Finite Group Databasebeta

← #30648230

Subgroups — #30648230

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
1C2×#16861235C211yesmaximal, characteristic, complemented
2#22550443C211yesmaximal, characteristic, complemented
3#22550351C211yesmaximal, characteristic, complemented
4#22550089C214yesmaximal, complemented
5#22550069C214yesmaximal, complemented
6#22549902C211yesmaximal, characteristic, complemented
7#22549881C211yesmaximal, characteristic, complemented
8#22549873C211yesmaximal, characteristic, complemented
9C22×D8×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11#16861235—24no—
12C2×#14523236C2212yescomplemented
13C2×#14523233C2212yescomplemented
14C2×#14523232C2211yescharacteristic
15#16861205C2212yescomplemented
16#16861196C2212yescomplemented
17#16861180—22no—
18#16861178—22no—
19D4×Dic127—22no—
20D4×Dic127C2214yescomplemented
21#16861168C2211yescharacteristic
22#16861168C2211yescharacteristic
23C2×C22⋊Dic127—22no—
24Dic127⋊C8C2214yescomplemented
25#16861134C2212yescomplemented
26#16861131C2212yescomplemented
27#16861130C2214yescomplemented
28C22×D4×C127C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic
30C2×D8×C127—24no—
31C2×D8×C127C2214yes—
32C22×C8×C127C2211yescharacteristic
33#14523248—22no—
34#14523248—22no—
35C2×Dic254C2311yescharacteristic
36C2×D508C2311yescharacteristic
37C2×C22⋊D127—24no—
38C2×C22⋊D127—21no—
39C22×Dic127—24no—
40C2×C4×D127—21no—
41#14523236—24no—
42#14523233—24no—
43#14523232—22no—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127D412yes—
46C4⋊Dic127C2312yes—
47C22⋊Dic127—24no—
48C22⋊Dic127—24no—
49C22⋊Dic127—21no—
50C22⋊Dic127D412yes—
51C4×Dic127—21no—
52C4×Dic127C2312yes—
53C24×C127—21no—
54C24×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—22no—
57C2×D4×C127—21no—
58C2×D4×C127—22no—
59C2×D4×C127—22no—
60C2×D4×C127D414yescomplemented
61C2×D4×C127C2312yes—
62C2×D4×C127C2312yes—
63C22×C4×C127C2311yescharacteristic
64D8×C127—28no—
65C2×C8×C127—22no—
66C2×C8×C127C2312yes—
67C22×D127—21no—
68Dic254—21no—
69Dic254—21no—
70D508—21no—
71D508—21no—
72C22⋊D127—42no—
73C22⋊D127—84no—
74C2×Dic127—22no—
75C2×Dic127—21no—
76C2×Dic127—84no—
77C2×Dic127C2×D414yes—
78C4×D127—42no—
79C127⋊C8—22no—
80C23×C127—22no—
81C23×C127—24no—
82C23×C127—22no—
83C23×C127—42no—
84C23×C127—82no—
85C23×C127—82no—
86C23×C127C2×D411yescharacteristic
87D4×C127—24no—
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—42no—
92D4×C127C2×D414yes—
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—42no—
102C22×C127—44no—
103C22×C127—84no—
104C22×C127—84no—
105C22×C127#12112yes—
106C22×C127C22×D411yescharacteristic
107C2×D127—41no—
108C2×D127—42no—
109C4×C127#12112yes—
110C4×C127C22×D411yescharacteristic
111C4×C127C2×#4111yescharacteristic
112Dic127—24no—
113Dic127—41no—
114Dic127—42no—
115C2×C127—22no—
116C2×C127—44no—
117C2×C127—82no—
118C2×C127C2×#12111yescharacteristic
119C2×C127#57612yes—
120D127—81no—
121#3041—1271nomaximal, Sylow
122C127#304111yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)—1271no—
124C22×D8D12711yescharacteristic, complemented
125#535—1271no—
126#518—1271no—
127#463—1274no—
128#459—1274no—
129#424—1271no—
130#418—1271no—
131#416—1271no—
132C2×#41—1271no—
133C22×D4C2×D12711yescharacteristic
134C22×D4C2×D12711yescharacteristic
135AGL(1,ℤ/8)—2544no—
136C2×SD16—1272no—
137C2×D8—24no—
138C2×D8—1272no—
139C2×D8C2×D12714yes—
140C2×M16—1271no—
141C22×C8C2×D12711yescharacteristic
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—1274no—
154#102—1272no—
155C24—21no—
156C24—21no—
157#41—2542no—
158#41—2542no—
159C2×Q8—1271no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—1271no—
166C2×D4—2544no—
167C2×D4—2541no—
168C2×D4C22⋊D12714yescomplemented
169C2×D4C22×D12712yes—
170C2×D4C22×D12712yes—
171C22×C4—2544no—
172C22×C4—2541no—
173C22×C4C22×D12711yescharacteristic
174SD16—2544no—
175D8—28no—
176D8—2544no—
177M16—2542no—
178C2×C8—22no—
179C2×C8—1272no—
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—22no—
192C23—42no—
193C23—82no—
194C23—82no—
195C23—2541no—
196C23D4×D12711yescharacteristic
197Q8—2541no—
198Q8—2541no—
199D4—22no—
200D4—22no—
201D4—22no—
202D4—24no—
203D4—42no—
204D4—10164no—
205D4—5082no—
206D4—2541no—
207D4—2541no—
208D4C2×C22⋊D12714yes—
209C2×C4—1274no—
210C2×C4—10164no—
211C2×C4—5082no—
212C2×C4—2542no—
213C2×C4—2541no—
214C2×C4D4×D12711yescharacteristic
215C2×C4C2×C22⋊D12712yes—
216C2×C4#1452324712yes—
217C2×C4C23×D12711yescharacteristic
218C8—22no—
219C8—2542no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—42no—
224C22—44no—
225C22—84no—
226C22—84no—
227C22—5081no—
228C22—5082no—
229C22#1686118112yes—
230C22C2×D4×D12711yescharacteristic
231C4—5081no—
232C4—5082no—
233C4—2544no—
234C4#1686118112yes—
235C4C22×C22⋊D12711yescharacteristic
236C4C2×#1452324711yescharacteristic
237C2—22no—
238C2—44no—
239C2—82no—
240C2—10161no—
241C2C2×#1686118111yescharacteristic
242C2#2255067712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC22×C127C22×D4
hypercenterC22×D8D127
hypocenterC127#3041
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. #22550443C2semidirect
3. #22550351C2semidirect
4. #22550089C2semidirect
5. #22550069C2semidirect
6. #22549902C2semidirect
7. #22549881C2semidirect
8. #22549873C2semidirect
9. C22×D8×C127C2semidirect
12. C2×#14523236C22semidirect
13. C2×#14523233C22semidirect
15. #16861205C22semidirect
16. #16861196C22semidirect
20. D4×Dic127C22semidirect
24. Dic127⋊C8C22semidirect
25. #16861134C22semidirect
26. #16861131C22semidirect
27. #16861130C22semidirect
60. C2×D4×C127D4semidirect
122. C127#3041semidirect
124. C22×D8D127semidirect
168. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225504432
#225503512
#225500892
#225500692
#225499022
#225498812
#225498732
C22×D8×C1272
#3041127