C22⋊C4—22no—
227C22⋊C4—2542no—
228C22⋊C4—2542no—
229C22⋊C4—2542no—
230C22⋊C4—2542no—
231C42—2542no—
232C23—22no—
233C23—1271no—
234C23—1271no—
235C23—10162no—
236C23—10162no—
237C23—5084no—
238C23—2542no—
239C23D4×D12711yescharacteristic
240Q8—2542no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—10162no—
245D4—10162no—
246D4—10162no—
247D4—5084no—
248D4—2542no—
249D4—2542no—
250D4—2542no—
251D4—2542no—
252D4—2542no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—1271no—
257C2×C4—1272no—
258C2×C4—1272no—
259C2×C4—1271no—
260C2×C4—2542no—
261C2×C4—2541no—
262C2×C4—2541no—
263C2×C4D4×D12711yescharacteristic
264C2×C4C23×D12711yescharacteristic
265C8—21no—
266C8—1272no—
267C8—2541no—
268C8D4×D12712yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—1272no—
276C22—10168no—
277C22—5082no—
278C22—5082no—
279C22—5082no—
280C22—2541no—
281C22—2541no—
282C22—2541no—
283C22—2541no—
284C22—2541no—
285C22C2×D4×D12711yescharacteristic
286C4—21no—
287C4—42no—
288C4—1272no—
289C4—5082no—
290C4—2541no—
291C4C2×D4×D12711yescharacteristic
292C4#41×D12711yescharacteristic
293C2—41no—
294C2—82no—
295C2—10162no—
296C2—5081no—
297C2—2542no—
298C2#121×D12711yescharacteristic
299C2#2255067711yescharacteristic
300C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#466×C127C2
socleC22×C127C22×D4
hypercenter#466D127
hypocenterC127#3058
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D8×D127C2semidirect
2. C2×#16861215C2semidirect
3. #22550442C2semidirect
4. #22550128C2semidirect
5. #22550127C2semidirect
6. #22550113C2semidirect
7. #22550112C2semidirect
8. #22550109C2semidirect
9. #22549913C2semidirect
10. #22549879C2semidirect
11. #466×C127C2semidirect
18. C2×#14523232C22semidirect
20. C2×#14523231C22semidirect
22. C2×C8×D127C22semidirect
25. #16861184C22semidirect
26. #16861183C22semidirect
27. #16861180C22semidirect
31. #16861143C22semidirect
32. #16861140C22semidirect
33. #16861131C22semidirect
34. #16861130C22semidirect
35. #16861129C22semidirect
37. C22×C8×C127C22semidirect
38. #121×C127C22semidirect
39. #107×C127C22semidirect
40. #102×C127C22semidirect
65. C8×D127D4semidirect
151. C127#3058semidirect
160. #466D127semidirect
175. C22×C8C2×D127semidirect
181. #121C2×D127semidirect
186. #107C2×D127semidirect
191. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1272
C2×#168612152
#225504422
#225501282
#225501272
#225501132
#225501122
#225501092
#225499132
#225498792
#466×C1272
#3058127