Finite Group Databasebeta

← #30650543

Subgroups — #30650543

Order 16256.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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
2#22550222C211yesmaximal, characteristic, complemented
3#22550187C212yesmaximal, complemented
4#22550156C211yesmaximal, characteristic, complemented
5#22550110C212yesmaximal, complemented
6#22550068C211yesmaximal, characteristic, complemented
7#22550010C212yesmaximal, complemented
8C4×#14523233C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10C2×#16861130C212yesmaximal, complemented
11D42×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C2×#14523233C2211yescharacteristic
14C2×#14523233C2212yescomplemented
15C22×C127⋊C8C2212yescomplemented
16#16861209C2211yescharacteristic, complemented
17#16861193—22no—
18#16861189—21no—
19#16861183C2212yescomplemented
20C4×D508C2211yescharacteristic, complemented
21C2×C4⋊Dic127C2212yescomplemented
22#16861142C2212yescomplemented
23#16861142C2211yescharacteristic, complemented
24C4⋊(C127⋊C8)C2211yescharacteristic, complemented
25#16861130C2211yescharacteristic
26#16861130C2212yes—
27#16861130C2212yescomplemented
28#16861130C2212yescomplemented
29#16861130C2212yescomplemented
30#16861129C2211yescharacteristic, complemented
31C22⋊(C127⋊C8)C2212yescomplemented
32C4×C127⋊C8C2211yescharacteristic, complemented
33C22×D4×C127—21no—
34C22×D4×C127C2212yes—
35#127×C127C2211yescharacteristic
36#121×C127—21no—
37#121×C127C2212yes—
38#120×C127—22no—
39C4×D4×C127C2211yescharacteristic
40C4×D4×C127C2211yescharacteristic
41#14523247—44no—
42C2×Dic254—21no—
43C2×D508C2311yescharacteristic
44C2×C22⋊D127—22no—
45C22×Dic127—22no—
46C2×C4×D127—21no—
47#14523233—22no—
48#14523233—42no—
49C2×C127⋊C8—42no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2312yes—
52C2×C127⋊C8C2311yescharacteristic
53C4⋊Dic127—22no—
54C4⋊Dic127—21no—
55C4⋊Dic127C2312yes—
56C4⋊Dic127C2311yescharacteristic
57(C2×D127)⋊C4—22no—
58(C2×D127)⋊C4—21no—
59C24×C127—22no—
60C2×D4×C127—24no—
61C2×D4×C127—22no—
62C2×D4×C127—22no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—42no—
66C2×D4×C127—44no—
67C2×D4×C127D411yescharacteristic
68C2×D4×C127C2312yes—
69C2×D4×C127C2311yescharacteristic
70C22×C4×C127—21no—
71C22×C4×C127C2312yes—
72C4⋊C4×C127D411yescharacteristic
73C4⋊C4×C127C2311yescharacteristic
74C22⋊C4×C127—21no—
75C22⋊C4×C127—22no—
76C22⋊C4×C127D412yes—
77C42×C127C2311yescharacteristic
78C22×D127—21no—
79Dic254—42no—
80D508—21no—
81D508—21no—
82C22⋊D127—44no—
83C2×Dic127—21no—
84C2×Dic127—22no—
85C2×Dic127—44no—
86C4×D127—42no—
87C127⋊C8—21no—
88C127⋊C8—21no—
89C127⋊C8—22no—
90C23×C127—21no—
91C23×C127—22no—
92C23×C127—44no—
93C23×C127—44no—
94C23×C127—82no—
95C23×C127C2×D412yes—
96D4×C127—22no—
97D4×C127—21no—
98D4×C127—22no—
99D4×C127—44no—
100D4×C127—44no—
101D4×C127—42no—
102D4×C127—81no—
103D4×C127D814yescomplemented
104D4×C127#4112yescomplemented
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D412yes—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C22×C127—22no—
114C22×C127—22no—
115C22×C127—22no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127—88no—
122C22×C127C2×D814yes—
123C22×C127C22×D411yescharacteristic
124C2×D127—41no—
125C2×D127—41no—
126C2×D127—41no—
127C4×C127—22no—
128C4×C127—21no—
129C4×C127—41no—
130C4×C127C2×D812yes—
131C4×C127C2×#4111yescharacteristic
132C4×C127H(5,𝔽2)11yescharacteristic
133Dic127—41no—
134Dic127—42no—
135C2×C127—24no—
136C2×C127—42no—
137C2×C127—82no—
138C2×C127#54611yescharacteristic
139C2×C127C22×D811yescharacteristic
140C2×C127#57711yescharacteristic
141D127—81no—
142#3294—1271nomaximal, Sylow
143C127#329411yescharacteristic, complemented, Sylow
144#547—1271no—
145D42D12711yescharacteristic, complemented
146#486—1271no—
147#480—1272no—
148#474—1271no—
149#466—1272no—
150#459—1271no—
151#447—1272no—
152C4×D8—1271no—
153C8×D4—1271no—
154C2×#102—1272no—
155C2×#41—2541no—
156C22×D4—21no—
157C22×D4C2×D12712yes—
158C2×D8—1272no—
159C2×D8—1271no—
160C22×C8—1272no—
161C4⋊Q8—1271no—
162#127C2×D12711yescharacteristic
163#123—2542no—
164C22⋊Q8—2541no—
165#121—21no—
166#121—1272no—
167#121C2×D12712yes—
168#120—22no—
169C4×D4—1271no—
170C4×D4C2×D12711yescharacteristic
171C4×D4C2×D12711yescharacteristic
172C2×C4⋊C4—1272no—
173#107—1271no—
174#107—1272no—
175C4⋊C8—1271no—
176#102—1271no—
177#102—1272no—
178#102—1272no—
179#102—1272no—
180#102—1272no—
181#102—1271no—
182C22⋊C8—1272no—
183C4×C8—1271no—
184C24—22no—
185#41—5084no—
186C2×Q8—2541no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—1271no—
195C2×D4—2542no—
196C2×D4C22⋊D12711yescharacteristic
197C2×D4C22×D12712yes—
198C2×D4C22×D12711yescharacteristic
199C22×C4—21no—
200C22×C4—2542no—
201C22×C4—2541no—
202C22×C4C22×D12712yes—
203D8—5082no—
204D8—2542no—
205C2×C8—1271no—
206C2×C8—1271no—
207C2×C8—1272no—
208C2×C8—5082no—
209C4⋊C4—1271no—
210C4⋊C4—1272no—
211C4⋊C4—2542no—
212C4⋊C4—2541no—
213C4⋊C4C22⋊D12711yescharacteristic
214C4⋊C4C22×D12711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2541no—
218C22⋊C4—2542no—
219C22⋊C4C22⋊D12712yes—
220C42C22×D12711yescharacteristic
221C23—21no—
222C23—22no—
223C23—44no—
224C23—44no—
225C23—82no—
226C23—2541no—
227C23C2×C22⋊D12712yes—
228Q8—5082no—
229D4—22no—
230D4—21no—
231D4—22no—
232D4—44no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—5084no—
237D4—2541no—
238D4—2541no—
239D4#1452323314yescomplemented
240D4#1452324712yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—5084no—
246C2×C4—5082no—
247C2×C4—2541no—
248C2×C4—2542no—
249C2×C4C2×C22⋊D12711yescharacteristic
250C2×C4C2×C22⋊D12712yes—
251C2×C4C2×C22⋊D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—2541no—
254C8—2541no—
255C8—2542no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—88no—
265C22—5081no—
266C22—5081no—
267C22—5081no—
268C22C2×#1452323314yes—
269C22C22×C22⋊D12711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—5081no—
274C4—5082no—
275C4C2×#1452323312yes—
276C4C2×#1452324711yescharacteristic
277C4#1686125411yescharacteristic
278C2—24no—
279C2—42no—
280C2—82no—
281C2—10161no—
282C2#2255052011yescharacteristic
283C2C22×#1452323311yescharacteristic
284C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD42×C127C2
socleC22×C127C22×D4
hypercenterD42D127
hypocenterC127#3294
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #22550530C2semidirect
2. #22550222C2semidirect
3. #22550187C2semidirect
4. #22550156C2semidirect
5. #22550110C2semidirect
6. #22550068C2semidirect
7. #22550010C2semidirect
8. C4×#14523233C2semidirect
9. D4×C127⋊C8C2semidirect
10. C2×#16861130C2semidirect
11. D42×C127C2semidirect
14. C2×#14523233C22semidirect
15. C22×C127⋊C8C22semidirect
16. #16861209C22semidirect
19. #16861183C22semidirect
20. C4×D508C22semidirect
21. C2×C4⋊Dic127C22semidirect
22. #16861142C22semidirect
23. #16861142C22semidirect
24. C4⋊(C127⋊C8)C22semidirect
27. #16861130C22semidirect
28. #16861130C22semidirect
29. #16861130C22semidirect
30. #16861129C22semidirect
31. C22⋊(C127⋊C8)C22semidirect
32. C4×C127⋊C8C22semidirect
103. D4×C127D8semidirect
104. D4×C127#41semidirect
143. C127#3294semidirect
145. D42D127semidirect
239. D4#14523233semidirect
240. D4#14523247semidirect

Maximal subgroups

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

subgroupindex
#225505302
#225502222
#225501872
#225501562
#225501102
#225500682
#225500102
C4×#145232332
D4×C127⋊C82
C2×#168611302
D42×C1272
#3294127