Finite Group Databasebeta

← #30647957

Subgroups — #30647957

Order 16256.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16861235C212yesmaximal, complemented
2#22550387C211yesmaximal, characteristic, complemented
3#22550027C214yesmaximal, complemented
4#22550010C214yesmaximal, complemented
5#22549881C212yesmaximal, complemented
6#22549840C211yesmaximal, characteristic, complemented
7C23×D4×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic, complemented
9#16861235—28no—
10C2×#14523236C2214yescomplemented
11C2×#14523233C2214yescomplemented
12C2×#14523232C2212yescomplemented
13#16861192—21no—
14#16861188C2212yescomplemented
15#16861183C2212yescomplemented
16#16861178—21no—
17C4×C22⋊D127—22no—
18#16861168C2211yescharacteristic, complemented
19#16861130C2218yescomplemented
20C22⋊(C127⋊C8)C2214yescomplemented
21C25×C127—21no—
22C22×D4×C127—28no—
23C22×D4×C127—21no—
24C22×D4×C127—22no—
25C22×D4×C127C2214yes—
26C22×D4×C127C2212yes—
27C23×C4×C127C2211yescharacteristic
28#14523248—24no—
29C2×Dic254C2311yescharacteristic, complemented
30C2×D508C2311yescharacteristic, complemented
31C2×C22⋊D127—21no—
32C2×C22⋊D127—22no—
33C2×C4×D127—21no—
34#14523236—28no—
35#14523233—28no—
36#14523232—24no—
37C2×C127⋊C8C2314yescomplemented
38C4⋊Dic127C2312yescomplemented
39Dic127⋊C4—22no—
40C22⋊Dic127—22no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—22no—
43C4×Dic127—21no—
44C24×C127—28no—
45C24×C127—24no—
46C24×C127—22no—
47C24×C127—48no—
48C24×C127D411yescharacteristic
49C2×D4×C127—28no—
50C2×D4×C127—216no—
51C2×D4×C127—216no—
52C2×D4×C127—24no—
53C2×D4×C127—22no—
54C2×D4×C127—24no—
55C2×D4×C127D418yescomplemented
56C2×D4×C127C2314yes—
57C22×C4×C127—24no—
58C22×C4×C127D411yescharacteristic
59C22×C4×C127D412yes—
60C22×C4×C127C2312yes—
61C22×C4×C127C2311yescharacteristic
62C22×D127—21no—
63Dic254—22no—
64D508—22no—
65C22⋊D127—42no—
66C22⋊D127—44no—
67C2×Dic127—22no—
68C2×Dic127—21no—
69C4×D127—42no—
70C127⋊C8—24no—
71C23×C127—22no—
72C23×C127—24no—
73C23×C127—216no—
74C23×C127—28no—
75C23×C127—24no—
76C23×C127—432no—
77C23×C127—48no—
78C23×C127—48no—
79C23×C127—48no—
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D412yes—
82D4×C127—216no—
83D4×C127—28no—
84D4×C127—24no—
85D4×C127C2×D418yes—
86C2×C4×C127—22no—
87C2×C4×C127—24no—
88C2×C4×C127—24no—
89C2×C4×C127C2×D412yes—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D414yes—
92C2×C4×C127C2411yescharacteristic
93C22×C127—28no—
94C22×C127—22no—
95C22×C127—22no—
96C22×C127—22no—
97C22×C127—28no—
98C22×C127—416no—
99C22×C127—432no—
100C22×C127—48no—
101C22×C127#12012yes—
102C22×C127AGL(1,ℤ/8)14yescomplemented
103C22×C127C22×D411yescharacteristic
104C2×D127—41no—
105C2×D127—42no—
106C4×C127—22no—
107C4×C127#12012yes—
108C4×C127C22×D412yes—
109Dic127—41no—
110Dic127—42no—
111C2×C127—22no—
112C2×C127—24no—
113C2×C127—48no—
114C2×C127C2×#12011yescharacteristic
115C2×C127C2×AGL(1,ℤ/8)12yes—
116D127—81no—
117#3003—1271nomaximal, Sylow
118C127#300311yescharacteristic, complemented, Sylow
119C23×D4D12711yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1272no—
121#525—1271no—
122#450—1274no—
123#447—1274no—
124#418—1272no—
125#407—1271no—
126C25—21no—
127C2×#41—1271no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D12714yes—
132C22×D4C2×D12712yes—
133C23×C4C2×D12711yescharacteristic
134AGL(1,ℤ/8)—2548no—
135C2×SD16—1274no—
136C2×D8—1274no—
137C2×M16—1272no—
138#123—2541no—
139C22⋊Q8—1272no—
140#121—1272no—
141#120—2541no—
142C4×D4—2542no—
143#117—1271no—
144#102—1278no—
145C22⋊C8—1274no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D12711yescharacteristic
151#41—2544no—
152C2×Q8—1271no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—1271no—
160C2×D4—2542no—
161C2×D4—2541no—
162C2×D4C22⋊D12718yescomplemented
163C2×D4C22×D12714yes—
164C22×C4—24no—
165C22×C4—2541no—
166C22×C4C22⋊D12712yes—
167C22×C4C22⋊D12711yescharacteristic
168C22×C4C22×D12712yes—
169C22×C4C22×D12711yescharacteristic
170SD16—2548no—
171D8—2548no—
172M16—2544no—
173C2×C8—1274no—
174C4⋊C4—1272no—
175C4⋊C4—2542no—
176C22⋊C4—2541no—
177C22⋊C4—2542no—
178C22⋊C4—2542no—
179C42—2541no—
180C23—24no—
181C23—22no—
182C23—216no—
183C23—24no—
184C23—28no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—2541no—
190C23C2×C22⋊D12712yes—
191C23C2×C22⋊D12711yescharacteristic
192Q8—2542no—
193D4—216no—
194D4—24no—
195D4—28no—
196D4—5084no—
197D4—5082no—
198D4—2542no—
199D4C2×C22⋊D12718yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—5082no—
204C2×C4—2542no—
205C2×C4—2541no—
206C2×C4C2×C22⋊D12712yes—
207C2×C4C2×C22⋊D12714yes—
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C23×D12711yescharacteristic
210C8—2544no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—432no—
217C22—416no—
218C22—48no—
219C22—5081no—
220C22—5082no—
221C22#1686117812yes—
222C22#1686123514yescomplemented
223C22C22×C22⋊D12711yescharacteristic
224C4—22no—
225C4—5081no—
226C4—5082no—
227C4#1686117812yes—
228C4C22×C22⋊D12712yes—
229C2—24no—
230C2—22no—
231C2—48no—
232C2—10161no—
233C2C2×#1686117811yescharacteristic
234C2C2×#1686123512yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC23×D4×C127C2
socleC22×C127C22×D4
hypercenterC23×D4D127
hypocenterC127#3003
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. #22550387C2semidirect
3. #22550027C2semidirect
4. #22550010C2semidirect
5. #22549881C2semidirect
6. #22549840C2semidirect
7. C23×D4×C127C2semidirect
8. C2×#14523248C22semidirect
10. C2×#14523236C22semidirect
11. C2×#14523233C22semidirect
12. C2×#14523232C22semidirect
14. #16861188C22semidirect
15. #16861183C22semidirect
18. #16861168C22semidirect
19. #16861130C22semidirect
20. C22⋊(C127⋊C8)C22semidirect
29. C2×Dic254C23semidirect
30. C2×D508C23semidirect
37. C2×C127⋊C8C23semidirect
38. C4⋊Dic127C23semidirect
55. C2×D4×C127D4semidirect
102. C22×C127AGL(1,ℤ/8)semidirect
118. C127#3003semidirect
119. C23×D4D127semidirect
162. C2×D4C22⋊D127semidirect
222. C22#16861235semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225503872
#225500272
#225500102
#225498812
#225498402
C23×D4×C1272
#3003127