Finite Group Databasebeta

← #30650540

Subgroups — #30650540

Order 16256.

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

normal subgroups only

This group has 494 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#22550529C211yesmaximal, characteristic, complemented
2D4×D508C211yesmaximal, characteristic, complemented
3#22550224C211yesmaximal, characteristic, complemented
4#22550191C212yesmaximal, complemented
5#22550113C212yesmaximal, complemented
6#22550072C211yesmaximal, characteristic, complemented
7#22550011C212yesmaximal, complemented
8C4×#14523233C211yesmaximal, characteristic, complemented
9#22549941C211yesmaximal, characteristic, complemented
10#102×D127C212yesmaximal, complemented
11#474×C127C211yesmaximal, characteristic, complemented
12C2×#14523249—21no—
13C22×D508—21no—
14C2×D4×D127C2212yes—
15C2×D1016C2212yescomplemented
16C2×#14523233C2211yescharacteristic
17C2×C8×D127C2212yescomplemented
18#16861206C2211yescharacteristic
19#16861194—22no—
20#16861185—21no—
21#16861184C2212yes—
22#16861184C2212yescomplemented
23#16861183—21no—
24#16861179—22no—
25C4×D508C2211yescharacteristic
26C4×D508C2211yescharacteristic, complemented
27C4⋊C4×D127C2212yescomplemented
28#16861142C2211yescharacteristic, complemented
29#16861131C2212yescomplemented
30#16861130C2211yescharacteristic
31#16861129C2211yescharacteristic, complemented
32#16861129C2212yescomplemented
33#16861129C2212yescomplemented
34(C2×D127)⋊C8C2212yescomplemented
35C4×C127⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C127C2211yescharacteristic, complemented
37C4×D4×C127C2211yescharacteristic
38#107×C127C2212yescomplemented
39C4⋊C8×C127C2211yescharacteristic, complemented
40#102×C127C2212yes—
41C23×D127—22no—
42#14523249—44no—
43C2×D508—22no—
44C2×D508—22no—
45C2×D508—21no—
46C2×D508—21no—
47C2×D508—44no—
48C2×D508D411yescharacteristic
49C2×D508C2312yes—
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—22no—
52D4×D127—24no—
53D4×D127—42no—
54C2×C4×D127—22no—
55C2×C4×D127—21no—
56C2×C4×D127C2312yes—
57D1016—42no—
58#14523233—22no—
59C2×C127⋊C8C2311yescharacteristic
60C2×C127⋊C8C2311yescharacteristic
61C8×D127—42no—
62C4⋊Dic127D411yescharacteristic
63C4⋊Dic127C2311yescharacteristic
64Dic127⋊C4—22no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4D412yes—
69C2×Q8×C127—21no—
70C2×D4×C127C2311yescharacteristic
71C22×C4×C127—21no—
72C2×C8×C127C2312yes—
73C4⋊C4×C127—21no—
74C4⋊C4×C127C2311yescharacteristic
75C4⋊C4×C127C2312yes—
76C22⋊C4×C127—21no—
77C42×C127C2311yescharacteristic
78C22×D127—21no—
79C22×D127—22no—
80C22×D127—44no—
81C22×D127—44no—
82C22×D127—82no—
83C22×D127C2×D412yes—
84D508—21no—
85D508—22no—
86D508—21no—
87D508—22no—
88D508—44no—
89D508—44no—
90D508—42no—
91D508—81no—
92D508D814yescomplemented
93C22⋊D127—44no—
94C2×Dic127—21no—
95C2×Dic127C2×D412yes—
96C4×D127—22no—
97C4×D127—22no—
98C4×D127—44no—
99C4×D127—42no—
100C127⋊C8—21no—
101C127⋊C8—21no—
102C23×C127—21no—
103Q8×C127—42no—
104D4×C127—21no—
105D4×C127#4112yescomplemented
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—22no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127C22×D411yescharacteristic
117C2×D127—22no—
118C2×D127—22no—
119C2×D127—42no—
120C2×D127—41no—
121C2×D127—41no—
122C2×D127—42no—
123C2×D127—42no—
124C2×D127—41no—
125C2×D127—88no—
126C2×D127C2×D814yes—
127C4×C127—21no—
128C4×C127—41no—
129C4×C127—42no—
130C4×C127C2×D812yes—
131C4×C127C2×#4111yescharacteristic
132C4×C127H(5,𝔽2)11yescharacteristic
133Dic127—22no—
134Dic127—41no—
135C2×C127—42no—
136C2×C127#54611yescharacteristic
137C2×C127C22×D811yescharacteristic
138C2×C127#57711yescharacteristic
139D127—24no—
140D127—81no—
141D127—82no—
142#3294—1271nomaximal, Sylow
143C127#329411yescharacteristic, complemented, Sylow
144#547—1271no—
145D42—1271no—
146#486—1271no—
147#480—1272no—
148#474D12711yescharacteristic, complemented
149#466—1272no—
150#459—1271no—
151#447—1272no—
152C4×D8—1271no—
153C8×D4—1271no—
154C2×#102—1272no—
155C2×#41—2541no—
156C22×D4—1272no—
157C22×D4—2541no—
158C2×D8—1271no—
159C2×D8—1272no—
160C22×C8—1272no—
161C4⋊Q8C2×D12711yescharacteristic, complemented
162#127—1271no—
163#123—2542no—
164C22⋊Q8—2541no—
165#121—1272no—
166#121—1272no—
167#121—2541no—
168#120—2542no—
169C4×D4—1271no—
170C4×D4—1271no—
171C4×D4C2×D12711yescharacteristic
172C2×C4⋊C4—1272no—
173#107—1271no—
174#107C2×D12712yescomplemented
175C4⋊C8C2×D12711yescharacteristic, complemented
176#102—1271no—
177#102—1272no—
178#102—1272no—
179#102—1271no—
180#102—1272no—
181#102C2×D12712yes—
182C22⋊C8—1272no—
183C4×C8—1271no—
184C24—2542no—
185#41—5084no—
186C2×Q8—21no—
187C2×D4—1271no—
188C2×D4—1272no—
189C2×D4—1271no—
190C2×D4—5084no—
191C2×D4—5082no—
192C2×D4—2542no—
193C2×D4—2542no—
194C2×D4—2542no—
195C2×D4—2544no—
196C2×D4—2541no—
197C2×D4—2541no—
198C2×D4C22×D12711yescharacteristic
199C22×C4—21no—
200C22×C4—1272no—
201C22×C4—2542no—
202C22×C4—2541no—
203D8—5082no—
204D8—2542no—
205C2×C8—1271no—
206C2×C8—1271no—
207C2×C8—5082no—
208C2×C8C22×D12712yes—
209C4⋊C4—21no—
210C4⋊C4—1271no—
211C4⋊C4—1271no—
212C4⋊C4—2542no—
213C4⋊C4C22×D12712yes—
214C4⋊C4C22×D12711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1272no—
217C22⋊C4—2542no—
218C22⋊C4—2542no—
219C22⋊C4—2541no—
220C42C22×D12711yescharacteristic
221C23—21no—
222C23—1272no—
223C23—10162no—
224C23—5084no—
225C23—5084no—
226C23—2541no—
227C23—2542no—
228Q8—42no—
229D4—21no—
230D4—1274no—
231D4—10161no—
232D4—5084no—
233D4—5084no—
234D4—5084no—
235D4—5082no—
236D4—2541no—
237D4—2542no—
238D4—2541no—
239D4—2542no—
240D4#1452324912yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1272no—
245C2×C4—5084no—
246C2×C4—5082no—
247C2×C4—2541no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—22no—
254C8—2541no—
255C8—2541no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—1274no—
260C22—10168no—
261C22—5081no—
262C22—5082no—
263C22—5082no—
264C22—5081no—
265C22—5082no—
266C22—5081no—
267C22—2542no—
268C22—2542no—
269C22C2×D4×D12711yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—5081no—
274C4—2542no—
275C4D8×D12712yes—
276C4C2×#1452324911yescharacteristic
277C4#1686125511yescharacteristic
278C2—42no—
279C2—10162no—
280C2—10161no—
281C2—2544no—
282C2#2255051711yescharacteristic
283C2C2×D8×D12711yescharacteristic
284C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#474×C127C2
socleC22×C127C22×D4
hypercenter#474D127
hypocenterC127#3294
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #22550529C2semidirect
2. D4×D508C2semidirect
3. #22550224C2semidirect
4. #22550191C2semidirect
5. #22550113C2semidirect
6. #22550072C2semidirect
7. #22550011C2semidirect
8. C4×#14523233C2semidirect
9. #22549941C2semidirect
10. #102×D127C2semidirect
11. #474×C127C2semidirect
15. C2×D1016C22semidirect
17. C2×C8×D127C22semidirect
22. #16861184C22semidirect
26. C4×D508C22semidirect
27. C4⋊C4×D127C22semidirect
28. #16861142C22semidirect
29. #16861131C22semidirect
31. #16861129C22semidirect
32. #16861129C22semidirect
33. #16861129C22semidirect
34. (C2×D127)⋊C8C22semidirect
35. C4×C127⋊C8C22semidirect
36. C4⋊Q8×C127C22semidirect
38. #107×C127C22semidirect
39. C4⋊C8×C127C22semidirect
92. D508D8semidirect
105. D4×C127#41semidirect
143. C127#3294semidirect
148. #474D127semidirect
161. C4⋊Q8C2×D127semidirect
174. #107C2×D127semidirect
175. C4⋊C8C2×D127semidirect
240. D4#14523249semidirect

Maximal subgroups

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

subgroupindex
#225505292
D4×D5082
#225502242
#225501912
#225501132
#225500722
#225500112
C4×#145232332
#225499412
#102×D1272
#474×C1272
#3294127