Finite Group Databasebeta

← #30648870

Subgroups — #30648870

Order 16256.

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

normal subgroups only

This group has 450 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
1#22550410C211yesmaximal, characteristic, complemented
2C2×C4×D508C211yesmaximal, characteristic, complemented
3#22550207C211yesmaximal, characteristic, complemented
4#22550190C211yesmaximal, characteristic, complemented
5#22550177C212yesmaximal, complemented
6#22550092C212yesmaximal, complemented
7#22550029C211yesmaximal, characteristic, complemented
8#22550012C211yesmaximal, characteristic
9#22549989C212yesmaximal, complemented
10#22549978C212yesmaximal, complemented
11#432×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C4×D127—21no—
14C2×#14523237C2212yes—
15C2×D1016C2212yes—
16#16861203—21no—
17#16861201C2211yescharacteristic, complemented
18#16861197C2211yescharacteristic, complemented
19#16861192—21no—
20#16861188C2211yescharacteristic
21#16861183C2211yescharacteristic
22C4×Dic254C2211yescharacteristic, complemented
23C4×D508—21no—
24C4×D508—21no—
25C4×D508C2212yes—
26C4×D508C2212yes—
27C4×D508C2211yescharacteristic, complemented
28C4×C22⋊D127—21no—
29#16861166—21no—
30C2×C4⋊Dic127C2211yescharacteristic
31C2×(C2×D127)⋊C4—21no—
32#16861143C2212yescomplemented
33#16861141C2212yescomplemented
34#16861134C2212yescomplemented
35#16861131C2212yes—
36#16861131C2212yes—
37#16861131C2212yescomplemented
38C2×C42×C127C2211yescharacteristic
39C4⋊C8×C127C2212yescomplemented
40C22⋊C8×C127C2211yescharacteristic
41C22⋊C8×C127C2211yescharacteristic
42#97×C127C2212yescomplemented
43C23×D127—21no—
44C2×Dic254C2311yescharacteristic
45C2×D508—24no—
46C2×D508—21no—
47C2×D508C2311yescharacteristic
48C2×D508C2312yes—
49C2×C22⋊D127—21no—
50C22×Dic127—21no—
51C2×C4×D127—22no—
52C2×C4×D127—22no—
53C2×C4×D127—21no—
54C2×C4×D127—44no—
55#14523237—24no—
56D1016—24no—
57C4⋊Dic127—21no—
58C4⋊Dic127C2311yescharacteristic
59C4⋊Dic127C2311yescharacteristic
60C4⋊Dic127C2312yes—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64Dic127⋊C4—21no—
65C22⋊Dic127—21no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—21no—
69(C2×D127)⋊C4—21no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72C4×Dic127—21no—
73C22×C4×C127D411yescharacteristic
74C22×C4×C127D411yescharacteristic
75C22×C4×C127C2311yescharacteristic
76C2×C8×C127C2312yes—
77C2×C8×C127C2312yes—
78C42×C127D411yescharacteristic
79C42×C127D411yescharacteristic
80C42×C127C2311yescharacteristic
81C42×C127C2311yescharacteristic
82C22×D127—24no—
83C22×D127—22no—
84C22×D127—21no—
85C22×D127—42no—
86C22×D127—42no—
87Dic254—22no—
88D508—24no—
89D508—22no—
90D508—22no—
91D508#4114yes—
92C22⋊D127—42no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—21no—
97C2×Dic127—42no—
98C4×D127—42no—
99C4×D127—44no—
100C4×D127—44no—
101C23×C127C2×D411yescharacteristic
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—41no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C8×C127—22no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127AGL(1,ℤ/8)12yes—
116C22×C127C22×D411yescharacteristic
117C2×D127—24no—
118C2×D127—48no—
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—41no—
124C4×C127—21no—
125C4×C127—21no—
126C4×C127—21no—
127C4×C127—21no—
128C4×C127—21no—
129C4×C127C2×#4112yes—
130Dic127—41no—
131Dic127—42no—
132Dic127—41no—
133Dic127—41no—
134C2×C127—22no—
135C2×C127#52511yescharacteristic
136C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C127#57511yescharacteristic
138D127—44no—
139D127—81no—
140#3114—1271nomaximal, Sylow
141C127#311411yescharacteristic, complemented, Sylow
142#529—1271no—
143C2×C4×D4—1271no—
144#483—1271no—
145#480—1271no—
146#478—1272no—
147#463—1272no—
148#450—1271no—
149#447—1271no—
150#442—1272no—
151#440—1272no—
152#432D12711yescharacteristic, complemented
153C22×D4—1271no—
154C23×C4—2541no—
155C2×SD16—1272no—
156C2×D8—1272no—
157#126—2541no—
158#125—1271no—
159#124—1271no—
160#123—2541no—
161C22⋊Q8—1271no—
162#121—1271no—
163C4×Q8—1271no—
164C4×D4—1272no—
165C4×D4—1272no—
166C4×D4—1271no—
167C4×D4—2541no—
168C4×D4—2541no—
169C4×D4—2541no—
170#117—2541no—
171C2×C4⋊C4—1271no—
172C2×C22⋊C4—2541no—
173C2×C42C2×D12711yescharacteristic
174#107—1272no—
175#106—1272no—
176C4⋊C8C2×D12712yescomplemented
177#103—1272no—
178#102—1272no—
179#102—1272no—
180#102—1272no—
181C22⋊C8C2×D12711yescharacteristic
182C22⋊C8C2×D12711yescharacteristic
183#97C2×D12712yescomplemented
184C24—2541no—
185C2×Q8—1271no—
186C2×D4—1272no—
187C2×D4—1271no—
188C2×D4—2544no—
189C2×D4—2541no—
190C2×D4—2541no—
191C22×C4—5084no—
192C22×C4—2541no—
193C22×C4—2541no—
194C22×C4—2542no—
195C22×C4—2542no—
196C22×C4D50811yescharacteristic
197C22×C4D50811yescharacteristic
198C22×C4C22×D12711yescharacteristic
199SD16—2544no—
200D8—2544no—
201C2×C8C22×D12712yes—
202C2×C8C22×D12712yes—
203C4⋊C4—1271no—
204C4⋊C4—1271no—
205C4⋊C4—1272no—
206C4⋊C4—2541no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4—2541no—
210C4⋊C4—2541no—
211C22⋊C4—2541no—
212C22⋊C4—2542no—
213C22⋊C4—2542no—
214C22⋊C4—2541no—
215C22⋊C4—2541no—
216C22⋊C4—2541no—
217C22⋊C4—2541no—
218C42—2541no—
219C42D50811yescharacteristic
220C42D50811yescharacteristic
221C42C22×D12711yescharacteristic
222C42C22×D12711yescharacteristic
223C23—5082no—
224C23—5082no—
225C23—2544no—
226C23—2541no—
227C23—2542no—
228C23C2×D50811yescharacteristic
229Q8—2542no—
230D4—1274no—
231D4—5082no—
232D4—2544no—
233D4—2542no—
234D4—2542no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—5084no—
239C2×C4—5084no—
240C2×C4—5082no—
241C2×C4—5082no—
242C2×C4—2541no—
243C2×C4—2541no—
244C2×C4—2541no—
245C2×C4—2542no—
246C2×C4C2×D50811yescharacteristic
247C2×C4C2×D50811yescharacteristic
248C2×C4C2×D50811yescharacteristic
249C2×C4C2×D50811yescharacteristic
250C2×C4C2×D50811yescharacteristic
251C2×C4C23×D12711yescharacteristic
252C8—22no—
253C8—22no—
254C22—21no—
255C22—21no—
256C22—5088no—
257C22—5081no—
258C22—5081no—
259C22—5081no—
260C22—5082no—
261C22—5082no—
262C22—2544no—
263C22#1686123712yes—
264C22C22×D50811yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—5081no—
271C4—5082no—
272C4—5081no—
273C4—5081no—
274C4#41×D12712yes—
275C2—22no—
276C2—10161no—
277C2—5084no—
278C2#2255038811yescharacteristic
279C2C2×#1686123711yescharacteristic
280C2#2255067611yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#432×C127C2
socleC22×C127C22×D4
prime radical#22550012C2
hypercenter#432D127
hypocenterC127#3114
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #22550410C2semidirect
2. C2×C4×D508C2semidirect
3. #22550207C2semidirect
4. #22550190C2semidirect
5. #22550177C2semidirect
6. #22550092C2semidirect
7. #22550029C2semidirect
9. #22549989C2semidirect
10. #22549978C2semidirect
11. #432×C127C2semidirect
17. #16861201C22semidirect
18. #16861197C22semidirect
22. C4×Dic254C22semidirect
27. C4×D508C22semidirect
32. #16861143C22semidirect
33. #16861141C22semidirect
34. #16861134C22semidirect
37. #16861131C22semidirect
39. C4⋊C8×C127C22semidirect
42. #97×C127C22semidirect
141. C127#3114semidirect
152. #432D127semidirect
176. C4⋊C8C2×D127semidirect
183. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
C2×C4×D5082
#225502072
#225501902
#225501772
#225500922
#225500292
#225500122
#225499892
#225499782
#432×C1272
#3114127