Finite Group Databasebeta

← C2×#22550038 #30647939

Subgroups — C2×#22550038 #30647939

Order 16256.

760 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 181 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
3C2×#16861188C211yesmaximal, characteristic, complemented
4#22550038C218yesmaximal, direct product factor
5C2×#16861133C211yesmaximal, characteristic, complemented
6C2×#16861130C211yesmaximal, characteristic, complemented
7C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
8C22×#41×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10C2×#14523238—24no—
11C2×#14523238C2214yescomplemented
12C2×#14523236—24no—
13C2×#14523236C2214yescomplemented
14C22×C127⋊C8C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16#16861188—22no—
17#16861188C2214yescomplemented
18C2×C4⋊Dic127C2211yescharacteristic
19C2×Dic127⋊C4—21no—
20C2×C22⋊Dic127—21no—
21#16861133C2214yescomplemented
22#16861130C2214yescomplemented
23C22⋊(C127⋊C8)C2214yescomplemented
24C2×#41×C127—24no—
25C2×#41×C127—24no—
26C2×#41×C127—22no—
27C2×#41×C127C2214yes—
28C22×Q8×C127C2211yescharacteristic
29C22×D4×C127—21no—
30C22×D4×C127C2211yescharacteristic
31C23×C4×C127—21no—
32C23×C4×C127C2211yescharacteristic
33C2×Dic254—21no—
34C2×Dic254—22no—
35C2×Dic254—22no—
36C2×Dic254C2312yes—
37C22×Dic127—21no—
38C22×Dic127—21no—
39#14523238—28no—
40#14523236—28no—
41C2×C127⋊C8—22no—
42C2×C127⋊C8—22no—
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127—21no—
46C4⋊Dic127C2312yes—
47Dic127⋊C4—24no—
48C22⋊Dic127—24no—
49C24×C127—21no—
50C24×C127D411yescharacteristic
51#41×C127—216no—
52#41×C127—28no—
53#41×C127—28no—
54C2×Q8×C127—22no—
55C2×Q8×C127—21no—
56C2×Q8×C127D414yes—
57C2×Q8×C127C2312yes—
58C2×D4×C127—28no—
59C2×D4×C127—24no—
60C2×D4×C127—21no—
61C2×D4×C127—22no—
62C2×D4×C127D414yes—
63C2×D4×C127C2312yes—
64C22×C4×C127—28no—
65C22×C4×C127—21no—
66C22×C4×C127—24no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127D412yes—
70C22×C4×C127D411yescharacteristic
71C22×C4×C127C2312yes—
72C22×C4×C127C2311yescharacteristic
73Dic254—22no—
74Dic254—22no—
75Dic254—42no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—42no—
79C2×Dic127—42no—
80C127⋊C8—22no—
81C127⋊C8—22no—
82C23×C127—22no—
83C23×C127—24no—
84C23×C127—22no—
85C23×C127—22no—
86C23×C127—22no—
87C23×C127—42no—
88C23×C127—42no—
89C23×C127C2×D412yes—
90C23×C127C2×D411yescharacteristic
91Q8×C127—24no—
92Q8×C127—22no—
93Q8×C127C2×D414yes—
94D4×C127—216no—
95D4×C127—24no—
96D4×C127—22no—
97D4×C127C2×D414yes—
98C2×C4×C127—216no—
99C2×C4×C127—22no—
100C2×C4×C127—24no—
101C2×C4×C127—22no—
102C2×C4×C127—24no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D412yes—
109C2×C4×C127C2411yescharacteristic
110C22×C127—22no—
111C22×C127—24no—
112C22×C127—22no—
113C22×C127—22no—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127—44no—
117C22×C127—48no—
118C22×C127#12012yes—
119C22×C127#13512yes—
120C22×C127#13712yes—
121C22×C127C22×D411yescharacteristic
122C4×C127—22no—
123C4×C127—24no—
124C4×C127#12012yes—
125C4×C127C22×D411yescharacteristic
126C4×C127C22×D411yescharacteristic
127Dic127—42no—
128Dic127—42no—
129C2×C127—42no—
130C2×C127—44no—
131C2×C127#45214yescomplemented
132C2×C127C2×#12011yescharacteristic
133C2×C127C2×#13511yescharacteristic
134C2×C127C2×#13711yescharacteristic
135C2×#452—1271nomaximal, Sylow
136C127C2×#45211yescharacteristic, complemented, Sylow
137C22×#41D12711yescharacteristic
138C22×Q16—1271no—
139C22×SD16—1271no—
140C2×C22⋊Q8—1271no—
141#452—1278no—
142C2×#103—1271no—
143C2×#102—1271no—
144C2×C22⋊C8—1271no—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—22no—
148C2×#41C2×D12714yes—
149C22×Q8—1271no—
150C22×Q8C2×D12711yescharacteristic
151C22×D4—21no—
152C22×D4C2×D12711yescharacteristic
153C23×C4—21no—
154C23×C4C2×D12711yescharacteristic
155C2×Q16—1274no—
156C2×Q16—2544no—
157C2×SD16—1274no—
158C2×SD16—2544no—
159C22×C8—1271no—
160C22×C8—1271no—
161C22⋊Q8—1274no—
162C22⋊Q8—2542no—
163C2×C4⋊C4—1271no—
164C2×C4⋊C4—2541no—
165C2×C22⋊C4—2541no—
166#103—1274no—
167#102—1274no—
168C22⋊C8—1274no—
169C24—21no—
170C24C22⋊D12711yescharacteristic
171#41—216no—
172#41—28no—
173#41—28no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—1272no—
177C2×Q8—2542no—
178C2×Q8—2542no—
179C2×Q8—2541no—
180C2×Q8C22⋊D12714yes—
181C2×Q8C22×D12712yes—
182C2×D4—28no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4C22⋊D12714yes—
187C2×D4C22×D12712yes—
188C22×C4—28no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—2541no—
194C22×C4—2541no—
195C22×C4C22⋊D12712yes—
196C22×C4C22⋊D12711yescharacteristic
197C22×C4C22×D12712yes—
198C22×C4C22×D12711yescharacteristic
199Q16—2548no—
200SD16—2548no—
201C2×C8—1272no—
202C2×C8—1272no—
203C2×C8—2542no—
204C2×C8—2542no—
205C4⋊C4—1272no—
206C4⋊C4—2544no—
207C4⋊C4—2541no—
208C22⋊C4—2544no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—24no—
214C23—42no—
215C23—42no—
216C23C2×C22⋊D12712yes—
217C23C2×C22⋊D12711yescharacteristic
218Q8—22no—
219Q8—24no—
220Q8—5082no—
221Q8—2542no—
222Q8—2542no—
223Q8C2×C22⋊D12714yes—
224D4—216no—
225D4—22no—
226D4—24no—
227D4C2×C22⋊D12714yes—
228C2×C4—216no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—5082no—
236C2×C4—5082no—
237C2×C4—2542no—
238C2×C4—2542no—
239C2×C4C2×C22⋊D12712yes—
240C2×C4C2×C22⋊D12712yes—
241C2×C4C2×C22⋊D12712yes—
242C2×C4C2×C22⋊D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—2542no—
245C8—2542no—
246C22—22no—
247C22—24no—
248C22—22no—
249C22—22no—
250C22—44no—
251C22—48no—
252C22—42no—
253C22—42no—
254C22#1686117812yes—
255C22#1686122912yes—
256C22#1686123912yes—
257C22C22×C22⋊D12711yescharacteristic
258C4—22no—
259C4—24no—
260C4—5082no—
261C4—5082no—
262C4#1686117812yes—
263C4C22×C22⋊D12711yescharacteristic
264C4C22×C22⋊D12711yescharacteristic
265C2—42no—
266C2—44no—
267C2#2255003814yesdirect product factor
268C2C2×#1686117811yescharacteristic
269C2C2×#1686122911yescharacteristic
270C2C2×#1686123911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC22×#41×C127C2
socleC23×C127C2×D4
hypercenterC22×#41D127
hypocenterC127C2×#452
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
2. C22×#14523236C2semidirect
3. C2×#16861188C2semidirect
4. #22550038C2direct (N × H)
5. C2×#16861133C2semidirect
6. C2×#16861130C2semidirect
7. C2×C22⋊(C127⋊C8)C2semidirect
11. C2×#14523238C22semidirect
13. C2×#14523236C22semidirect
17. #16861188C22semidirect
21. #16861133C22semidirect
22. #16861130C22semidirect
23. C22⋊(C127⋊C8)C22semidirect
131. C2×C127#452semidirect
136. C127C2×#452semidirect
267. C2#22550038direct (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
C2×#168611882
#225500382
C2×#168611332
C2×#168611302
C2×C22⋊(C127⋊C8)2
C22×#41×C1272
C2×#452127