Finite Group Databasebeta

← #30648657

Subgroups — #30648657

Order 16256.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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×C4⋊C4×D127C211yesmaximal, characteristic, complemented
3#22550209C211yesmaximal, characteristic, complemented
4#22550205C212yesmaximal, complemented
5#22550195C211yesmaximal, characteristic, complemented
6#22550194C211yesmaximal, characteristic, complemented
7#22550191C212yesmaximal, complemented
8#22549917C212yesmaximal, complemented
9#22549876C212yesmaximal, complemented
10#22549838C211yesmaximal, characteristic, complemented
11#484×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2211yescharacteristic
13C22×C4×D127—21no—
14C22×C4×D127C2211yescharacteristic
15C2×#14523231C2212yes—
16#16861194—21no—
17#16861185C2211yescharacteristic
18#16861184C2211yescharacteristic
19#16861183C2211yescharacteristic
20#16861179—21no—
21#16861171—21no—
22(C22⋊D127)⋊C4—21no—
23#16861167C2211yescharacteristic
24C2×C4⋊Dic127C2211yescharacteristic
25C2×Dic127⋊C4—21no—
26C4⋊C4×D127—21no—
27C4⋊C4×D127—21no—
28C4⋊C4×D127C2212yes—
29C4⋊C4×D127C2212yes—
30#16861142C2212yescomplemented
31#16861141C2212yescomplemented
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—22no—
53C2×C4×D127—22no—
54C2×C4×D127—21no—
55C2×C4×D127—44no—
56C2×C4×D127D411yescharacteristic
57C2×C4×D127D411yescharacteristic
58C2×C4×D127C2311yescharacteristic
59C2×C4×D127C2311yescharacteristic
60#14523231—24no—
61C2×C127⋊C8C2312yes—
62C4⋊Dic127—21no—
63C4⋊Dic127C2311yescharacteristic
64C4⋊Dic127C2312yes—
65Dic127⋊C4—21no—
66Dic127⋊C4—22no—
67Dic127⋊C4—22no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70(C2×D127)⋊C4—21no—
71(C2×D127)⋊C4—21no—
72C4×Dic127—21no—
73C2×Q8×C127C2311yescharacteristic
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2312yes—
81C22⋊C4×C127—21no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127—22no—
85C22×D127—21no—
86C22×D127C2×D411yescharacteristic
87C22×D127C2×D411yescharacteristic
88D508—22no—
89D508—42no—
90D508—42no—
91C22⋊D127—42no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—22no—
95C2×Dic127—42no—
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C4×D127—22no—
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—44no—
102C4×D127—44no—
103C4×D127—42no—
104C4×D127#4114yes—
105C127⋊C8—22no—
106C23×C127C2×D411yescharacteristic
107Q8×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—42no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2411yescharacteristic
115C8×C127—22no—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127#13712yes—
119C22×C127C22×D411yescharacteristic
120C2×D127—28no—
121C2×D127—21no—
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—21no—
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—41no—
128C2×D127#12312yes—
129C2×D127AGL(1,ℤ/8)12yes—
130C4×C127—21no—
131C4×C127—41no—
132C4×C127—42no—
133C4×C127—41no—
134C4×C127C2×#4112yes—
135Dic127—21no—
136Dic127—42no—
137Dic127—41no—
138Dic127#12312yes—
139C2×C127—22no—
140C2×C127C2×#12311yescharacteristic
141C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C127C2×#13711yescharacteristic
143D127—22no—
144D127—22no—
145D127—81no—
146#3092—1271nomaximal, Sylow
147C127#309211yescharacteristic, complemented, Sylow
148#525—1271no—
149C22×C4⋊C4—1271no—
150#484—1271no—
151#484D12711yescharacteristic, complemented
152#483—1272no—
153#481—1271no—
154#481—1271no—
155#480—1272no—
156#428—1272no—
157#417—1272no—
158#407—1271no—
159C2×#41—1271no—
160C23×C4—1271no—
161C23×C4—2541no—
162C2×M16—1272no—
163#123—2541no—
164C22⋊Q8—1271no—
165C22⋊Q8C2×D12711yescharacteristic
166#121—1271no—
167#121—1271no—
168#120—2541no—
169C4×D4—2541no—
170C4×D4—2541no—
171#117—1271no—
172C2×C4⋊C4—1271no—
173C2×C4⋊C4—1272no—
174C2×C4⋊C4—1272no—
175C2×C4⋊C4—2541no—
176C2×C4⋊C4—2541no—
177C2×C4⋊C4—2541no—
178C2×C4⋊C4C2×D12711yescharacteristic
179#107—1272no—
180#107C2×D12712yescomplemented
181#106—1272no—
182#106—1272no—
183#103—1272no—
184#103C2×D12712yescomplemented
185#102—1272no—
186#102—1272no—
187C22⋊C8—1271no—
188C22⋊C8—1272no—
189C22⋊C8C2×D12711yescharacteristic
190C24—1271no—
191#41—2544no—
192C2×Q8C22×D12711yescharacteristic
193C2×D4—1271no—
194C2×D4—2541no—
195C2×D4—2541no—
196C2×D4—2541no—
197C22×C4—21no—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—1271no—
201C22×C4—1271no—
202C22×C4—1271no—
203C22×C4—5084no—
204C22×C4—2541no—
205C22×C4—2544no—
206C22×C4—2542no—
207C22×C4—2542no—
208C22×C4—2541no—
209C22×C4C22×D12711yescharacteristic
210M16—2544no—
211C2×C8—1272no—
212C2×C8C22×D12712yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—1272no—
216C4⋊C4—1271no—
217C4⋊C4—2542no—
218C4⋊C4—2542no—
219C4⋊C4—2541no—
220C4⋊C4—2541no—
221C4⋊C4C22×D12712yes—
222C4⋊C4C22×D12711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2541no—
225C22⋊C4—2541no—
226C22⋊C4—2541no—
227C22⋊C4—2541no—
228C42—2541no—
229C23—1271no—
230C23—1271no—
231C23—2542no—
232C23—2542no—
233C23—2541no—
234C23—2542no—
235C23D4×D12711yescharacteristic
236Q8—22no—
237D4—5082no—
238D4—5082no—
239D4—5082no—
240D4—2542no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—1271no—
247C2×C4—1274no—
248C2×C4—1271no—
249C2×C4—5082no—
250C2×C4—5084no—
251C2×C4—5084no—
252C2×C4—5082no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4—2542no—
256C2×C4—2541no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4D4×D12711yescharacteristic
260C2×C4C23×D12711yescharacteristic
261C8—22no—
262C8—2542no—
263C22—21no—
264C22—21no—
265C22—1272no—
266C22—1272no—
267C22—5081no—
268C22—5081no—
269C22—5081no—
270C22—2548no—
271C22—2541no—
272C22—2541no—
273C22—2541no—
274C22—2541no—
275C22#1686124112yes—
276C22C2×D4×D12711yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—1272no—
282C4—5082no—
283C4—5081no—
284C4—2541no—
285C4#41×D12712yes—
286C2—22no—
287C2—10161no—
288C2—2542no—
289C2—2542no—
290C2#123×D12711yescharacteristic
291C2AGL(1,ℤ/8)×D12711yescharacteristic
292C2C2×#1686124111yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#484×C127C2
socleC22×C127C22×D4
prime radical#22550389C2
hypercenter#484D127
hypocenterC127#3092
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×C4⋊C4×D127C2semidirect
3. #22550209C2semidirect
4. #22550205C2semidirect
5. #22550195C2semidirect
6. #22550194C2semidirect
7. #22550191C2semidirect
8. #22549917C2semidirect
9. #22549876C2semidirect
10. #22549838C2semidirect
11. #484×C127C2semidirect
30. #16861142C22semidirect
31. #16861141C22semidirect
32. #16861140C22semidirect
33. #16861133C22semidirect
34. #16861131C22semidirect
35. #16861129C22semidirect
40. #107×C127C22semidirect
41. #103×C127C22semidirect
147. C127#3092semidirect
151. #484D127semidirect
180. #107C2×D127semidirect
184. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503892
C2×C4⋊C4×D1272
#225502092
#225502052
#225501952
#225501942
#225501912
#225499172
#225498762
#225498382
#484×C1272
#3092127