Finite Group Databasebeta

← #30650544

Subgroups — #30650544

Order 16256.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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#22550530C211yesmaximal, characteristic, complemented
2D4×D508C211yesmaximal, characteristic, complemented
3#22550223C211yesmaximal, characteristic, complemented
4#22550190C212yesmaximal, complemented
5#22550157C211yesmaximal, characteristic, complemented
6#22550112C212yesmaximal, complemented
7#22550072C211yesmaximal, characteristic, complemented
8#22550012C212yesmaximal, complemented
9C4×D1016C211yesmaximal, characteristic, complemented
10C2×#16861131C212yesmaximal, complemented
11C8×D4×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C22×D508C2212yes—
14C2×D4×D127—21no—
15C2×D1016C2211yescharacteristic
16C2×D1016C2212yescomplemented
17#16861209C2211yescharacteristic, complemented
18#16861206C2211yescharacteristic
19#16861193—22no—
20#16861189—21no—
21#16861184—21no—
22#16861183C2212yes—
23#16861183C2212yescomplemented
24#16861179—22no—
25C4×D508C2211yescharacteristic
26C4×D508C2211yescharacteristic, complemented
27C2×C4⋊Dic127C2212yescomplemented
28#16861143C2212yescomplemented
29#16861143C2211yescharacteristic, complemented
30#16861131C2211yescharacteristic
31#16861131C2212yes—
32#16861131C2212yescomplemented
33#16861131C2211yescharacteristic, complemented
34#16861131C2212yescomplemented
35#16861131C2212yescomplemented
36C22×C8×C127C2212yescomplemented
37C4×D4×C127C2211yescharacteristic
38C4⋊C8×C127C2211yescharacteristic, complemented
39C22⋊C8×C127C2212yescomplemented
40C4×C8×C127C2211yescharacteristic, complemented
41C23×D127—22no—
42#14523247—44no—
43C2×Dic254—21no—
44C2×D508—24no—
45C2×D508—21no—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—42no—
49C2×D508C2312yes—
50C2×D508C2311yescharacteristic
51C2×D508C2311yescharacteristic
52C2×C22⋊D127—22no—
53C2×C22⋊D127—22no—
54D4×D127—44no—
55C22×Dic127—22no—
56C2×C4×D127—21no—
57C2×C4×D127—21no—
58D1016—22no—
59D1016—42no—
60C4⋊Dic127—22no—
61C4⋊Dic127—21no—
62C4⋊Dic127C2312yes—
63C4⋊Dic127C2311yescharacteristic
64C4⋊Dic127C2311yescharacteristic
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—21no—
69C2×D4×C127D411yescharacteristic
70C22×C4×C127C2312yes—
71C2×C8×C127—42no—
72C2×C8×C127C2311yescharacteristic
73C2×C8×C127C2312yes—
74C2×C8×C127C2311yescharacteristic
75C4⋊C4×C127D411yescharacteristic
76C22⋊C4×C127D412yes—
77C42×C127C2311yescharacteristic
78C22×D127—21no—
79C22×D127—21no—
80C22×D127—22no—
81C22×D127—44no—
82C22×D127—44no—
83C22×D127—82no—
84Dic254—42no—
85D508—21no—
86D508—22no—
87D508—21no—
88D508—21no—
89D508—22no—
90D508—44no—
91D508—42no—
92D508—81no—
93D508#4112yescomplemented
94C22⋊D127—44no—
95C22⋊D127—44no—
96C2×Dic127—21no—
97C2×Dic127—21no—
98C2×Dic127—22no—
99C2×Dic127—44no—
100C4×D127—42no—
101C4×D127—42no—
102C23×C127C2×D412yes—
103D4×C127D814yescomplemented
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2411yescharacteristic
110C8×C127—21no—
111C8×C127—21no—
112C8×C127—22no—
113C22×C127—22no—
114C22×C127—22no—
115C22×C127C2×D814yes—
116C22×C127C22×D411yescharacteristic
117C2×D127—22no—
118C2×D127—41no—
119C2×D127—41no—
120C2×D127—42no—
121C2×D127—41no—
122C2×D127—41no—
123C2×D127—42no—
124C2×D127—42no—
125C2×D127—41no—
126C2×D127—88no—
127C4×C127—22no—
128C4×C127—21no—
129C4×C127C2×D812yes—
130C4×C127C2×#4111yescharacteristic
131C4×C127H(5,𝔽2)11yescharacteristic
132Dic127—41no—
133Dic127—41no—
134Dic127—42no—
135C2×C127—24no—
136C2×C127#54611yescharacteristic
137C2×C127C22×D811yescharacteristic
138C2×C127#57711yescharacteristic
139D127—42no—
140D127—81no—
141D127—82no—
142#3294—1271nomaximal, Sylow
143C127#329411yescharacteristic, complemented, Sylow
144#547—1271no—
145D42—1271no—
146#486—1271no—
147#480—1272no—
148#474—1271no—
149#466—1272no—
150#459—1271no—
151#447—1272no—
152C4×D8—1271no—
153C8×D4D12711yescharacteristic, complemented
154C2×#102—1272no—
155C2×#41—2541no—
156C22×D4—1272no—
157C22×D4—2541no—
158C2×D8—1272no—
159C2×D8—1271no—
160C22×C8C2×D12712yescomplemented
161C4⋊Q8—1271no—
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#107—1272no—
175C4⋊C8C2×D12711yescharacteristic, complemented
176#102—1271no—
177#102—1272no—
178#102—1272no—
179#102—1272no—
180#102—1271no—
181#102—1272no—
182C22⋊C8C2×D12712yescomplemented
183C4×C8C2×D12711yescharacteristic, complemented
184C24—2542no—
185#41—5084no—
186C2×Q8—2541no—
187C2×D4—1271no—
188C2×D4—1272no—
189C2×D4—1271no—
190C2×D4—5082no—
191C2×D4—5084no—
192C2×D4—2544no—
193C2×D4—2542no—
194C2×D4—2542no—
195C2×D4—2541no—
196C2×D4—2542no—
197C2×D4—2541no—
198C2×D4D50811yescharacteristic
199C22×C4—2542no—
200C22×C4—2541no—
201C22×C4—2541no—
202C22×C4C22×D12712yes—
203D8—5082no—
204D8—2542no—
205C2×C8—42no—
206C2×C8C22×D12711yescharacteristic
207C2×C8C22×D12711yescharacteristic
208C2×C8C22×D12712yes—
209C4⋊C4—1271no—
210C4⋊C4—1272no—
211C4⋊C4—1271no—
212C4⋊C4—2542no—
213C4⋊C4—2541no—
214C4⋊C4D50811yescharacteristic
215C22⋊C4—2541no—
216C22⋊C4—2541no—
217C22⋊C4—2542no—
218C22⋊C4—2542no—
219C22⋊C4D50812yes—
220C42C22×D12711yescharacteristic
221C23—10162no—
222C23—5084no—
223C23—5084no—
224C23—2541no—
225C23—2542no—
226C23—2541no—
227C23C2×D50812yes—
228Q8—5082no—
229D4—1272no—
230D4—10161no—
231D4—5084no—
232D4—5084no—
233D4—5084no—
234D4—5082no—
235D4—2541no—
236D4—2542no—
237D4—2541no—
238D4—2542no—
239D4—2541no—
240D4D101614yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—5084no—
244C2×C4—5082no—
245C2×C4—5082no—
246C2×C4—2541no—
247C2×C4—2541no—
248C2×C4—2542no—
249C2×C4C2×D50811yescharacteristic
250C2×C4C2×D50812yes—
251C2×C4C2×D50811yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—10168no—
259C22—5081no—
260C22—5082no—
261C22—5082no—
262C22—5081no—
263C22—5082no—
264C22—5081no—
265C22—5081no—
266C22—5081no—
267C22—2542no—
268C22C2×D101614yes—
269C22C22×D50811yescharacteristic
270C4—21no—
271C4—22no—
272C4—5081no—
273C4—5081no—
274C4—5082no—
275C4C2×D101612yes—
276C4#41×D12711yescharacteristic
277C4#1686125411yescharacteristic
278C2—24no—
279C2—10162no—
280C2—10161no—
281C2—5082no—
282C2#2255052111yescharacteristic
283C2C22×D101611yescharacteristic
284C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC8×D4×C127C2
socleC22×C127C22×D4
hypercenterC8×D4D127
hypocenterC127#3294
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22550530C2semidirect
2. D4×D508C2semidirect
3. #22550223C2semidirect
4. #22550190C2semidirect
5. #22550157C2semidirect
6. #22550112C2semidirect
7. #22550072C2semidirect
8. #22550012C2semidirect
9. C4×D1016C2semidirect
10. C2×#16861131C2semidirect
11. C8×D4×C127C2semidirect
16. C2×D1016C22semidirect
17. #16861209C22semidirect
23. #16861183C22semidirect
26. C4×D508C22semidirect
27. C2×C4⋊Dic127C22semidirect
28. #16861143C22semidirect
29. #16861143C22semidirect
32. #16861131C22semidirect
33. #16861131C22semidirect
34. #16861131C22semidirect
35. #16861131C22semidirect
36. C22×C8×C127C22semidirect
38. C4⋊C8×C127C22semidirect
39. C22⋊C8×C127C22semidirect
40. C4×C8×C127C22semidirect
93. D508#41semidirect
103. D4×C127D8semidirect
143. C127#3294semidirect
153. C8×D4D127semidirect
160. C22×C8C2×D127semidirect
175. C4⋊C8C2×D127semidirect
182. C22⋊C8C2×D127semidirect
183. C4×C8C2×D127semidirect
240. D4D1016semidirect

Maximal subgroups

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

subgroupindex
#225505302
D4×D5082
#225502232
#225501902
#225501572
#225501122
#225500722
#225500122
C4×D10162
C2×#168611312
C8×D4×C1272
#3294127