Finite Group Databasebeta

← #30650545

Subgroups — #30650545

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#22550531C211yesmaximal, characteristic, complemented
2D4×D508C211yesmaximal, characteristic, complemented
3#22550223C211yesmaximal, characteristic, complemented
4#22550191C212yesmaximal, complemented
5#22550156C211yesmaximal, characteristic, complemented
6#22550111C212yesmaximal, complemented
7#22550071C211yesmaximal, characteristic, complemented
8#22550008C212yesmaximal, complemented
9C8×D508C211yesmaximal, characteristic, complemented
10#102×D127C212yesmaximal, complemented
11C4×D8×C127C211yesmaximal, characteristic, complemented
12C2×#14523248—21no—
13C22×D508—21no—
14C2×D4×D127C2212yes—
15C2×#14523233C2212yescomplemented
16C2×C8×D127C2212yescomplemented
17#16861209C2211yescharacteristic, complemented
18#16861206C2211yescharacteristic
19#16861191—22no—
20#16861188—21no—
21#16861184C2212yes—
22#16861183—21no—
23#16861180C2212yescomplemented
24#16861179—22no—
25C4×D508C2211yescharacteristic
26C4⋊C4×D127C2212yescomplemented
27#16861142C2212yescomplemented
28C4⋊(C127⋊C8)C2211yescharacteristic, complemented
29#16861131C2212yescomplemented
30#16861131C2212yescomplemented
31#16861130C2212yes—
32#16861129C2212yescomplemented
33(C2×D127)⋊C8C2212yescomplemented
34C2×D8×C127C2211yescharacteristic
35C4×D4×C127C2211yescharacteristic
36C4×D4×C127C2211yescharacteristic, complemented
37#107×C127C2211yescharacteristic, complemented
38#102×C127C2211yescharacteristic
39#102×C127C2211yescharacteristic, complemented
40C4×C8×C127C2211yescharacteristic, complemented
41C23×D127—22no—
42#14523248—44no—
43C2×Dic254—21no—
44C2×D508—22no—
45C2×D508—21no—
46C2×D508—21no—
47C2×D508—44no—
48C2×D508D411yescharacteristic
49C2×D508C2312yes—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—22no—
52D4×D127—24no—
53D4×D127—42no—
54C2×C4×D127—22no—
55C2×C4×D127C2312yes—
56#14523233—42no—
57C2×C127⋊C8C2312yes—
58C8×D127—42no—
59C4⋊Dic127—21no—
60C4⋊Dic127D411yescharacteristic
61C4⋊Dic127C2312yes—
62Dic127⋊C4—22no—
63C22⋊Dic127—22no—
64(C2×D127)⋊C4—22no—
65(C2×D127)⋊C4D412yes—
66C2×D4×C127C2311yescharacteristic
67C2×D4×C127C2311yescharacteristic
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70D8×C127—22no—
71C2×C8×C127C2311yescharacteristic
72C2×C8×C127C2311yescharacteristic
73C4⋊C4×C127C2311yescharacteristic
74C4⋊C4×C127C2311yescharacteristic
75C22⋊C4×C127—21no—
76C22⋊C4×C127—21no—
77C42×C127C2311yescharacteristic
78C22×D127—22no—
79C22×D127—44no—
80C22×D127—44no—
81C22×D127—82no—
82C22×D127C2×D412yes—
83Dic254—42no—
84D508—22no—
85D508—22no—
86D508—44no—
87D508—42no—
88D508—81no—
89D508D814yescomplemented
90C22⋊D127—44no—
91C22⋊D127—44no—
92C2×Dic127—22no—
93C2×Dic127C2×D412yes—
94C4×D127—22no—
95C4×D127—22no—
96C4×D127—44no—
97C127⋊C8—22no—
98C23×C127—21no—
99C23×C127—21no—
100D4×C127—21no—
101D4×C127—21no—
102D4×C127—21no—
103D4×C127#4112yescomplemented
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2411yescharacteristic
111C8×C127—21no—
112C8×C127—21no—
113C22×C127—22no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—41no—
119C22×C127C22×D411yescharacteristic
120C2×D127—22no—
121C2×D127—22no—
122C2×D127—42no—
123C2×D127—42no—
124C2×D127—42no—
125C2×D127—88no—
126C2×D127C2×D814yes—
127C4×C127—21no—
128C4×C127—41no—
129C4×C127—41no—
130C4×C127C2×D812yes—
131C4×C127C2×#4111yescharacteristic
132C4×C127H(5,𝔽2)11yescharacteristic
133Dic127—22no—
134Dic127—42no—
135C2×C127—42no—
136C2×C127—81no—
137C2×C127#54611yescharacteristic
138C2×C127C22×D811yescharacteristic
139C2×C127#57711yescharacteristic
140D127—24no—
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×D8D12711yescharacteristic, complemented
153C8×D4—1271no—
154C2×#102—1272no—
155C2×#41—2541no—
156C22×D4—1272no—
157C22×D4—2541no—
158C2×D8—1272no—
159C2×D8C2×D12711yescharacteristic
160C22×C8—1272no—
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×D4C2×D12711yescharacteristic
171C4×D4C2×D12711yescharacteristic, complemented
172C2×C4⋊C4—1272no—
173#107—1272no—
174#107C2×D12711yescharacteristic, complemented
175C4⋊C8—1271no—
176#102—1272no—
177#102—1272no—
178#102—1272no—
179#102—1272no—
180#102C2×D12711yescharacteristic
181#102C2×D12711yescharacteristic, complemented
182C22⋊C8—1272no—
183C4×C8C2×D12711yescharacteristic, complemented
184C24—2542no—
185#41—5084no—
186C2×Q8—2541no—
187C2×D4—1271no—
188C2×D4—1272no—
189C2×D4—5084no—
190C2×D4—5082no—
191C2×D4—2542no—
192C2×D4—2542no—
193C2×D4—2542no—
194C2×D4—2544no—
195C2×D4—2541no—
196C2×D4—2541no—
197C2×D4C22×D12711yescharacteristic
198C2×D4C22×D12711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1272no—
202C22×C4—2542no—
203D8—22no—
204D8—5082no—
205C2×C8—1272no—
206C2×C8—5082no—
207C2×C8C22×D12711yescharacteristic
208C2×C8C22×D12711yescharacteristic
209C4⋊C4—1271no—
210C4⋊C4—1272no—
211C4⋊C4—2542no—
212C4⋊C4—2541no—
213C4⋊C4C22×D12711yescharacteristic
214C4⋊C4C22×D12711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—1272no—
218C22⋊C4—2542no—
219C22⋊C4—2542no—
220C42C22×D12711yescharacteristic
221C23—21no—
222C23—21no—
223C23—1272no—
224C23—10162no—
225C23—5084no—
226C23—5084no—
227C23—2542no—
228Q8—5082no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—1274no—
233D4—10161no—
234D4—5084no—
235D4—5084no—
236D4—5084no—
237D4—5082no—
238D4—2542no—
239D4—2542no—
240D4#1452324812yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—1272no—
246C2×C4—5084no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—21no—
254C8—21no—
255C8—2542no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—1274no—
263C22—10168no—
264C22—5082no—
265C22—5082no—
266C22—5082no—
267C22—2542no—
268C22—2542no—
269C22C2×D4×D12711yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—5082no—
274C4—2542no—
275C4D8×D12712yes—
276C4C2×#1452324811yescharacteristic
277C4#1686125511yescharacteristic
278C2—42no—
279C2—81no—
280C2—10162no—
281C2—2544no—
282C2#2255052211yescharacteristic
283C2C2×D8×D12711yescharacteristic
284C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×D8×C127C2
socleC22×C127C22×D4
hypercenterC4×D8D127
hypocenterC127#3294
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22550531C2semidirect
2. D4×D508C2semidirect
3. #22550223C2semidirect
4. #22550191C2semidirect
5. #22550156C2semidirect
6. #22550111C2semidirect
7. #22550071C2semidirect
8. #22550008C2semidirect
9. C8×D508C2semidirect
10. #102×D127C2semidirect
11. C4×D8×C127C2semidirect
15. C2×#14523233C22semidirect
16. C2×C8×D127C22semidirect
17. #16861209C22semidirect
23. #16861180C22semidirect
26. C4⋊C4×D127C22semidirect
27. #16861142C22semidirect
28. C4⋊(C127⋊C8)C22semidirect
29. #16861131C22semidirect
30. #16861131C22semidirect
32. #16861129C22semidirect
33. (C2×D127)⋊C8C22semidirect
36. C4×D4×C127C22semidirect
37. #107×C127C22semidirect
39. #102×C127C22semidirect
40. C4×C8×C127C22semidirect
89. D508D8semidirect
103. D4×C127#41semidirect
143. C127#3294semidirect
152. C4×D8D127semidirect
171. C4×D4C2×D127semidirect
174. #107C2×D127semidirect
181. #102C2×D127semidirect
183. C4×C8C2×D127semidirect
240. D4#14523248semidirect

Maximal subgroups

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

subgroupindex
#225505312
D4×D5082
#225502232
#225501912
#225501562
#225501112
#225500712
#225500082
C8×D5082
#102×D1272
C4×D8×C1272
#3294127