Finite Group Databasebeta

← #30648235

Subgroups — #30648235

Order 16256.

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

normal subgroups only

This group has 526 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
1C22×#14523233C211yesmaximal, characteristic, complemented
2#22550440C211yesmaximal, characteristic, complemented
3#22550351C211yesmaximal, characteristic, complemented
4#22550093C214yesmaximal, complemented
5#22550069C214yesmaximal, complemented
6#22549904C211yesmaximal, characteristic, complemented
7#22549882C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×#14523233—24no—
12C2×#14523233C2214yes—
13C22×C127⋊C8C2211yescharacteristic
14#16861205C2212yescomplemented
15#16861198C2212yescomplemented
16#16861183—22no—
17#16861177—22no—
18D4×Dic127—22no—
19D4×Dic127C2214yescomplemented
20#16861168C2211yescharacteristic
21#16861168C2211yescharacteristic
22C2×C22⋊Dic127—22no—
23Dic127⋊C8C2214yescomplemented
24#16861133C2212yescomplemented
25#16861131C2214yescomplemented
26#16861130C2212yescomplemented
27C2×#41×C127C2211yescharacteristic
28C22×D4×C127C2211yescharacteristic
29AGL(1,ℤ/8)×C127—24no—
30C2×SD16×C127C2212yescomplemented
31C2×D8×C127C2212yescomplemented
32C2×M16×C127C2211yescharacteristic
33C23×D127—21no—
34C2×D508—22no—
35C2×D508—21no—
36C2×D508—22no—
37C2×D508C2312yes—
38C2×C22⋊D127—24no—
39C22×Dic127—24no—
40#14523233—28no—
41C2×C127⋊C8—22no—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127D412yes—
44C4⋊Dic127C2312yes—
45C22⋊Dic127—24no—
46C22⋊Dic127—21no—
47C22⋊Dic127D412yes—
48(C2×D127)⋊C4—24no—
49C4×Dic127—21no—
50C4×Dic127C2312yes—
51C24×C127—21no—
52#41×C127—22no—
53#41×C127—22no—
54C2×Q8×C127C2311yescharacteristic
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—22no—
58C2×D4×C127D414yescomplemented
59C2×D4×C127C2311yescharacteristic
60C2×D4×C127C2312yes—
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63SD16×C127—24no—
64D8×C127—24no—
65M16×C127—22no—
66C2×C8×C127C2312yes—
67C22×D127—22no—
68C22×D127—42no—
69C22×D127—82no—
70D508—22no—
71D508—22no—
72D508—42no—
73C22⋊D127—84no—
74C2×Dic127—22no—
75C2×Dic127—84no—
76C2×Dic127C2×D414yes—
77C127⋊C8—22no—
78C23×C127—22no—
79C23×C127—24no—
80C23×C127—21no—
81C23×C127—82no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—21no—
84Q8×C127—21no—
85D4×C127—24no—
86D4×C127—21no—
87D4×C127—22no—
88D4×C127—21no—
89D4×C127—42no—
90D4×C127C2×D414yes—
91C2×C4×C127—21no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127—22no—
99C22×C127—24no—
100C22×C127—44no—
101C22×C127—41no—
102C22×C127—42no—
103C22×C127—84no—
104C22×C127#12112yes—
105C22×C127C22×D411yescharacteristic
106C2×D127—42no—
107C2×D127—44no—
108C2×D127—84no—
109C4×C127—41no—
110C4×C127#12112yes—
111C4×C127C22×D411yescharacteristic
112C4×C127C2×#4111yescharacteristic
113Dic127—24no—
114Dic127—42no—
115C2×C127—22no—
116C2×C127—44no—
117C2×C127—81no—
118C2×C127C2×#12111yescharacteristic
119C2×C127#57612yes—
120D127—82no—
121#3041—1271nomaximal, Sylow
122C127#304111yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
124C22×D8—1271no—
125#535—1271no—
126#518—1271no—
127#463—1274no—
128#459—1274no—
129#424—1271no—
130#418—1271no—
131#416—1271no—
132C2×#41C2×D12711yescharacteristic
133C22×D4—1271no—
134C22×D4C2×D12711yescharacteristic
135AGL(1,ℤ/8)—24no—
136C2×SD16C2×D12712yescomplemented
137C2×D8—1274no—
138C2×D8—2544no—
139C2×D8C2×D12712yescomplemented
140C2×M16C2×D12711yescharacteristic
141C22×C8—1271no—
142#127—1272no—
143#124—1272no—
144#121—2542no—
145#120—2542no—
146C4×D4—1274no—
147C4×D4—2542no—
148#117—1271no—
149#117—1271no—
150C2×C22⋊C4—2542no—
151C4⋊C8—1274no—
152#103—1272no—
153#102—1272no—
154#102—1274no—
155C24—21no—
156C24—2541no—
157#41—22no—
158#41—22no—
159C2×Q8C22×D12711yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—1272no—
164C2×D4—2544no—
165C2×D4—2541no—
166C2×D4—2542no—
167C2×D4—2542no—
168C2×D4C22⋊D12714yescomplemented
169C2×D4C22×D12711yescharacteristic
170C2×D4C22×D12712yes—
171C22×C4—21no—
172C22×C4—2544no—
173C22×C4C22×D12711yescharacteristic
174SD16—24no—
175D8—24no—
176D8—2548no—
177M16—22no—
178C2×C8—1272no—
179C2×C8—2542no—
180C2×C8C22×D12712yes—
181C4⋊C4—1272no—
182C4⋊C4—1272no—
183C22⋊C4—1272no—
184C22⋊C4—2544no—
185C22⋊C4—2544no—
186C22⋊C4—2541no—
187C42—1272no—
188C42—2541no—
189C23—22no—
190C23—24no—
191C23—21no—
192C23—82no—
193C23—10162no—
194C23—5082no—
195C23—2542no—
196C23D4×D12711yescharacteristic
197Q8—21no—
198Q8—21no—
199D4—21no—
200D4—22no—
201D4—21no—
202D4—24no—
203D4—42no—
204D4—10164no—
205D4—5082no—
206D4—2542no—
207D4—2542no—
208D4C2×C22⋊D12714yes—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—1274no—
212C2×C4—10164no—
213C2×C4—2542no—
214C2×C4D4×D12711yescharacteristic
215C2×C4C2×C22⋊D12712yes—
216C2×C4#1452324712yes—
217C2×C4C23×D12711yescharacteristic
218C8—22no—
219C8—2542no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—41no—
224C22—42no—
225C22—84no—
226C22—10164no—
227C22—5082no—
228C22—5084no—
229C22#1686118112yes—
230C22C2×D4×D12711yescharacteristic
231C4—41no—
232C4—5082no—
233C4—2544no—
234C4#1686118112yes—
235C4C22×C22⋊D12711yescharacteristic
236C4C2×#1452324711yescharacteristic
237C2—22no—
238C2—44no—
239C2—81no—
240C2—10162no—
241C2C2×#1686118111yescharacteristic
242C2#2255068012yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3041
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×#14523233C2semidirect
2. #22550440C2semidirect
3. #22550351C2semidirect
4. #22550093C2semidirect
5. #22550069C2semidirect
6. #22549904C2semidirect
7. #22549882C2semidirect
8. #22549872C2semidirect
9. C2×AGL(1,ℤ/8)×C127C2semidirect
14. #16861205C22semidirect
15. #16861198C22semidirect
19. D4×Dic127C22semidirect
23. Dic127⋊C8C22semidirect
24. #16861133C22semidirect
25. #16861131C22semidirect
26. #16861130C22semidirect
30. C2×SD16×C127C22semidirect
31. C2×D8×C127C22semidirect
58. C2×D4×C127D4semidirect
122. C127#3041semidirect
123. C2×AGL(1,ℤ/8)D127semidirect
136. C2×SD16C2×D127semidirect
139. C2×D8C2×D127semidirect
168. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
#225504402
#225503512
#225500932
#225500692
#225499042
#225498822
#225498722
C2×AGL(1,ℤ/8)×C1272
#3041127