Finite Group Databasebeta

← #30650554

Subgroups — #30650554

Order 16256.

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

normal subgroups only

This group has 462 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#22550583C211yesmaximal, characteristic, complemented
2#22550222C211yesmaximal, characteristic, complemented
3#22550192C212yesmaximal, complemented
4#22550166C211yesmaximal, characteristic, complemented
5#22550103C212yesmaximal, complemented
6#22550073C211yesmaximal, characteristic, complemented
7#22550027C212yesmaximal, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10C2×#16861130C212yesmaximal, complemented
11D42×C127C211yesmaximal, characteristic
12C2×#14523236C2211yescharacteristic
13C2×#14523236C2212yescomplemented
14C22×C127⋊C8C2212yescomplemented
15#16861209C2211yescharacteristic, complemented
16#16861200—21no—
17#16861188C2212yescomplemented
18#16861186—22no—
19C4×Dic254C2211yescharacteristic, complemented
20D4×Dic127—21no—
21C2×C4⋊Dic127C2212yescomplemented
22#16861140C2212yescomplemented
23#16861140C2211yescharacteristic, complemented
24C4⋊(C127⋊C8)C2211yescharacteristic, complemented
25#16861132C2211yescharacteristic, complemented
26#16861130C2212yes—
27#16861130C2211yescharacteristic
28#16861130C2212yescomplemented
29#16861130C2212yescomplemented
30#16861130C2212yescomplemented
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
41C2×Dic254—21no—
42C2×Dic254C2311yescharacteristic
43C22×Dic127—22no—
44#14523236—22no—
45#14523236—42no—
46C2×C127⋊C8—42no—
47C2×C127⋊C8C2311yescharacteristic
48C2×C127⋊C8C2312yes—
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127—22no—
51C4⋊Dic127—21no—
52C4⋊Dic127C2312yes—
53C4⋊Dic127C2311yescharacteristic
54Dic127⋊C4—21no—
55Dic127⋊C4—22no—
56C22⋊Dic127—22no—
57C4×Dic127—21no—
58C24×C127—22no—
59C2×D4×C127—24no—
60C2×D4×C127—22no—
61C2×D4×C127—22no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—42no—
65C2×D4×C127—44no—
66C2×D4×C127D411yescharacteristic
67C2×D4×C127C2312yes—
68C2×D4×C127C2311yescharacteristic
69C22×C4×C127—21no—
70C22×C4×C127C2312yes—
71C4⋊C4×C127D411yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—21no—
74C22⋊C4×C127—22no—
75C22⋊C4×C127D412yes—
76C42×C127C2311yescharacteristic
77Dic254—21no—
78Dic254—21no—
79Dic254—81no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—22no—
83C2×Dic127—82no—
84C127⋊C8—21no—
85C127⋊C8—21no—
86C127⋊C8—22no—
87C23×C127—21no—
88C23×C127—22no—
89C23×C127—44no—
90C23×C127—44no—
91C23×C127—82no—
92C23×C127C2×D412yes—
93D4×C127—22no—
94D4×C127—21no—
95D4×C127—22no—
96D4×C127—44no—
97D4×C127—44no—
98D4×C127—42no—
99D4×C127—81no—
100D4×C127SD1614yescomplemented
101D4×C127#4112yes—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—42no—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2411yescharacteristic
110C22×C127—22no—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127—41no—
114C22×C127—41no—
115C22×C127—42no—
116C22×C127—42no—
117C22×C127—42no—
118C22×C127—88no—
119C22×C127C2×SD1614yes—
120C22×C127C22×D411yescharacteristic
121C4×C127—22no—
122C4×C127—21no—
123C4×C127—41no—
124C4×C127C2×SD1612yes—
125C4×C127C2×#4111yescharacteristic
126C4×C127H(5,𝔽2)11yescharacteristic
127Dic127—41no—
128Dic127—41no—
129Dic127—42no—
130C2×C127—24no—
131C2×C127—42no—
132C2×C127—82no—
133C2×C127#54611yescharacteristic
134C2×C127C22×SD1611yescharacteristic
135C2×C127#57611yescharacteristic
136#3295—1271nomaximal, Sylow
137C127#329511yescharacteristic, complemented, Sylow
138#554—1271no—
139D42D12711yescharacteristic
140#486—1271no—
141#481—1272no—
142#476—1271no—
143#465—1272no—
144#460—1271no—
145#450—1272no—
146C4×SD16—1271no—
147C8×D4—1271no—
148C2×#102—1272no—
149C22×D4—21no—
150C22×D4C2×D12712yes—
151C2×SD16—1272no—
152C2×SD16—1271no—
153C22×C8—1272no—
154C4⋊Q8—1271no—
155#127C2×D12711yescharacteristic
156#125—2541no—
157C22⋊Q8—1272no—
158C22⋊Q8—2542no—
159#121—21no—
160#121C2×D12712yes—
161#120—22no—
162C4×Q8—1271no—
163C4×D4—2541no—
164C4×D4C2×D12711yescharacteristic
165C4×D4C2×D12711yescharacteristic
166C2×C4⋊C4—1272no—
167#106—1271no—
168#106—1272no—
169C4⋊C8—1271no—
170#103—1271no—
171#102—1271no—
172#102—1272no—
173#102—1272no—
174#102—1272no—
175#102—1272no—
176C22⋊C8—1272no—
177C4×C8—1271no—
178C24—22no—
179C2×Q8—1271no—
180C2×Q8—2541no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4C22⋊D12711yescharacteristic
189C2×D4C22×D12712yes—
190C2×D4C22×D12711yescharacteristic
191C22×C4—21no—
192C22×C4—2542no—
193C22×C4C22×D12712yes—
194SD16—5082no—
195SD16—2542no—
196C2×C8—1271no—
197C2×C8—1271no—
198C2×C8—1272no—
199C2×C8—5082no—
200C4⋊C4—1271no—
201C4⋊C4—1272no—
202C4⋊C4—2541no—
203C4⋊C4—2542no—
204C4⋊C4—2542no—
205C4⋊C4—2541no—
206C4⋊C4C22⋊D12711yescharacteristic
207C4⋊C4C22×D12711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—2542no—
211C22⋊C4C22⋊D12712yes—
212C42—2541no—
213C42C22×D12711yescharacteristic
214C23—21no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—82no—
219C23C2×C22⋊D12712yes—
220Q8—10161no—
221Q8—2541no—
222Q8—2541no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1452323614yescomplemented
231D4#1452324712yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—10162no—
237C2×C4—2541no—
238C2×C4—2541no—
239C2×C4—2542no—
240C2×C4C2×C22⋊D12711yescharacteristic
241C2×C4C2×C22⋊D12712yes—
242C2×C4C2×C22⋊D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—2541no—
245C8—2541no—
246C8—2542no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—88no—
256C22C2×#1452323614yes—
257C22C22×C22⋊D12711yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—5081no—
262C4—5082no—
263C4—5081no—
264C4C2×#1452323612yes—
265C4C2×#1452324711yescharacteristic
266C4#1686125411yescharacteristic
267C2—24no—
268C2—42no—
269C2—82no—
270C2#2255052011yescharacteristic
271C2C22×#1452323611yescharacteristic
272C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalD42×C127C2
socleC22×C127C22×D4
hypercenterD42D127
hypocenterC127#3295
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. #22550583C2semidirect
2. #22550222C2semidirect
3. #22550192C2semidirect
4. #22550166C2semidirect
5. #22550103C2semidirect
6. #22550073C2semidirect
7. #22550027C2semidirect
8. C4×#14523236C2semidirect
9. D4×C127⋊C8C2semidirect
10. C2×#16861130C2semidirect
13. C2×#14523236C22semidirect
14. C22×C127⋊C8C22semidirect
15. #16861209C22semidirect
17. #16861188C22semidirect
19. C4×Dic254C22semidirect
21. C2×C4⋊Dic127C22semidirect
22. #16861140C22semidirect
23. #16861140C22semidirect
24. C4⋊(C127⋊C8)C22semidirect
25. #16861132C22semidirect
28. #16861130C22semidirect
29. #16861130C22semidirect
30. #16861130C22semidirect
31. C22⋊(C127⋊C8)C22semidirect
32. C4×C127⋊C8C22semidirect
100. D4×C127SD16semidirect
137. C127#3295semidirect
230. D4#14523236semidirect

Maximal subgroups

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

subgroupindex
#225505832
#225502222
#225501922
#225501662
#225501032
#225500732
#225500272
C4×#145232362
D4×C127⋊C82
C2×#168611302
D42×C1272
#3295127