Finite Group Databasebeta

← C2×#22550258 #30649149

Subgroups — C2×#22550258 #30649149

Order 16256.

592 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14523238C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3C22×#14523235C211yesmaximal, characteristic, complemented
4C22×#14523233C211yesmaximal, characteristic
5C2×#16861197C211yesmaximal, characteristic, complemented
6#22550258C218yesmaximal, direct product factor
7C2×C4×C127⋊C8C211yesmaximal, characteristic, complemented
8C2×#124×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2211yescharacteristic
10C22×D508C2211yescharacteristic
11C2×#14523238—24no—
12C2×#14523238C2214yescomplemented
13C2×#14523236—24no—
14C2×#14523236C2214yescomplemented
15C2×#14523235—24no—
16C2×#14523235C2214yescomplemented
17C2×#14523233—24no—
18C2×#14523233C2214yes—
19C22×C127⋊C8C2211yescharacteristic
20C22×C127⋊C8C2211yescharacteristic
21#16861197—22no—
22#16861197C2214yescomplemented
23C2×(C2×D127)⋊C4—22no—
24C4×C127⋊C8C2214yescomplemented
25C22×Q8×C127C2211yescharacteristic
26C22×D4×C127C2211yescharacteristic
27#124×C127—22no—
28#124×C127C2214yescomplemented
29C2×C22⋊C4×C127—22no—
30C2×C42×C127C2211yescharacteristic
31C23×D127—21no—
32C2×Dic254—21no—
33C2×Dic254—22no—
34C2×Dic254—22no—
35C2×Dic254C2312yes—
36C2×D508—22no—
37C2×D508—22no—
38C2×D508—21no—
39C2×D508C2312yes—
40C22×Dic127—21no—
41#14523238—28no—
42#14523236—28no—
43#14523235—28no—
44#14523233—28no—
45C2×C127⋊C8D414yes—
46C2×C127⋊C8D414yes—
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2312yes—
49(C2×D127)⋊C4—28no—
50C24×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—22no—
53C2×Q8×C127—22no—
54C2×Q8×C127C2312yes—
55C2×D4×C127—22no—
56C2×D4×C127—22no—
57C2×D4×C127—21no—
58C2×D4×C127C2312yes—
59C22×C4×C127—21no—
60C22×C4×C127D412yes—
61C22×C4×C127C2311yescharacteristic
62C22⋊C4×C127—28no—
63C42×C127D412yes—
64C42×C127C2312yes—
65C22×D127—22no—
66C22×D127—42no—
67C22×D127—44no—
68Dic254—22no—
69Dic254—22no—
70Dic254—42no—
71D508—22no—
72D508—22no—
73D508—42no—
74C2×Dic127—22no—
75C2×Dic127—42no—
76C127⋊C8C2×D414yes—
77C127⋊C8C2×D414yes—
78C23×C127—22no—
79C23×C127—42no—
80C23×C127—44no—
81C23×C127C2×D411yescharacteristic
82Q8×C127—22no—
83Q8×C127—22no—
84Q8×C127—42no—
85D4×C127—22no—
86D4×C127—22no—
87D4×C127—42no—
88C2×C4×C127—24no—
89C2×C4×C127—22no—
90C2×C4×C127—42no—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D414yes—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2411yescharacteristic
96C22×C127—42no—
97C22×C127—44no—
98C22×C127—84no—
99C22×C127#12712yes—
100C22×C127#13514yes—
101C22×C127C22×D411yescharacteristic
102C2×D127—42no—
103C2×D127—44no—
104C2×D127—84no—
105C4×C127—24no—
106C4×C127—42no—
107C4×C127#12712yes—
108C4×C127C22×D411yescharacteristic
109C4×C127C22×D411yescharacteristic
110Dic127—42no—
111C2×C127—82no—
112C2×C127#49514yescomplemented
113C2×C127C2×#12711yescharacteristic
114C2×C127C2×#13512yes—
115D127—82no—
116C2×#495—1271nomaximal, Sylow
117C127C2×#49511yescharacteristic, complemented, Sylow
118C22×Q16—1271no—
119C22×SD16—1271no—
120C22×SD16—1271no—
121C22×D8—1271no—
122C2×#124—1271no—
123C2×#124D12711yescharacteristic, complemented
124#495—1278no—
125C2×C4×C8—1271no—
126C22×Q8—1271no—
127C22×Q8C2×D12711yescharacteristic
128C22×D4—1271no—
129C22×D4C2×D12711yescharacteristic
130C2×Q16—1274no—
131C2×Q16—2544no—
132C2×SD16—1274no—
133C2×SD16—1274no—
134C2×SD16—2544no—
135C2×SD16—2544no—
136C2×D8—1274no—
137C2×D8—2544no—
138C22×C8—1271no—
139C22×C8—1271no—
140#124—22no—
141#124—1274no—
142#124—2542no—
143#124C2×D12714yescomplemented
144C2×C22⋊C4—22no—
145C2×C22⋊C4—2542no—
146C2×C42C2×D12711yescharacteristic
147C4×C8—1274no—
148C24—21no—
149C24—2541no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—1272no—
154C2×Q8—2542no—
155C2×Q8—2542no—
156C2×Q8—2541no—
157C2×Q8C22×D12712yes—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—21no—
161C2×D4—1272no—
162C2×D4—2542no—
163C2×D4—2542no—
164C2×D4—2541no—
165C2×D4C22×D12712yes—
166C22×C4—21no—
167C22×C4—2541no—
168C22×C4C22⋊D12712yes—
169C22×C4C22×D12711yescharacteristic
170Q16—2548no—
171SD16—2548no—
172SD16—2548no—
173D8—2548no—
174C2×C8—1274no—
175C2×C8—1274no—
176C2×C8—1272no—
177C2×C8—1272no—
178C22⋊C4—28no—
179C22⋊C4—2548no—
180C42C22⋊D12712yes—
181C42C22×D12712yes—
182C23—22no—
183C23—42no—
184C23—44no—
185C23—5082no—
186C23—5084no—
187C23—2542no—
188C23C2×C22⋊D12711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—5082no—
193Q8—2542no—
194Q8—2542no—
195D4—22no—
196D4—22no—
197D4—42no—
198D4—5082no—
199D4—2542no—
200D4—2542no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—5082no—
205C2×C4—2542no—
206C2×C4D4×D12712yes—
207C2×C4D4×D12712yes—
208C2×C4C2×C22⋊D12714yes—
209C2×C4C2×C22⋊D12711yescharacteristic
210C2×C4C23×D12711yescharacteristic
211C8—1274no—
212C8—1274no—
213C22—44no—
214C22—42no—
215C22—84no—
216C22—10164no—
217C22—5084no—
218C22—5082no—
219C22#1686120512yes—
220C22#1686122914yes—
221C22C22×C22⋊D12711yescharacteristic
222C4—24no—
223C4—42no—
224C4—5082no—
225C4#1686120512yes—
226C4C2×D4×D12711yescharacteristic
227C4C2×D4×D12711yescharacteristic
228C2—82no—
229C2—10162no—
230C2#2255025814yesdirect product factor
231C2C2×#1686120511yescharacteristic
232C2C2×#1686122912yes—
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#124×C127C2
socleC23×C127C2×D4
prime radicalC22×#14523233C2
hypercenterC2×#124D127
hypocenterC127C2×#495
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
2. C22×#14523236C2semidirect
3. C22×#14523235C2semidirect
5. C2×#16861197C2semidirect
6. #22550258C2direct (N × H)
7. C2×C4×C127⋊C8C2semidirect
8. C2×#124×C127C2semidirect
12. C2×#14523238C22semidirect
14. C2×#14523236C22semidirect
16. C2×#14523235C22semidirect
22. #16861197C22semidirect
24. C4×C127⋊C8C22semidirect
28. #124×C127C22semidirect
112. C2×C127#495semidirect
117. C127C2×#495semidirect
123. C2×#124D127semidirect
143. #124C2×D127semidirect
230. C2#22550258direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232382
C22×#145232362
C22×#145232352
C22×#145232332
C2×#168611972
#225502582
C2×C4×C127⋊C82
C2×#124×C1272
C2×#495127