Finite Group Databasebeta

← #30650556

Subgroups — #30650556

Order 16256.

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

normal subgroups only

This group has 462 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#22550586C211yesmaximal, characteristic, complemented
2D4×D508C211yesmaximal, characteristic
3#22550223C211yesmaximal, characteristic, complemented
4#22550196C212yesmaximal, complemented
5#22550166C211yesmaximal, characteristic, complemented
6#22550108C212yesmaximal, complemented
7#22550076C211yesmaximal, characteristic, complemented
8#22550025C212yesmaximal, complemented
9C8×D508C211yesmaximal, characteristic, complemented
10#102×D127C212yesmaximal, complemented
11C4×SD16×C127C211yesmaximal, characteristic, complemented
12C22×D508—21no—
13C2×D4×D127C2212yes—
14C2×#14523235C2212yescomplemented
15C2×C8×D127C2212yescomplemented
16#16861209C2211yescharacteristic, complemented
17#16861206C2211yescharacteristic
18#16861201—21no—
19#16861187—22no—
20#16861185C2212yescomplemented
21#16861184C2212yes—
22#16861183—21no—
23#16861179—22no—
24C4×D508—21no—
25C4×D508C2211yescharacteristic
26C4⋊C4×D127C2212yescomplemented
27#16861140C2212yescomplemented
28C4⋊(C127⋊C8)C2211yescharacteristic, complemented
29#16861131C2212yescomplemented
30#16861131C2212yescomplemented
31#16861130C2212yes—
32#16861129C2212yescomplemented
33(C2×D127)⋊C8C2212yescomplemented
34C2×SD16×C127C2211yescharacteristic
35C4×Q8×C127C2211yescharacteristic, complemented
36C4×D4×C127C2211yescharacteristic
37#106×C127C2211yescharacteristic, complemented
38#103×C127C2211yescharacteristic, complemented
39#102×C127C2211yescharacteristic
40C4×C8×C127C2211yescharacteristic, complemented
41C23×D127—22no—
42C2×Dic254—21no—
43C2×D508—22no—
44C2×D508—21no—
45C2×D508—21no—
46C2×D508—44no—
47C2×D508D411yescharacteristic
48C2×D508C2312yes—
49C2×C22⋊D127—22no—
50D4×D127—24no—
51D4×D127—42no—
52C2×C4×D127—22no—
53C2×C4×D127C2312yes—
54#14523235—42no—
55C2×C127⋊C8C2312yes—
56C8×D127—42no—
57C4⋊Dic127—21no—
58C4⋊Dic127D411yescharacteristic
59C4⋊Dic127C2312yes—
60Dic127⋊C4—22no—
61Dic127⋊C4—22no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—22no—
64(C2×D127)⋊C4D412yes—
65C2×Q8×C127C2311yescharacteristic
66C2×D4×C127C2311yescharacteristic
67C22×C4×C127—21no—
68SD16×C127—22no—
69C2×C8×C127C2311yescharacteristic
70C2×C8×C127C2311yescharacteristic
71C4⋊C4×C127—21no—
72C4⋊C4×C127C2311yescharacteristic
73C4⋊C4×C127C2311yescharacteristic
74C22⋊C4×C127—21no—
75C42×C127—21no—
76C42×C127C2311yescharacteristic
77C22×D127—22no—
78C22×D127—44no—
79C22×D127—44no—
80C22×D127—82no—
81C22×D127C2×D412yes—
82Dic254—81no—
83D508—22no—
84D508—22no—
85D508—44no—
86D508—42no—
87D508—81no—
88D508SD1614yescomplemented
89C22⋊D127—44no—
90C2×Dic127—22no—
91C2×Dic127C2×D412yes—
92C4×D127—22no—
93C4×D127—22no—
94C4×D127—82no—
95C127⋊C8—22no—
96C23×C127—21no—
97Q8×C127—21no—
98Q8×C127—21no—
99D4×C127—21no—
100D4×C127#4112yes—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—42no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2411yescharacteristic
108C8×C127—21no—
109C8×C127—21no—
110C22×C127—22no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127C22×D411yescharacteristic
114C2×D127—22no—
115C2×D127—22no—
116C2×D127—42no—
117C2×D127—42no—
118C2×D127—42no—
119C2×D127—88no—
120C2×D127C2×SD1614yes—
121C4×C127—21no—
122C4×C127—41no—
123C4×C127—41no—
124C4×C127—41no—
125C4×C127C2×SD1612yes—
126C4×C127C2×#4111yescharacteristic
127C4×C127H(5,𝔽2)11yescharacteristic
128Dic127—22no—
129Dic127—42no—
130C2×C127—42no—
131C2×C127#54611yescharacteristic
132C2×C127C22×SD1611yescharacteristic
133C2×C127#57611yescharacteristic
134D127—24no—
135D127—82no—
136#3295—1271nomaximal, Sylow
137C127#329511yescharacteristic, complemented, Sylow
138#554—1271no—
139D42—1271no—
140#486—1271no—
141#481—1272no—
142#476—1271no—
143#465—1272no—
144#460—1271no—
145#450—1272no—
146C4×SD16D12711yescharacteristic, complemented
147C8×D4—1271no—
148C2×#102—1272no—
149C22×D4—1272no—
150C22×D4—2541no—
151C2×SD16—1272no—
152C2×SD16C2×D12711yescharacteristic
153C22×C8—1272no—
154C4⋊Q8—1271no—
155#127—1271no—
156#125—2541no—
157C22⋊Q8—1272no—
158C22⋊Q8—2542no—
159#121—1272no—
160#121—2541no—
161#120—2542no—
162C4×Q8C2×D12711yescharacteristic, complemented
163C4×D4—1271no—
164C4×D4—2541no—
165C4×D4C2×D12711yescharacteristic
166C2×C4⋊C4—1272no—
167#106—1272no—
168#106C2×D12711yescharacteristic, complemented
169C4⋊C8—1271no—
170#103C2×D12711yescharacteristic, complemented
171#102—1272no—
172#102—1272no—
173#102—1272no—
174#102—1272no—
175#102C2×D12711yescharacteristic
176C22⋊C8—1272no—
177C4×C8C2×D12711yescharacteristic, complemented
178C24—2542no—
179C2×Q8—2541no—
180C2×Q8C22×D12711yescharacteristic
181C2×D4—1271no—
182C2×D4—1272no—
183C2×D4—5084no—
184C2×D4—5082no—
185C2×D4—2542no—
186C2×D4—2542no—
187C2×D4—2544no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4C22×D12711yescharacteristic
191C22×C4—21no—
192C22×C4—1272no—
193C22×C4—2542no—
194SD16—22no—
195SD16—5082no—
196C2×C8—1272no—
197C2×C8—5082no—
198C2×C8C22×D12711yescharacteristic
199C2×C8C22×D12711yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—1271no—
202C4⋊C4—1272no—
203C4⋊C4—2542no—
204C4⋊C4—2542no—
205C4⋊C4—2541no—
206C4⋊C4C22×D12711yescharacteristic
207C4⋊C4C22×D12711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—1272no—
210C22⋊C4—2542no—
211C22⋊C4—2542no—
212C42—21no—
213C42C22×D12711yescharacteristic
214C23—21no—
215C23—1272no—
216C23—10162no—
217C23—5084no—
218C23—5084no—
219C23—2542no—
220Q8—21no—
221Q8—21no—
222Q8—10161no—
223D4—21no—
224D4—1274no—
225D4—10161no—
226D4—5084no—
227D4—5084no—
228D4—5082no—
229D4—2542no—
230D4—2542no—
231D4#1452324812yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—1272no—
237C2×C4—10162no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4—2542no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—21no—
245C8—21no—
246C8—2542no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—1274no—
251C22—10168no—
252C22—5082no—
253C22—5082no—
254C22—5082no—
255C22—2542no—
256C22—2542no—
257C22C2×D4×D12711yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—5082no—
263C4—2542no—
264C4SD16×D12712yes—
265C4C2×#1452324811yescharacteristic
266C4#1686125511yescharacteristic
267C2—42no—
268C2—10162no—
269C2—2544no—
270C2#2255052211yescharacteristic
271C2C2×SD16×D12711yescharacteristic
272C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
prime radicalD4×D508C2
hypercenterC4×SD16D127
hypocenterC127#3295
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #22550586C2semidirect
3. #22550223C2semidirect
4. #22550196C2semidirect
5. #22550166C2semidirect
6. #22550108C2semidirect
7. #22550076C2semidirect
8. #22550025C2semidirect
9. C8×D508C2semidirect
10. #102×D127C2semidirect
11. C4×SD16×C127C2semidirect
14. C2×#14523235C22semidirect
15. C2×C8×D127C22semidirect
16. #16861209C22semidirect
20. #16861185C22semidirect
26. C4⋊C4×D127C22semidirect
27. #16861140C22semidirect
28. C4⋊(C127⋊C8)C22semidirect
29. #16861131C22semidirect
30. #16861131C22semidirect
32. #16861129C22semidirect
33. (C2×D127)⋊C8C22semidirect
35. C4×Q8×C127C22semidirect
37. #106×C127C22semidirect
38. #103×C127C22semidirect
40. C4×C8×C127C22semidirect
88. D508SD16semidirect
137. C127#3295semidirect
146. C4×SD16D127semidirect
162. C4×Q8C2×D127semidirect
168. #106C2×D127semidirect
170. #103C2×D127semidirect
177. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505862
D4×D5082
#225502232
#225501962
#225501662
#225501082
#225500762
#225500252
C8×D5082
#102×D1272
C4×SD16×C1272
#3295127