Finite Group Databasebeta

← #30649115

Subgroups — #30649115

Order 16256.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550431C211yesmaximal, characteristic, complemented
2#124×D127C211yesmaximal, characteristic
3#22550238C211yesmaximal, characteristic, complemented
4#22550236C211yesmaximal, characteristic, complemented
5#22550235C211yesmaximal, characteristic, complemented
6#22550232C212yesmaximal, complemented
7#22550229C212yesmaximal, complemented
8#22549883C212yesmaximal, complemented
9#22549879C212yesmaximal
10#22549828C211yesmaximal, characteristic, complemented
11#489×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2211yescharacteristic
13C2×D4×D127C2211yescharacteristic
14C2×#14523231C2212yes—
15#16861209C2211yescharacteristic, complemented
16#16861200C2212yescomplemented
17#16861198C2211yescharacteristic
18#16861197C2211yescharacteristic
19#16861196C2211yescharacteristic
20#16861195—21no—
21#16861195—21no—
22#16861189—21no—
23#16861185—21no—
24Q8×Dic127—21no—
25#16861174—21no—
26#16861167C2212yes—
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
37C8×Dic127C2212yescomplemented
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—21no—
46C2×Dic254C2311yescharacteristic
47Q8×D127—24no—
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—21no—
50D4×D127—24no—
51C2×C4×D127—21no—
52C2×C4×D127D411yescharacteristic
53C2×C4×D127D411yescharacteristic
54C2×C4×D127C2311yescharacteristic
55#14523231—24no—
56C2×C127⋊C8C2312yes—
57C4⋊Dic127—22no—
58C4⋊Dic127—21no—
59C4⋊Dic127C2312yes—
60Dic127⋊C4—22no—
61C22⋊Dic127—21no—
62C22⋊Dic127—21no—
63(C2×D127)⋊C4—22no—
64(C2×D127)⋊C4—21no—
65(C2×D127)⋊C4—21no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68C4×Dic127—22no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127—21no—
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127C2311yescharacteristic
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127C2312yes—
78C22⋊C4×C127—21no—
79C22⋊C4×C127—21no—
80C42×C127C2311yescharacteristic
81C22×D127—21no—
82C22×D127—42no—
83C22×D127—42no—
84C22×D127—42no—
85C22×D127C2×D411yescharacteristic
86Dic254—22no—
87Dic254—42no—
88Dic254—42no—
89D508—22no—
90C22⋊D127—42no—
91C2×Dic127—22no—
92C2×Dic127—21no—
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—41no—
97C4×D127—41no—
98C4×D127—42no—
99C4×D127#4114yes—
100C127⋊C8—22no—
101C23×C127—21no—
102Q8×C127—22no—
103Q8×C127—42no—
104D4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2411yescharacteristic
110C8×C127—22no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127—41no—
114C22×C127C22×D411yescharacteristic
115C2×D127—21no—
116C2×D127—21no—
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—41no—
120C2×D127—84no—
121C2×D127#12412yes—
122C4×C127—21no—
123C4×C127—21no—
124C4×C127—41no—
125C4×C127—42no—
126C4×C127C2×#4112yes—
127Dic127—21no—
128Dic127—21no—
129Dic127—41no—
130Dic127—42no—
131Dic127#12412yes—
132C2×C127—81no—
133C2×C127C2×#12411yescharacteristic
134C2×C127#57511yescharacteristic
135C2×C127#57511yescharacteristic
136D127—22no—
137D127—81no—
138#3140—1271nomaximal, Sylow
139C127#314011yescharacteristic, complemented, Sylow
140#533—1271no—
141C2×#124—1271no—
142#489—1271no—
143#489—1271no—
144#489—1271no—
145#489D12711yescharacteristic, complemented
146#488—1272no—
147#488—1272no—
148#419—1272no—
149#418—1272no—
150C4×M16—1271no—
151C22×Q8—1271no—
152C22×D4—1271no—
153C2×M16—1272no—
154C4⋊Q8—1271no—
155C4⋊Q8C2×D12711yescharacteristic, complemented
156#125—1272no—
157#124—1271no—
158#124—1271no—
159#124—1271no—
160#124—2541no—
161#124—2541no—
162#124C2×D12711yescharacteristic
163C22⋊Q8—2541no—
164C22⋊Q8—2541no—
165C4×Q8—2541no—
166C4×Q8—2541no—
167#117—1272no—
168C2×C22⋊C4—2541no—
169C2×C22⋊C4—2541no—
170C2×C42—1271no—
171#103—1272no—
172#103—1272no—
173#103—1272no—
174#103C2×D12712yescomplemented
175#102—1272no—
176#102—1272no—
177#102—1272no—
178#102C2×D12712yes—
179#97—1271no—
180#97C2×D12711yescharacteristic, complemented
181C4×C8—1272no—
182C24—2541no—
183C2×Q8—21no—
184C2×Q8—1271no—
185C2×Q8—2541no—
186C2×Q8—2544no—
187C2×Q8—2541no—
188C2×Q8C22×D12711yescharacteristic
189C2×D4—1271no—
190C2×D4—2544no—
191C2×D4—2541no—
192C2×D4C22×D12711yescharacteristic
193C22×C4—1271no—
194C22×C4—1271no—
195C22×C4—1271no—
196C22×C4—2541no—
197M16—2544no—
198C2×C8—1272no—
199C2×C8C22×D12712yes—
200C4⋊C4—21no—
201C4⋊C4—1272no—
202C4⋊C4—2542no—
203C4⋊C4—2542no—
204C4⋊C4—2541no—
205C4⋊C4C22×D12712yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—2541no—
209C22⋊C4—2541no—
210C22⋊C4—2542no—
211C22⋊C4—2541no—
212C22⋊C4—2541no—
213C22⋊C4—2541no—
214C22⋊C4—2541no—
215C42—1271no—
216C42—1271no—
217C42—1271no—
218C42—2542no—
219C42C22×D12711yescharacteristic
220C23—21no—
221C23—1271no—
222C23—5082no—
223C23—5082no—
224C23—5082no—
225C23—2541no—
226Q8—22no—
227Q8—42no—
228Q8—5082no—
229Q8—5082no—
230Q8—2542no—
231D4—22no—
232D4—5082no—
233D4—2542no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—1271no—
237C2×C4—1274no—
238C2×C4—1271no—
239C2×C4—1271no—
240C2×C4—5081no—
241C2×C4—5081no—
242C2×C4—5082no—
243C2×C4—2542no—
244C2×C4—2541no—
245C2×C4D4×D12711yescharacteristic
246C2×C4D4×D12711yescharacteristic
247C2×C4C23×D12711yescharacteristic
248C8—22no—
249C8—2542no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—1272no—
254C22—10164no—
255C22—5081no—
256C22—5081no—
257C22—5081no—
258C22—2541no—
259C22—2541no—
260C22C2×D4×D12711yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—1272no—
266C4—5081no—
267C4—5082no—
268C4—2541no—
269C4—2541no—
270C4#41×D12712yes—
271C2—81no—
272C2—10161no—
273C2—2542no—
274C2#124×D12711yescharacteristic
275C2#2255067211yescharacteristic
276C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#489×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
hypercenter#489D127
hypocenterC127#3140
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. #22550431C2semidirect
3. #22550238C2semidirect
4. #22550236C2semidirect
5. #22550235C2semidirect
6. #22550232C2semidirect
7. #22550229C2semidirect
8. #22549883C2semidirect
10. #22549828C2semidirect
11. #489×C127C2semidirect
15. #16861209C22semidirect
16. #16861200C22semidirect
30. #16861134C22semidirect
31. #16861133C22semidirect
32. #16861132C22semidirect
36. #16861120C22semidirect
37. C8×Dic127C22semidirect
38. C4⋊Q8×C127C22semidirect
40. #103×C127C22semidirect
42. #97×C127C22semidirect
139. C127#3140semidirect
145. #489D127semidirect
155. C4⋊Q8C2×D127semidirect
174. #103C2×D127semidirect
180. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#124×D1272
#225502382
#225502362
#225502352
#225502322
#225502292
#225498832
#225498792
#225498282
#489×C1272
#3140127