Finite Group Databasebeta

← #30648672

Subgroups — #30648672

Order 16256.

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

normal subgroups only

This group has 462 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×#16861184C211yesmaximal, characteristic
2#117×D127C211yesmaximal, characteristic, complemented
3#22550203C212yesmaximal, complemented
4#22550202C212yesmaximal, complemented
5#22550196C212yesmaximal, complemented
6#22550191C212yesmaximal, complemented
7#22549916C211yesmaximal, characteristic, complemented
8#22549882C211yesmaximal, characteristic, complemented
9#22549880C211yesmaximal, characteristic, complemented
10#22549839C211yesmaximal, characteristic, complemented
11#428×C127C211yesmaximal, characteristic, complemented
12C22×D508—21no—
13C22×D508C2211yescharacteristic
14C22×C4×D127C2211yescharacteristic
15C2×#14523232C2211yescharacteristic
16#16861184—21no—
17#16861184—21no—
18#16861184C2212yes—
19#16861184C2212yes—
20#16861168C2211yescharacteristic
21#16861167C2211yescharacteristic, complemented
22#16861167C2211yescharacteristic, complemented
23#16861166—21no—
24#16861165—21no—
25C4⋊C4×D127C2211yescharacteristic, complemented
26C4⋊C4×D127C2211yescharacteristic, complemented
27C2×(C2×D127)⋊C4—21no—
28C22⋊C4×D127—21no—
29C42×D127—21no—
30#16861142C2212yescomplemented
31#16861140C2212yescomplemented
32#16861131C2212yescomplemented
33#16861131C2212yescomplemented
34#16861129C2212yescomplemented
35#16861129C2212yescomplemented
36(C2×D127)⋊C8C2212yescomplemented
37(C2×D127)⋊C8C2212yescomplemented
38C2×M16×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#107×C127C2212yescomplemented
42#106×C127C2212yescomplemented
43C23×D127—21no—
44C23×D127D411yescharacteristic
45C2×D508—22no—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—22no—
49C2×D508—22no—
50C2×D508—44no—
51C2×D508C2312yes—
52C22×Dic127D411yescharacteristic
53C2×C4×D127—21no—
54C2×C4×D127—21no—
55C2×C4×D127—22no—
56C2×C4×D127—22no—
57C2×C4×D127D411yescharacteristic
58C2×C4×D127D411yescharacteristic
59C2×C4×D127C2311yescharacteristic
60C2×C4×D127C2311yescharacteristic
61#14523232—22no—
62C2×C127⋊C8C2312yes—
63C4⋊Dic127C2311yescharacteristic
64C4⋊Dic127C2311yescharacteristic
65Dic127⋊C4—21no—
66Dic127⋊C4—21no—
67C22⋊Dic127—21no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72C4×Dic127—21no—
73C4×Dic127—21no—
74C4×Dic127—21no—
75C22×C4×C127—21no—
76C22×C4×C127C2311yescharacteristic
77M16×C127—22no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127—21no—
80C4⋊C4×C127C2312yes—
81C4⋊C4×C127C2311yescharacteristic
82C4⋊C4×C127C2311yescharacteristic
83C22⋊C4×C127—21no—
84C42×C127—21no—
85C22×D127—22no—
86C22×D127—22no—
87C22×D127—22no—
88C22×D127—41no—
89C22×D127—42no—
90C22×D127—42no—
91C22×D127—42no—
92C22×D127C2×D411yescharacteristic
93C22×D127C2×D411yescharacteristic
94D508—22no—
95D508—22no—
96D508—44no—
97D508—44no—
98D508—42no—
99C2×Dic127—22no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127C2×D411yescharacteristic
103C2×Dic127C2×D411yescharacteristic
104C4×D127—22no—
105C4×D127—22no—
106C4×D127—21no—
107C4×D127—21no—
108C4×D127—21no—
109C4×D127—21no—
110C4×D127—42no—
111C4×D127—42no—
112C127⋊C8—22no—
113C23×C127C2×D411yescharacteristic
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—22no—
117C2×C4×C127—42no—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127#4112yes—
120C2×C4×C127#4112yes—
121C2×C4×C127C2411yescharacteristic
122C8×C127—22no—
123C22×C127—21no—
124C22×C127—21no—
125C22×C127#12312yes—
126C22×C127C22×D411yescharacteristic
127C2×D127—21no—
128C2×D127—21no—
129C2×D127—21no—
130C2×D127—21no—
131C2×D127—21no—
132C2×D127—44no—
133C2×D127—42no—
134C2×D127—42no—
135C2×D127—42no—
136C2×D127—84no—
137C2×D127AGL(1,ℤ/8)12yes—
138C4×C127—42no—
139C4×C127—41no—
140C4×C127—41no—
141C4×C127#12312yes—
142C4×C127C2×#4111yescharacteristic
143C4×C127C2×#4111yescharacteristic
144Dic127—21no—
145Dic127—21no—
146Dic127—41no—
147Dic127—41no—
148C2×C127—22no—
149C2×C127C2×#12311yescharacteristic
150C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C127#57511yescharacteristic
152D127—22no—
153D127—41no—
154D127—82no—
155#3094—1271nomaximal, Sylow
156C127#309411yescharacteristic, complemented, Sylow
157C2×#121—1271no—
158C2×#117—1271no—
159#482—1272no—
160#482—1272no—
161#481—1272no—
162#480—1272no—
163#428—1271no—
164#428D12711yescharacteristic, complemented
165#418—1271no—
166#418—1271no—
167#407—1271no—
168C22×D4—1271no—
169C22×D4—2541no—
170C23×C4—1271no—
171C2×M16—1271no—
172C2×M16C2×D12711yescharacteristic
173#121—1272no—
174#121—1272no—
175#121—2541no—
176#121—2541no—
177#117—1271no—
178#117—1271no—
179#117—1271no—
180#117—2541no—
181#117—2541no—
182#117C2×D12711yescharacteristic
183C2×C4⋊C4—1271no—
184C2×C4⋊C4—1271no—
185C2×C4⋊C4C2×D12711yescharacteristic
186C2×C22⋊C4—2541no—
187C2×C22⋊C4—2541no—
188C2×C42—2541no—
189#107—1272no—
190#107C2×D12712yescomplemented
191#106—1272no—
192#106C2×D12712yescomplemented
193#102—1272no—
194#102—1272no—
195#102—1272no—
196#102—1272no—
197C22⋊C8—1272no—
198C22⋊C8—1272no—
199C24—1271no—
200C24—2541no—
201C2×D4—1272no—
202C2×D4—5084no—
203C2×D4—2542no—
204C2×D4—2542no—
205C2×D4—2541no—
206C2×D4—2542no—
207C2×D4—2542no—
208C22×C4—21no—
209C22×C4—1271no—
210C22×C4—1271no—
211C22×C4—1271no—
212C22×C4—1271no—
213C22×C4—1271no—
214C22×C4—2541no—
215C22×C4—2541no—
216C22×C4—2542no—
217C22×C4—2542no—
218C22×C4C22×D12711yescharacteristic
219M16—22no—
220M16—2542no—
221C2×C8—1272no—
222C2×C8C22×D12712yes—
223C4⋊C4—21no—
224C4⋊C4—1271no—
225C4⋊C4—1271no—
226C4⋊C4—2541no—
227C4⋊C4—2541no—
228C4⋊C4C22×D12711yescharacteristic
229C4⋊C4C22×D12711yescharacteristic
230C4⋊C4C22×D12712yes—
231C22⋊C4—21no—
232C22⋊C4—2542no—
233C22⋊C4—2542no—
234C22⋊C4—2541no—
235C22⋊C4—2541no—
236C22⋊C4—2541no—
237C42—21no—
238C42—2541no—
239C42—2541no—
240C42—2541no—
241C23—1271no—
242C23—1271no—
243C23—5082no—
244C23—5082no—
245C23—5081no—
246C23—5082no—
247C23—2542no—
248C23—2542no—
249C23—2542no—
250C23D4×D12711yescharacteristic
251D4—5084no—
252D4—5084no—
253D4—5082no—
254D4—2542no—
255D4—2542no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—1271no—
261C2×C4—1271no—
262C2×C4—5082no—
263C2×C4—5082no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4—2542no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4—2541no—
270C2×C4—2541no—
271C2×C4—2541no—
272C2×C4—2541no—
273C2×C4D4×D12711yescharacteristic
274C2×C4#1452324812yes—
275C2×C4#1452324912yes—
276C2×C4C23×D12711yescharacteristic
277C8—22no—
278C8—2542no—
279C22—21no—
280C22—21no—
281C22—1272no—
282C22—10164no—
283C22—5084no—
284C22—5082no—
285C22—5082no—
286C22—5082no—
287C22—2541no—
288C22—2541no—
289C22—2541no—
290C22—2541no—
291C22—2541no—
292C22#1686119412yes—
293C22C2×D4×D12711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—5081no—
298C4—5081no—
299C4—2541no—
300C4—2541no—
301C4#1686119412yes—
302C4C2×#1452324811yescharacteristic
303C4C2×#1452324911yescharacteristic
304C2—22no—
305C2—10162no—
306C2—5081no—
307C2—2542no—
308C2C2×#1686119411yescharacteristic
309C2AGL(1,ℤ/8)×D12711yescharacteristic
310C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861184C2
hypercenter#428D127
hypocenterC127#3094
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #117×D127C2semidirect
3. #22550203C2semidirect
4. #22550202C2semidirect
5. #22550196C2semidirect
6. #22550191C2semidirect
7. #22549916C2semidirect
8. #22549882C2semidirect
9. #22549880C2semidirect
10. #22549839C2semidirect
11. #428×C127C2semidirect
21. #16861167C22semidirect
22. #16861167C22semidirect
25. C4⋊C4×D127C22semidirect
26. C4⋊C4×D127C22semidirect
30. #16861142C22semidirect
31. #16861140C22semidirect
32. #16861131C22semidirect
33. #16861131C22semidirect
34. #16861129C22semidirect
35. #16861129C22semidirect
36. (C2×D127)⋊C8C22semidirect
37. (C2×D127)⋊C8C22semidirect
41. #107×C127C22semidirect
42. #106×C127C22semidirect
156. C127#3094semidirect
164. #428D127semidirect
190. #107C2×D127semidirect
192. #106C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611842
#117×D1272
#225502032
#225502022
#225501962
#225501912
#225499162
#225498822
#225498802
#225498392
#428×C1272
#3094127