Finite Group Databasebeta

← #489×D127 #30649056

Subgroups — #489×D127 #30649056

Order 16256.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C4⋊Q8×D127C211yesmaximal, characteristic, complemented
2#124×D127C211yesmaximal, characteristic
3#22550238C211yesmaximal, characteristic, complemented
4#22550237C212yesmaximal, complemented
5#22550236C211yesmaximal, characteristic, complemented
6#22550235C211yesmaximal, characteristic, complemented
7#22550234C212yesmaximal, complemented
8#103×D127C212yesmaximal, complemented
9#102×D127C212yesmaximal
10#97×D127C211yesmaximal, characteristic, complemented
11#489×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127—21no—
13C2×Q8×D127C2211yescharacteristic
14C2×D4×D127C2211yescharacteristic
15C2×C8×D127C2212yes—
16#16861209C2211yescharacteristic, complemented
17#16861208—21no—
18#16861208C2212yescomplemented
19#16861207—21no—
20#16861198C2211yescharacteristic
21#16861197C2211yescharacteristic
22#16861196C2211yescharacteristic
23#16861195—21no—
24#16861195—21no—
25C4⋊C4×D127—21no—
26C4⋊C4×D127C2212yes—
27C22⋊C4×D127—21no—
28C22⋊C4×D127—21no—
29C42×D127C2211yescharacteristic
30#16861134C2212yescomplemented
31#16861133C2212yescomplemented
32#16861132C2212yescomplemented
33#16861131C2212yes—
34#16861130C2212yes—
35#16861129C2212yes—
36#16861120C2211yescharacteristic, complemented
37#16861119C2212yescomplemented
38C4⋊Q8×C127C2211yescharacteristic, complemented
39#124×C127C2211yescharacteristic
40#103×C127C2212yescomplemented
41#102×C127C2212yes—
42#97×C127C2211yescharacteristic, complemented
43C23×D127—21no—
44C2×Dic254—21no—
45C2×Dic254—22no—
46C2×Dic254—21no—
47C2×Dic254C2311yescharacteristic
48Q8×D127—24no—
49Q8×D127—44no—
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—21no—
52D4×D127—24no—
53C2×C4×D127—22no—
54C2×C4×D127—21no—
55C2×C4×D127D411yescharacteristic
56C2×C4×D127D411yescharacteristic
57C2×C4×D127C2311yescharacteristic
58C2×C127⋊C8C2312yes—
59C8×D127—24no—
60C4⋊Dic127—21no—
61C4⋊Dic127C2312yes—
62Dic127⋊C4—22no—
63Dic127⋊C4—22no—
64C22⋊Dic127—21no—
65C22⋊Dic127—21no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70C4×Dic127D411yescharacteristic
71C4×Dic127D411yescharacteristic
72C4×Dic127C2311yescharacteristic
73C2×Q8×C127—21no—
74C2×Q8×C127C2311yescharacteristic
75C2×D4×C127C2311yescharacteristic
76C2×C8×C127C2312yes—
77C4⋊C4×C127—21no—
78C4⋊C4×C127C2312yes—
79C22⋊C4×C127—21no—
80C22⋊C4×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127—21no—
83C22×D127—42no—
84C22×D127—42no—
85C22×D127—42no—
86C22×D127C2×D411yescharacteristic
87Dic254—22no—
88Dic254—44no—
89Dic254—42no—
90Dic254—42no—
91D508—22no—
92C22⋊D127—42no—
93C2×Dic127—21no—
94C2×Dic127—22no—
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C4×D127—22no—
99C4×D127—22no—
100C4×D127—42no—
101C4×D127—44no—
102C4×D127#4114yes—
103C127⋊C8—22no—
104C23×C127—21no—
105Q8×C127—22no—
106Q8×C127—42no—
107D4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C8×C127—22no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127C22×D411yescharacteristic
118C2×D127—41no—
119C2×D127—41no—
120C2×D127—41no—
121C2×D127—84no—
122C2×D127#12412yes—
123C2×D127AGL(1,ℤ/8)12yes—
124C2×D127#13712yes—
125C4×C127—21no—
126C4×C127—21no—
127C4×C127—41no—
128C4×C127—42no—
129C4×C127C2×#4112yes—
130Dic127—21no—
131Dic127—21no—
132Dic127—41no—
133Dic127—42no—
134Dic127#12412yes—
135C2×C127—81no—
136C2×C127C2×#12411yescharacteristic
137C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C127C2×#13711yescharacteristic
139D127—81no—
140D127#48914yesdirect product factor
141C2×#489—1271nomaximal, Sylow
142C127C2×#48911yescharacteristic, complemented, Sylow
143C2×C4⋊Q8—1271no—
144C2×#124—1271no—
145#489—1271no—
146#489—1271no—
147#489—1272no—
148#489—1272no—
149#489—1271no—
150#489D12711yescharacteristic, direct product factor
151C2×#103—1272no—
152C2×#102—1272no—
153C2×#97—1271no—
154C22×Q8—1271no—
155C22×Q8—2541no—
156C22×D4—1271no—
157C22×C8—1272no—
158C4⋊Q8—1272no—
159C4⋊Q8—1271no—
160C4⋊Q8—2541no—
161C4⋊Q8—2541no—
162C4⋊Q8C2×D12711yescharacteristic, complemented
163#124—1271no—
164#124—1271no—
165#124—1271no—
166#124—2541no—
167#124—2541no—
168#124C2×D12711yescharacteristic
169C2×C4⋊C4—1272no—
170C2×C4⋊C4—2541no—
171C2×C22⋊C4—2541no—
172C2×C22⋊C4—2541no—
173C2×C42—1271no—
174#103—1272no—
175#103—1272no—
176#103—1272no—
177#103C2×D12712yescomplemented
178#102—1272no—
179#102—1272no—
180#102—1272no—
181#102C2×D12712yes—
182#97—1272no—
183#97—1271no—
184#97C2×D12711yescharacteristic, complemented
185C24—2541no—
186C2×Q8—21no—
187C2×Q8—1271no—
188C2×Q8—5084no—
189C2×Q8—2541no—
190C2×Q8—2542no—
191C2×Q8—2544no—
192C2×Q8—2541no—
193C2×Q8C22×D12711yescharacteristic
194C2×D4—1271no—
195C2×D4—2544no—
196C2×D4—2541no—
197C2×D4C22×D12711yescharacteristic
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—1271no—
201C22×C4—2541no—
202C22×C4—2542no—
203C2×C8—1272no—
204C2×C8—2544no—
205C2×C8C22×D12712yes—
206C4⋊C4—21no—
207C4⋊C4—1272no—
208C4⋊C4—2542no—
209C4⋊C4—2542no—
210C4⋊C4—2541no—
211C4⋊C4C22×D12712yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—2541no—
215C22⋊C4—2541no—
216C22⋊C4—2541no—
217C22⋊C4—2541no—
218C22⋊C4—2541no—
219C22⋊C4—2541no—
220C42—1271no—
221C42—1271no—
222C42—1271no—
223C42C22×D12711yescharacteristic
224C23—21no—
225C23—1271no—
226C23—5082no—
227C23—5082no—
228C23—5082no—
229C23—2541no—
230Q8—22no—
231Q8—42no—
232Q8—5082no—
233Q8—5082no—
234Q8—5084no—
235Q8—2542no—
236D4—22no—
237D4—5082no—
238D4—2542no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—1271no—
242C2×C4—1274no—
243C2×C4—1271no—
244C2×C4—1271no—
245C2×C4—5084no—
246C2×C4—5082no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2541no—
250C2×C4—2542no—
251C2×C4D4×D12711yescharacteristic
252C2×C4D4×D12711yescharacteristic
253C2×C4C23×D12711yescharacteristic
254C8—22no—
255C8—2542no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—1272no—
260C22—1272no—
261C22—1272no—
262C22—10164no—
263C22—5081no—
264C22—5081no—
265C22—5081no—
266C22C2×D4×D12711yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—1272no—
272C4—5081no—
273C4—5082no—
274C4—2541no—
275C4—2541no—
276C4#41×D12712yes—
277C2—81no—
278C2—1274no—
279C2—10161no—
280C2#124×D12711yescharacteristic
281C2AGL(1,ℤ/8)×D12711yescharacteristic
282C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#489×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
hypercenter#489D127
hypocenterC127C2×#489
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D127C2semidirect
3. #22550238C2semidirect
4. #22550237C2semidirect
5. #22550236C2semidirect
6. #22550235C2semidirect
7. #22550234C2semidirect
8. #103×D127C2semidirect
10. #97×D127C2semidirect
11. #489×C127C2semidirect
16. #16861209C22semidirect
18. #16861208C22semidirect
30. #16861134C22semidirect
31. #16861133C22semidirect
32. #16861132C22semidirect
36. #16861120C22semidirect
37. #16861119C22semidirect
38. C4⋊Q8×C127C22semidirect
40. #103×C127C22semidirect
42. #97×C127C22semidirect
140. D127#489direct (N × H)
142. C127C2×#489semidirect
150. #489D127direct (N × H)
162. C4⋊Q8C2×D127semidirect
177. #103C2×D127semidirect
184. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1272
#124×D1272
#225502382
#225502372
#225502362
#225502352
#225502342
#103×D1272
#102×D1272
#97×D1272
#489×C1272
C2×#489127