Finite Group Databasebeta

← #30648646

Subgroups — #30648646

Order 16256.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#22550389C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic, complemented
3#22550216C212yesmaximal, complemented
4#22550204C211yesmaximal, characteristic, complemented
5#22550203C212yesmaximal, complemented
6#22550194C211yesmaximal, characteristic, complemented
7#22550190C211yesmaximal, characteristic, complemented
8#22549913C212yesmaximal, complemented
9#22549871C212yesmaximal, complemented
10C22⋊C8×D127C211yesmaximal, characteristic, complemented
11#484×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14C2×C8×D127C2212yes—
15#16861194—21no—
16#16861185C2211yescharacteristic
17#16861184C2211yescharacteristic
18#16861183C2211yescharacteristic
19#16861179—21no—
20#16861171—21no—
21(C22⋊D127)⋊C4—21no—
22#16861167—21no—
23#16861167—21no—
24#16861167C2212yes—
25#16861167C2212yes—
26#16861167C2211yescharacteristic
27C2×C4⋊Dic127C2211yescharacteristic
28C2×(C2×D127)⋊C4—21no—
29C2×C4×Dic127—21no—
30#16861143C2212yescomplemented
31#16861140C2212yescomplemented
32#16861140C2212yescomplemented
33#16861133C2212yescomplemented
34#16861131C2212yescomplemented
35#16861129C2212yescomplemented
36C22⋊(C127⋊C8)C2211yescharacteristic
37(C2×D127)⋊C8C2212yes—
38C22⋊Q8×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic
40#107×C127C2212yescomplemented
41#103×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic
43C23×D127D411yescharacteristic
44#14523249—24no—
45C2×D508—21no—
46C2×D508—21no—
47C2×D508C2311yescharacteristic
48C2×C22⋊D127—21no—
49C22×Dic127—21no—
50C22×Dic127D411yescharacteristic
51C2×C4×D127—24no—
52C2×C4×D127—21no—
53C2×C4×D127D411yescharacteristic
54C2×C4×D127D411yescharacteristic
55C2×C4×D127C2311yescharacteristic
56C2×C4×D127C2311yescharacteristic
57C2×C127⋊C8C2312yes—
58C8×D127—24no—
59C4⋊Dic127—21no—
60C4⋊Dic127C2311yescharacteristic
61C4⋊Dic127C2312yes—
62Dic127⋊C4—21no—
63(C2×D127)⋊C4—21no—
64(C2×D127)⋊C4—21no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—21no—
69C4×Dic127—22no—
70C4×Dic127—22no—
71C4×Dic127—21no—
72C2×Q8×C127C2311yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127C2311yescharacteristic
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C22×D127—22no—
82C22×D127—22no—
83C22×D127—21no—
84C22×D127—41no—
85C22×D127C2×D411yescharacteristic
86C22×D127C2×D411yescharacteristic
87D508—22no—
88D508—42no—
89D508—42no—
90C22⋊D127—42no—
91C2×Dic127—22no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—42no—
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C4×D127—22no—
98C4×D127—22no—
99C4×D127—22no—
100C4×D127—42no—
101C4×D127#4114yes—
102C127⋊C8—22no—
103C23×C127C2×D411yescharacteristic
104Q8×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—42no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127#13512yes—
116C22×C127C22×D411yescharacteristic
117C2×D127—21no—
118C2×D127—21no—
119C2×D127—21no—
120C2×D127—21no—
121C2×D127—21no—
122C2×D127—44no—
123C2×D127—41no—
124C2×D127—41no—
125C2×D127—41no—
126C2×D127#12312yes—
127C4×C127—21no—
128C4×C127—41no—
129C4×C127—42no—
130C4×C127—41no—
131C4×C127C2×#4112yes—
132Dic127—21no—
133Dic127—42no—
134Dic127—41no—
135Dic127#12312yes—
136C2×C127—22no—
137C2×C127C2×#12311yescharacteristic
138C2×C127C2×#13511yescharacteristic
139C2×C127#57511yescharacteristic
140D127—22no—
141D127—41no—
142D127—81no—
143#3091—1271nomaximal, Sylow
144C127#309111yescharacteristic, complemented, Sylow
145#525—1271no—
146C2×#117—1271no—
147#485—1272no—
148#484D12711yescharacteristic, complemented
149#483—1271no—
150#482—1272no—
151#481—1271no—
152#480—1271no—
153#427—1272no—
154#416—1272no—
155C2×C22⋊C8—1271no—
156C2×#41—1271no—
157C23×C4—1271no—
158C22×C8—1272no—
159#123—2541no—
160C22⋊Q8—1271no—
161C22⋊Q8C2×D12711yescharacteristic
162#121—1271no—
163#121—1271no—
164#120—2541no—
165C4×D4—2541no—
166C4×D4—2541no—
167#117—1272no—
168#117—1272no—
169#117—1271no—
170#117—2541no—
171#117—2541no—
172C2×C4⋊C4—1271no—
173C2×C4⋊C4C2×D12711yescharacteristic
174C2×C22⋊C4—2541no—
175C2×C42—2541no—
176#107—1272no—
177#107C2×D12712yescomplemented
178#106—1272no—
179#106—1272no—
180#103—1272no—
181#103C2×D12712yescomplemented
182#102—1272no—
183#102—1272no—
184C22⋊C8—1271no—
185C22⋊C8—1272no—
186C22⋊C8C2×D12711yescharacteristic
187C24—1271no—
188#41—2544no—
189C2×Q8C22×D12711yescharacteristic
190C2×D4—1271no—
191C2×D4—2541no—
192C2×D4—2541no—
193C2×D4—2541no—
194C22×C4—21no—
195C22×C4—1271no—
196C22×C4—1271no—
197C22×C4—1271no—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—2541no—
201C22×C4—2544no—
202C22×C4—2541no—
203C22×C4C22×D12711yescharacteristic
204C2×C8—1272no—
205C2×C8—2544no—
206C2×C8C22×D12712yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—1272no—
210C4⋊C4—1271no—
211C4⋊C4—2541no—
212C4⋊C4—2541no—
213C4⋊C4C22×D12712yes—
214C4⋊C4C22×D12711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—2541no—
217C22⋊C4—2542no—
218C22⋊C4—2542no—
219C22⋊C4—2541no—
220C22⋊C4—2541no—
221C22⋊C4—2541no—
222C42—2542no—
223C42—2542no—
224C42—2541no—
225C23—1271no—
226C23—1271no—
227C23—5081no—
228C23—2542no—
229C23—2541no—
230C23—2542no—
231C23D4×D12711yescharacteristic
232Q8—22no—
233D4—5082no—
234D4—5082no—
235D4—5082no—
236D4—2542no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—1271no—
243C2×C4—1274no—
244C2×C4—1271no—
245C2×C4—5082no—
246C2×C4—5082no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4—2542no—
251C2×C4—2541no—
252C2×C4—2542no—
253C2×C4D4×D12711yescharacteristic
254C2×C4C23×D12711yescharacteristic
255C8—22no—
256C8—2542no—
257C22—21no—
258C22—21no—
259C22—1272no—
260C22—5084no—
261C22—5081no—
262C22—5081no—
263C22—5081no—
264C22—2541no—
265C22—2541no—
266C22—2541no—
267C22—2541no—
268C22—2541no—
269C22#1686123212yes—
270C22C2×D4×D12711yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—1272no—
276C4—5082no—
277C4—5081no—
278C4—2541no—
279C4#41×D12712yes—
280C2—22no—
281C2—10161no—
282C2—5081no—
283C2—2542no—
284C2#123×D12711yescharacteristic
285C2C2×#1686123211yescharacteristic
286C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#484×C127C2
socleC22×C127C22×D4
prime radical#22550389C2
hypercenter#484D127
hypocenterC127#3091
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#16861167C2semidirect
3. #22550216C2semidirect
4. #22550204C2semidirect
5. #22550203C2semidirect
6. #22550194C2semidirect
7. #22550190C2semidirect
8. #22549913C2semidirect
9. #22549871C2semidirect
10. C22⋊C8×D127C2semidirect
11. #484×C127C2semidirect
30. #16861143C22semidirect
31. #16861140C22semidirect
32. #16861140C22semidirect
33. #16861133C22semidirect
34. #16861131C22semidirect
35. #16861129C22semidirect
40. #107×C127C22semidirect
41. #103×C127C22semidirect
144. C127#3091semidirect
148. #484D127semidirect
177. #107C2×D127semidirect
181. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503892
C2×#168611672
#225502162
#225502042
#225502032
#225501942
#225501902
#225499132
#225498712
C22⋊C8×D1272
#484×C1272
#3091127