Finite Group Databasebeta

← #30650414

Subgroups — #30650414

Order 16256.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#16861232C211yesmaximal, characteristic, complemented
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550519C212yesmaximal, complemented
4#22550114C211yesmaximal, characteristic, complemented
5#22550109C211yesmaximal, characteristic, complemented
6#22550088C212yesmaximal, complemented
7#22550025C212yesmaximal, complemented
8#22550020C212yesmaximal, complemented
9C4×D1016C211yesmaximal, characteristic, complemented
10C8×D508C211yesmaximal, characteristic, complemented
11#495×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yescomplemented
13C2×D4×D127C2212yes—
14#16861232—24no—
15C2×#14523235C2212yescomplemented
16C2×D1016C2211yescharacteristic
17C2×#14523233C2212yes—
18D8×D127—24no—
19C2×C8×D127C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic, complemented
21#16861191—22no—
22#16861185C2212yes—
23#16861182—22no—
24#16861180C2212yescomplemented
25#16861179—22no—
26C4×D508C2211yescharacteristic
27C4×D508C2212yescomplemented
28C22⋊C4×D127—22no—
29#16861143C2211yescharacteristic, complemented
30C4⋊(C127⋊C8)C2211yescharacteristic, complemented
31#16861133C2212yescomplemented
32#16861131C2212yescomplemented
33#16861130C2212yescomplemented
34(C2×D127)⋊C8C2211yescharacteristic
35(C2×D127)⋊C8C2211yescharacteristic, complemented
36C2×Q16×C127C2211yescharacteristic, complemented
37C2×SD16×C127C2212yescomplemented
38C2×D8×C127C2211yescharacteristic
39#124×C127C2212yescomplemented
40C4×C8×C127C2211yescharacteristic, complemented
41C23×D127—22no—
42#14523249—24no—
43#14523249—42no—
44C2×D508—22no—
45C2×D508D411yescharacteristic
46C2×D508C2312yes—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—22no—
49D4×D127—24no—
50D4×D127—42no—
51C2×C4×D127—22no—
52C2×C4×D127—22no—
53C2×C4×D127C2311yescharacteristic
54C2×C4×D127C2311yescharacteristic
55#14523235—24no—
56D1016D414yescomplemented
57#14523233—24no—
58C2×C127⋊C8C2311yescharacteristic
59C2×C127⋊C8C2311yescharacteristic
60C8×D127—22no—
61C8×D127—22no—
62C4⋊Dic127D411yescharacteristic
63C4⋊Dic127C2312yes—
64Dic127⋊C4—22no—
65C22⋊Dic127—22no—
66(C2×D127)⋊C4—22no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4D411yescharacteristic
69(C2×D127)⋊C4D411yescharacteristic
70C2×Q8×C127C2312yes—
71C2×D4×C127C2312yes—
72Q16×C127—22no—
73SD16×C127—42no—
74D8×C127—22no—
75C2×C8×C127C2311yescharacteristic
76C2×C8×C127C2311yescharacteristic
77C22⋊C4×C127—22no—
78C22⋊C4×C127—22no—
79C42×C127C2311yescharacteristic
80C22×D127—22no—
81C22×D127—44no—
82C22×D127—44no—
83C22×D127—82no—
84C22×D127C2×D411yescharacteristic
85C22×D127C2×D411yescharacteristic
86D508—22no—
87D508—22no—
88D508—44no—
89D508C2×D414yes—
90C22⋊D127—44no—
91C22⋊D127—82no—
92C2×Dic127—22no—
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C4×D127—21no—
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—21no—
99C4×D127—44no—
100C4×D127—82no—
101C127⋊C8—21no—
102C127⋊C8—21no—
103C23×C127—22no—
104Q8×C127—22no—
105Q8×C127—22no—
106D4×C127—22no—
107D4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—21no—
113C8×C127C2×D412yes—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127—42no—
117C22×C127C22×D411yescharacteristic
118C2×D127—21no—
119C2×D127—21no—
120C2×D127—21no—
121C2×D127—24no—
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—42no—
125C2×D127—42no—
126C2×D127—88no—
127C2×D127#13512yescomplemented
128C4×C127—21no—
129C4×C127—21no—
130C4×C127—42no—
131C4×C127C22×D411yescharacteristic
132C4×C127H(5,𝔽2)11yescharacteristic
133Dic127—21no—
134Dic127—21no—
135Dic127—42no—
136C2×C127—82no—
137C2×C127D4211yescharacteristic
138C2×C127C2×#13511yescharacteristic
139C2×C127#57611yescharacteristic
140D127—22no—
141D127—41no—
142D127—44no—
143#3280—1271nomaximal, Sylow
144C127#328011yescharacteristic, complemented, Sylow
145C2×#135—1271no—
146C22×D8—1271no—
147#546—1272no—
148#495D12711yescharacteristic, complemented
149#467—1271no—
150#466—1271no—
151#463—1272no—
152#450—1272no—
153#449—1272no—
154C4×D8—1271no—
155C8×D4—1271no—
156C2×#41—1272no—
157C22×D4—1272no—
158#135—2544no—
159C2×Q16C2×D12711yescharacteristic, complemented
160C2×SD16—1272no—
161C2×SD16C2×D12712yescomplemented
162C2×D8—1272no—
163C2×D8—1271no—
164C2×D8—2544no—
165C2×D8C2×D12711yescharacteristic
166C22×C8—1271no—
167C22×C8—1271no—
168#124C2×D12712yescomplemented
169#123—2542no—
170C22⋊Q8—1272no—
171#121—1272no—
172#121—2542no—
173#120—2542no—
174C4×D4—1271no—
175C4×D4—1272no—
176C2×C22⋊C4—2542no—
177#107—1271no—
178C4⋊C8—1271no—
179#103—1272no—
180#102—1272no—
181#102—1272no—
182C22⋊C8—1271no—
183C22⋊C8—1271no—
184C4×C8C2×D12711yescharacteristic, complemented
185C24—2542no—
186#41—5082no—
187#41—2544no—
188C2×Q8C22×D12712yes—
189C2×D4—1271no—
190C2×D4—1272no—
191C2×D4—5082no—
192C2×D4—2542no—
193C2×D4—2542no—
194C2×D4—2544no—
195C2×D4—2542no—
196C2×D4C22×D12712yes—
197C22×C4—1271no—
198C22×C4—1271no—
199C22×C4—2542no—
200C22×C4—2542no—
201Q16—22no—
202SD16—42no—
203SD16—2544no—
204D8—22no—
205D8—1274no—
206D8—2544no—
207C2×C8—1271no—
208C2×C8—1271no—
209C2×C8—2542no—
210C2×C8—2542no—
211C2×C8C22×D12711yescharacteristic
212C2×C8C22×D12711yescharacteristic
213C4⋊C4—1271no—
214C4⋊C4—1272no—
215C4⋊C4—2542no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—1271no—
219C22⋊C4—1271no—
220C22⋊C4—2542no—
221C22⋊C4—2542no—
222C22⋊C4—2542no—
223C42C22×D12711yescharacteristic
224C23—22no—
225C23—1271no—
226C23—1271no—
227C23—10162no—
228C23—5084no—
229C23—5084no—
230C23—2542no—
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—22no—
235D4—1274no—
236D4—10162no—
237D4—5084no—
238D4—5084no—
239D4—2542no—
240D4—2542no—
241C2×C4—22no—
242C2×C4—1271no—
243C2×C4—1271no—
244C2×C4—10162no—
245C2×C4—5084no—
246C2×C4—2542no—
247C2×C4—2541no—
248C2×C4—2541no—
249C2×C4—2541no—
250C2×C4—2541no—
251C2×C4D4×D12711yescharacteristic
252C2×C4D4×D12711yescharacteristic
253C2×C4C23×D12711yescharacteristic
254C8—21no—
255C8—2541no—
256C8—2541no—
257C8D4×D12712yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—1272no—
262C22—10168no—
263C22—5082no—
264C22—5082no—
265C22—2541no—
266C22—2541no—
267C22—2544no—
268C22—2541no—
269C22—2541no—
270C22—2541no—
271C22C2×D4×D12711yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—5082no—
276C4—2541no—
277C4—2541no—
278C4C2×D4×D12711yescharacteristic
279C4#1686125411yescharacteristic
280C2—82no—
281C2—5084no—
282C2—5081no—
283C2—2542no—
284C2#2255051111yescharacteristic
285C2#135×D12711yescharacteristic
286C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#495×C127C2
socleC22×C127C22×D4
hypercenter#495D127
hypocenterC127#3280
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. C2×D8×D127C2semidirect
3. #22550519C2semidirect
4. #22550114C2semidirect
5. #22550109C2semidirect
6. #22550088C2semidirect
7. #22550025C2semidirect
8. #22550020C2semidirect
9. C4×D1016C2semidirect
10. C8×D508C2semidirect
11. #495×C127C2semidirect
12. C2×#14523249C22semidirect
15. C2×#14523235C22semidirect
20. C2×C8×D127C22semidirect
24. #16861180C22semidirect
27. C4×D508C22semidirect
29. #16861143C22semidirect
30. C4⋊(C127⋊C8)C22semidirect
31. #16861133C22semidirect
32. #16861131C22semidirect
33. #16861130C22semidirect
35. (C2×D127)⋊C8C22semidirect
36. C2×Q16×C127C22semidirect
37. C2×SD16×C127C22semidirect
39. #124×C127C22semidirect
40. C4×C8×C127C22semidirect
56. D1016D4semidirect
127. C2×D127#135semidirect
144. C127#3280semidirect
148. #495D127semidirect
159. C2×Q16C2×D127semidirect
161. C2×SD16C2×D127semidirect
168. #124C2×D127semidirect
184. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612322
C2×D8×D1272
#225505192
#225501142
#225501092
#225500882
#225500252
#225500202
C4×D10162
C8×D5082
#495×C1272
#3280127