Finite Group Databasebeta

← #30648376

Subgroups — #30648376

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×#16861217C211yesmaximal, characteristic, complemented
3#22550442C212yesmaximal, complemented
4#22550130C211yesmaximal, characteristic, complemented
5#22550129C211yesmaximal, characteristic, complemented
6#22550126C211yesmaximal, characteristic, complemented
7#22550111C212yesmaximal, complemented
8#22550110C212yesmaximal, complemented
9#22549911C211yesmaximal, characteristic, complemented
10#22549879C212yesmaximal, complemented
11#469×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×D4×D127C2212yes—
14C2×D1016C2211yescharacteristic
15C2×#14523233C2212yes—
16D8×D127—24no—
17#16861217—24no—
18C2×#14523231C2211yescharacteristic
19C2×#14523231C2211yescharacteristic, complemented
20C22×C127⋊C8C2211yescharacteristic, 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#16861142C2212yescomplemented
32#16861141C2211yescharacteristic, complemented
33#16861131C2212yescomplemented
34#16861130C2212yescomplemented
35#16861129C2212yescomplemented
36C2×D8×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic, complemented
38#121×C127C2212yescomplemented
39#106×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#14523231—22no—
60#14523231—22no—
61C2×C127⋊C8—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—
76M16×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—22no—
120C22×C127—42no—
121C22×C127—42no—
122C22×C127—42no—
123C22×C127C22×D411yescharacteristic
124C2×D127—22no—
125C2×D127—21no—
126C2×D127—22no—
127C2×D127—42no—
128C2×D127—42no—
129C2×D127—42no—
130C2×D127—88no—
131C2×D127#12112yes—
132C4×C127—21no—
133C4×C127—42no—
134C4×C127C22×D411yescharacteristic
135C4×C127C2×#4111yescharacteristic
136Dic127—21no—
137Dic127—42no—
138Dic127#12112yes—
139C2×C127—41no—
140C2×C127—82no—
141C2×C127C2×#12111yescharacteristic
142C2×C127#57612yes—
143D127—22no—
144D127—41no—
145D127—82no—
146#3058—1271nomaximal, Sylow
147C127#305811yescharacteristic, complemented, Sylow
148C22×D8—1271no—
149C2×#131—1271no—
150#535—1272no—
151#469—1271no—
152#469—1271no—
153#469—1271no—
154#469D12711yescharacteristic, complemented
155#466—1272no—
156#466—1272no—
157#427—1271no—
158#418—1272no—
159C2×#41—1271no—
160C22×D4—1272no—
161C2×D8—1272no—
162C2×D8—1271no—
163C2×D8—2544no—
164C2×D8C2×D12711yescharacteristic
165#131—2544no—
166C2×M16—1271no—
167C2×M16—1271no—
168C2×M16C2×D12711yescharacteristic, complemented
169C22×C8—1271no—
170C22×C8—1271no—
171C22×C8—1271no—
172#127—2542no—
173#124—2542no—
174#121—1272no—
175#121—1272no—
176#121—1272no—
177#121C2×D12712yescomplemented
178#120—2542no—
179#120—2542no—
180#117—1272no—
181#107—1272no—
182#106—1271no—
183#106C2×D12711yescharacteristic, complemented
184#102—1272no—
185#102—1272no—
186#102—1272no—
187#102C2×D12712yescomplemented
188C24—2542no—
189#41—2542no—
190#41—2542no—
191C2×Q8—1271no—
192C2×D4—22no—
193C2×D4—1271no—
194C2×D4—1271no—
195C2×D4—1271no—
196C2×D4—1272no—
197C2×D4—2542no—
198C2×D4—2544no—
199C2×D4—2544no—
200C2×D4—2542no—
201C2×D4—2542no—
202C2×D4—2542no—
203C2×D4C22×D12712yes—
204C22×C4—1271no—
205C22×C4—1271no—
206C22×C4C22×D12711yescharacteristic
207D8—22no—
208D8—2544no—
209D8—2542no—
210M16—22no—
211M16—2542no—
212M16—2542no—
213C2×C8—1271no—
214C2×C8—1274no—
215C2×C8—1271no—
216C2×C8—2542no—
217C2×C8—2542no—
218C2×C8C22×D12711yescharacteristic
219C2×C8C22×D12711yescharacteristic
220C4⋊C4—1272no—
221C4⋊C4C22×D12712yes—
222C22⋊C4—22no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C42—2542no—
228C23—22no—
229C23—1271no—
230C23—1271no—
231C23—10162no—
232C23—10162no—
233C23—5084no—
234C23—2542no—
235C23D4×D12711yescharacteristic
236Q8—2542no—
237D4—22no—
238D4—22no—
239D4—82no—
240D4—10162no—
241D4—10162no—
242D4—10162no—
243D4—5084no—
244D4—2542no—
245D4—2542no—
246D4—2542no—
247D4—2542no—
248D4—2542no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—1271no—
253C2×C4—1272no—
254C2×C4—1272no—
255C2×C4—1271no—
256C2×C4—2542no—
257C2×C4—2541no—
258C2×C4—2541no—
259C2×C4D4×D12711yescharacteristic
260C2×C4C23×D12711yescharacteristic
261C8—21no—
262C8—1272no—
263C8—2541no—
264C8D4×D12712yes—
265C22—21no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—1272no—
271C22—10168no—
272C22—5082no—
273C22—5082no—
274C22—5082no—
275C22—2541no—
276C22—2542no—
277C22—2542no—
278C22C2×D4×D12711yescharacteristic
279C4—21no—
280C4—42no—
281C4—1272no—
282C4—5082no—
283C4—2541no—
284C4C2×D4×D12711yescharacteristic
285C4#41×D12711yescharacteristic
286C2—41no—
287C2—82no—
288C2—10162no—
289C2—5081no—
290C2—2542no—
291C2#121×D12711yescharacteristic
292C2#2255068012yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127#3058
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D127C2semidirect
2. C2×#16861217C2semidirect
3. #22550442C2semidirect
4. #22550130C2semidirect
5. #22550129C2semidirect
6. #22550126C2semidirect
7. #22550111C2semidirect
8. #22550110C2semidirect
9. #22549911C2semidirect
10. #22549879C2semidirect
11. #469×C127C2semidirect
19. C2×#14523231C22semidirect
20. C22×C127⋊C8C22semidirect
22. C2×C8×D127C22semidirect
25. #16861184C22semidirect
26. #16861183C22semidirect
27. #16861180C22semidirect
31. #16861142C22semidirect
32. #16861141C22semidirect
33. #16861131C22semidirect
34. #16861130C22semidirect
35. #16861129C22semidirect
37. C2×M16×C127C22semidirect
38. #121×C127C22semidirect
39. #106×C127C22semidirect
40. #102×C127C22semidirect
65. C8×D127D4semidirect
147. C127#3058semidirect
154. #469D127semidirect
168. C2×M16C2×D127semidirect
177. #121C2×D127semidirect
183. #106C2×D127semidirect
187. #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×#168612172
#225504422
#225501302
#225501292
#225501262
#225501112
#225501102
#225499112
#225498792
#469×C1272
#3058127