Finite Group Databasebeta

← #30648691

Subgroups — #30648691

Order 16256.

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

normal subgroups only

This group has 418 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#22550445C211yesmaximal, characteristic
2#22550356C211yesmaximal, characteristic, complemented
3#22550203C212yesmaximal, complemented
4#22550202C212yesmaximal, complemented
5#22550196C212yesmaximal, complemented
6#22550191C212yesmaximal, complemented
7#22549919C211yesmaximal, characteristic, complemented
8#22549880C211yesmaximal, characteristic, complemented
9C2×#16861131C211yesmaximal, characteristic, complemented
10#22549842C211yesmaximal, characteristic, complemented
11#427×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861206—21no—
16#16861197—21no—
17#16861184C2212yes—
18#16861184C2212yes—
19#16861179—22no—
20C4×Dic254—21no—
21C4×D508—21no—
22(C22⋊D127)⋊C4—22no—
23#16861167C2212yescomplemented
24C2×C4⋊Dic127C2211yescharacteristic
25C4⋊C4×D127C2212yescomplemented
26#16861142C2212yescomplemented
27#16861140C2212yescomplemented
28#16861131C2212yes—
29#16861131C2212yes—
30#16861129C2212yes—
31#16861129C2212yes—
32(C2×D127)⋊C8C2212yescomplemented
33(C2×D127)⋊C8C2212yescomplemented
34C22×C8×C127C2211yescharacteristic
35#117×C127C2211yescharacteristic
36#117×C127C2211yescharacteristic
37#107×C127C2211yescharacteristic, complemented
38#107×C127C2211yescharacteristic, complemented
39#106×C127C2211yescharacteristic, complemented
40#106×C127C2211yescharacteristic, complemented
41C23×D127—21no—
42#14523248—22no—
43#14523248—22no—
44C2×Dic254D411yescharacteristic
45C2×D508—22no—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—22no—
49C2×D508—22no—
50C2×D508D411yescharacteristic
51C2×D508C2312yes—
52C2×C22⋊D127D412yes—
53C22×Dic127—21no—
54C2×C4×D127—22no—
55C2×C4×D127C2312yes—
56#14523232—22no—
57C2×C127⋊C8C2312yes—
58C4⋊Dic127—21no—
59C4⋊Dic127C2312yes—
60Dic127⋊C4—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—22no—
64C4×Dic127—22no—
65C22×C4×C127C2311yescharacteristic
66C2×C8×C127—22no—
67C2×C8×C127C2311yescharacteristic
68C2×C8×C127C2311yescharacteristic
69C4⋊C4×C127C2311yescharacteristic
70C4⋊C4×C127C2311yescharacteristic
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—21no—
74C22⋊C4×C127—21no—
75C42×C127—21no—
76C42×C127—21no—
77C22×D127—22no—
78C22×D127—42no—
79C22×D127—81no—
80C22×D127—81no—
81C22×D127C2×D412yes—
82Dic254—22no—
83D508—22no—
84D508—22no—
85D508—22no—
86D508—42no—
87D508—82no—
88D508—82no—
89C22⋊D127—24no—
90C2×Dic127—22no—
91C2×Dic127—81no—
92C2×Dic127C2×D412yes—
93C4×D127—22no—
94C4×D127—22no—
95C4×D127—82no—
96C127⋊C8—22no—
97C23×C127C2×D411yescharacteristic
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127#4112yes—
104C2×C4×C127#4112yes—
105C2×C4×C127C2411yescharacteristic
106C8×C127—21no—
107C8×C127—21no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127#12312yes—
111C22×C127C22×D411yescharacteristic
112C2×D127—22no—
113C2×D127—22no—
114C2×D127—22no—
115C2×D127—42no—
116C2×D127—42no—
117C2×D127—42no—
118C2×D127—84no—
119C4×C127—41no—
120C4×C127—41no—
121C4×C127—41no—
122C4×C127—41no—
123C4×C127#12312yes—
124C4×C127C2×#4111yescharacteristic
125C4×C127C2×#4111yescharacteristic
126Dic127—22no—
127Dic127—42no—
128C2×C127—22no—
129C2×C127C2×#12311yescharacteristic
130C2×C127#57611yescharacteristic
131C2×C127#57711yescharacteristic
132D127—42no—
133D127—82no—
134#3096—1271nomaximal, Sylow
135C127#309611yescharacteristic, complemented, Sylow
136#535—1271no—
137#520—1271no—
138#482—1272no—
139#482—1272no—
140#481—1272no—
141#480—1272no—
142#428—1271no—
143#427D12711yescharacteristic, complemented
144#418—1271no—
145C2×#102—1271no—
146#408—1271no—
147C2×#41—1271no—
148C22×D4—1271no—
149C2×M16—1271no—
150C22×C8C2×D12711yescharacteristic
151#127—2541no—
152#124—2541no—
153#121—1272no—
154#121—1272no—
155#120—2542no—
156C4×Q8—2541no—
157C4×D4—2541no—
158C4×D4—2542no—
159#117—1272no—
160#117C2×D12711yescharacteristic
161#117C2×D12711yescharacteristic
162C2×C4⋊C4—1271no—
163C2×C4⋊C4—1272no—
164#107—1272no—
165#107C2×D12711yescharacteristic, complemented
166#107C2×D12711yescharacteristic, complemented
167#106—1272no—
168#106C2×D12711yescharacteristic, complemented
169#106C2×D12711yescharacteristic, complemented
170#102—1272no—
171#102—1272no—
172#102—1272no—
173#102—1272no—
174C22⋊C8—1272no—
175C22⋊C8—1272no—
176C24—2541no—
177#41—2542no—
178#41—2542no—
179C2×Q8—1271no—
180C2×D4—1272no—
181C2×D4—1271no—
182C2×D4—1272no—
183C2×D4—2542no—
184C2×D4—2542no—
185C2×D4—2541no—
186C2×D4—2542no—
187C2×D4—2542no—
188C22×C4—1272no—
189C22×C4—2542no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192M16—2542no—
193C2×C8—22no—
194C2×C8—1272no—
195C2×C8C22×D12711yescharacteristic
196C2×C8C22×D12711yescharacteristic
197C4⋊C4—1272no—
198C4⋊C4—2542no—
199C4⋊C4—2541no—
200C4⋊C4C22×D12711yescharacteristic
201C4⋊C4C22×D12711yescharacteristic
202C4⋊C4C22×D12711yescharacteristic
203C4⋊C4C22×D12711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—2542no—
207C22⋊C4—2542no—
208C22⋊C4—2542no—
209C42—21no—
210C42—21no—
211C42—2542no—
212C23—1272no—
213C23—10161no—
214C23—10161no—
215C23—5082no—
216C23—2542no—
217C23D4×D12711yescharacteristic
218Q8—2542no—
219D4—10162no—
220D4—10162no—
221D4—5082no—
222D4—2544no—
223D4—2542no—
224D4—2542no—
225D4—2542no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—1272no—
231C2×C4—10162no—
232C2×C4—10161no—
233C2×C4—2542no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4D4×D12711yescharacteristic
237C2×C4#1452324812yes—
238C2×C4#1452324912yes—
239C2×C4C23×D12711yescharacteristic
240C8—21no—
241C8—21no—
242C8—2542no—
243C22—21no—
244C22—21no—
245C22—10164no—
246C22—5082no—
247C22—5082no—
248C22—5082no—
249C22—2542no—
250C22—2542no—
251C22—2542no—
252C22#1686119412yes—
253C22C2×D4×D12711yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—5082no—
259C4—2542no—
260C4#1686119412yes—
261C4C2×#1452324811yescharacteristic
262C4C2×#1452324911yescharacteristic
263C2—22no—
264C2—10162no—
265C2—5082no—
266C2C2×#1686119411yescharacteristic
267C2#2255067811yescharacteristic
268C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#427×C127C2
socleC22×C127C22×D4
prime radical#22550445C2
hypercenter#427D127
hypocenterC127#3096
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #22550356C2semidirect
3. #22550203C2semidirect
4. #22550202C2semidirect
5. #22550196C2semidirect
6. #22550191C2semidirect
7. #22549919C2semidirect
8. #22549880C2semidirect
9. C2×#16861131C2semidirect
10. #22549842C2semidirect
11. #427×C127C2semidirect
23. #16861167C22semidirect
25. C4⋊C4×D127C22semidirect
26. #16861142C22semidirect
27. #16861140C22semidirect
32. (C2×D127)⋊C8C22semidirect
33. (C2×D127)⋊C8C22semidirect
37. #107×C127C22semidirect
38. #107×C127C22semidirect
39. #106×C127C22semidirect
40. #106×C127C22semidirect
135. C127#3096semidirect
143. #427D127semidirect
165. #107C2×D127semidirect
166. #107C2×D127semidirect
168. #106C2×D127semidirect
169. #106C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504452
#225503562
#225502032
#225502022
#225501962
#225501912
#225499192
#225498802
C2×#168611312
#225498422
#427×C1272
#3096127