Finite Group Databasebeta

← #30650553

Subgroups — #30650553

Order 16256.

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

normal subgroups only

This group has 462 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
1#22550587C211yesmaximal, characteristic, complemented
2#22550511C211yesmaximal, characteristic
3#22550193C212yesmaximal, complemented
4#22550163C211yesmaximal, characteristic, complemented
5#22550104C212yesmaximal, complemented
6#22550073C211yesmaximal, characteristic, complemented
7#22550028C212yesmaximal, complemented
8C4×#14523237C211yesmaximal, characteristic, complemented
9C8×D508C211yesmaximal, characteristic, complemented
10#102×D127C212yesmaximal, complemented
11#486×C127C211yesmaximal, characteristic, complemented
12C2×D4×D127—21no—
13C2×D4×D127C2212yes—
14C2×#14523237C2211yescharacteristic
15C2×#14523236C2212yescomplemented
16C2×C8×D127C2212yescomplemented
17#16861200—21no—
18#16861189C2212yescomplemented
19#16861185—22no—
20#16861180—21no—
21#16861180C2212yes—
22#16861177—22no—
23C4×Dic254C2211yescharacteristic, complemented
24C4×D508C2211yescharacteristic
25C4×D508C2211yescharacteristic
26#16861171—21no—
27C4⋊C4×D127C2212yescomplemented
28#16861141C2211yescharacteristic, complemented
29#16861140C2212yescomplemented
30C4⋊(C127⋊C8)C2211yescharacteristic, complemented
31#16861134C2211yescharacteristic, complemented
32#16861131C2211yescharacteristic
33#16861130C2212yescomplemented
34#16861129C2212yes—
35(C2×D127)⋊C8C2212yescomplemented
36C4⋊Q8×C127C2211yescharacteristic, complemented
37#127×C127C2211yescharacteristic
38#102×C127C2212yescomplemented
39#102×C127C2212yescomplemented
40C4×C8×C127C2211yescharacteristic, complemented
41C23×D127—22no—
42C2×Dic254C2311yescharacteristic
43C2×D508D411yescharacteristic
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—22no—
46C2×C22⋊D127—22no—
47D4×D127—24no—
48D4×D127—42no—
49D4×D127—44no—
50C2×C4×D127—22no—
51C2×C4×D127—21no—
52C2×C4×D127C2312yes—
53#14523237—22no—
54#14523236—42no—
55C2×C127⋊C8C2312yes—
56C8×D127—42no—
57C4⋊Dic127—22no—
58C4⋊Dic127D411yescharacteristic
59C4⋊Dic127C2311yescharacteristic
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—22no—
62Dic127⋊C4—21no—
63C22⋊Dic127—22no—
64(C2×D127)⋊C4—22no—
65(C2×D127)⋊C4—21no—
66(C2×D127)⋊C4D412yes—
67C4×Dic127—21no—
68C2×Q8×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127C2312yes—
72C2×C8×C127C2311yescharacteristic
73C2×C8×C127C2311yescharacteristic
74C4⋊C4×C127—21no—
75C4⋊C4×C127C2312yes—
76C42×C127C2311yescharacteristic
77C22×D127—21no—
78C22×D127—44no—
79C22×D127—44no—
80C22×D127—82no—
81C22×D127C2×D412yes—
82Dic254—21no—
83Dic254—21no—
84D508—21no—
85D508SD1614yescomplemented
86D508#4112yes—
87C22⋊D127—44no—
88C22⋊D127—44no—
89C2×Dic127—21no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127C2×D412yes—
93C4×D127—22no—
94C4×D127—22no—
95C4×D127—42no—
96C4×D127—82no—
97C127⋊C8—22no—
98C23×C127—22no—
99Q8×C127—81no—
100D4×C127—22no—
101D4×C127—22no—
102D4×C127—42no—
103D4×C127—81no—
104C2×C4×C127—22no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2411yescharacteristic
108C8×C127—21no—
109C8×C127—21no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127—42no—
113C22×C127C22×D411yescharacteristic
114C2×D127—22no—
115C2×D127—22no—
116C2×D127—22no—
117C2×D127—41no—
118C2×D127—41no—
119C2×D127—88no—
120C2×D127C2×SD1614yes—
121C4×C127—21no—
122C4×C127—42no—
123C4×C127C2×SD1612yes—
124C4×C127C2×#4111yescharacteristic
125C4×C127H(5,𝔽2)11yescharacteristic
126Dic127—22no—
127Dic127—41no—
128Dic127—41no—
129Dic127—41no—
130C2×C127—82no—
131C2×C127#54611yescharacteristic
132C2×C127C22×SD1611yescharacteristic
133C2×C127#57611yescharacteristic
134D127—24no—
135D127—42no—
136#3295—1271nomaximal, Sylow
137C127#329511yescharacteristic, complemented, Sylow
138#554—1271no—
139D42—1271no—
140#486D12711yescharacteristic, complemented
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—1272no—
150C22×D4—2541no—
151C2×SD16—1272no—
152C2×SD16—1271no—
153C22×C8—1272no—
154C4⋊Q8C2×D12711yescharacteristic, complemented
155#127C2×D12711yescharacteristic
156#125—2541no—
157C22⋊Q8—1272no—
158C22⋊Q8—2542no—
159#121—1272no—
160#121—2541no—
161#120—2542no—
162C4×Q8—1271no—
163C4×D4—1271no—
164C4×D4—1271no—
165C4×D4—2541no—
166C2×C4⋊C4—1272no—
167#106—1271no—
168#106—1272no—
169C4⋊C8—1271no—
170#103—1271no—
171#102—1272no—
172#102—1271no—
173#102—1272no—
174#102C2×D12712yescomplemented
175#102C2×D12712yescomplemented
176C22⋊C8—1272no—
177C4×C8C2×D12711yescharacteristic, complemented
178C24—2542no—
179C2×Q8—21no—
180C2×Q8—1271no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—1271no—
184C2×D4—1271no—
185C2×D4—5084no—
186C2×D4—5082no—
187C2×D4—2542no—
188C2×D4—2544no—
189C2×D4—2542no—
190C2×D4C22×D12712yes—
191C22×C4—1272no—
192C22×C4—2542no—
193C22×C4—2541no—
194SD16—5082no—
195SD16—2542no—
196C2×C8—1272no—
197C2×C8—5082no—
198C2×C8C22×D12711yescharacteristic
199C2×C8C22×D12711yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—1271no—
202C4⋊C4—1271no—
203C4⋊C4—1271no—
204C4⋊C4—2542no—
205C4⋊C4—2542no—
206C4⋊C4—2541no—
207C4⋊C4C22×D12712yes—
208C22⋊C4—1272no—
209C22⋊C4—2542no—
210C22⋊C4—2542no—
211C22⋊C4—2541no—
212C42—2541no—
213C42C22×D12711yescharacteristic
214C23—22no—
215C23—1272no—
216C23—10162no—
217C23—5084no—
218C23—5084no—
219C23—2541no—
220Q8—81no—
221Q8—2541no—
222Q8—2541no—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—81no—
227D4—1274no—
228D4—1272no—
229D4—5084no—
230D4—5084no—
231D4—2541no—
232C2×C4—22no—
233C2×C4—1272no—
234C2×C4—10162no—
235C2×C4—5082no—
236C2×C4—2541no—
237C2×C4—2541no—
238C2×C4—2541no—
239C2×C4—2542no—
240C2×C4—2542no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—21no—
245C8—21no—
246C8—2542no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—1274no—
251C22—10168no—
252C22—5081no—
253C22—5081no—
254C22—2542no—
255C22—2542no—
256C22—2542no—
257C22C2×D4×D12711yescharacteristic
258C4—21no—
259C4—42no—
260C4—5081no—
261C4—5081no—
262C4—5081no—
263C4—2542no—
264C4SD16×D12712yes—
265C4#41×D12711yescharacteristic
266C4#1686125511yescharacteristic
267C2—82no—
268C2—5082no—
269C2—2544no—
270C2#2255051911yescharacteristic
271C2C2×SD16×D12711yescharacteristic
272C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#486×C127C2
socleC22×C127C22×D4
prime radical#22550511C2
hypercenter#486D127
hypocenterC127#3295
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #22550587C2semidirect
3. #22550193C2semidirect
4. #22550163C2semidirect
5. #22550104C2semidirect
6. #22550073C2semidirect
7. #22550028C2semidirect
8. C4×#14523237C2semidirect
9. C8×D508C2semidirect
10. #102×D127C2semidirect
11. #486×C127C2semidirect
15. C2×#14523236C22semidirect
16. C2×C8×D127C22semidirect
18. #16861189C22semidirect
23. C4×Dic254C22semidirect
27. C4⋊C4×D127C22semidirect
28. #16861141C22semidirect
29. #16861140C22semidirect
30. C4⋊(C127⋊C8)C22semidirect
31. #16861134C22semidirect
33. #16861130C22semidirect
35. (C2×D127)⋊C8C22semidirect
36. C4⋊Q8×C127C22semidirect
38. #102×C127C22semidirect
39. #102×C127C22semidirect
40. C4×C8×C127C22semidirect
85. D508SD16semidirect
137. C127#3295semidirect
140. #486D127semidirect
154. C4⋊Q8C2×D127semidirect
174. #102C2×D127semidirect
175. #102C2×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
#225505872
#225505112
#225501932
#225501632
#225501042
#225500732
#225500282
C4×#145232372
C8×D5082
#102×D1272
#486×C1272
#3295127