Finite Group Databasebeta

← C2×#22550264 #30649163

Subgroups — C2×#22550264 #30649163

Order 16256.

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

normal subgroups only

This group has 668 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×#14523237C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3C22×#14523233C211yesmaximal, characteristic, complemented
4C2×#16861205C211yesmaximal, characteristic, complemented
5C2×#16861196C211yesmaximal, characteristic, complemented
6#22550264C218yesmaximal, direct product factor
7C2×#16861119C211yesmaximal, characteristic, complemented
8C22×D8×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2211yescharacteristic
10C22×D508C2211yescharacteristic
11C22×C22⋊D127—22no—
12C2×#14523237—24no—
13C2×#14523237C2214yescomplemented
14C2×#14523236—24no—
15C2×#14523236C2214yescomplemented
16C2×#14523233—24no—
17C2×#14523233C2214yescomplemented
18C22×C127⋊C8C2211yescharacteristic
19#16861205—22no—
20#16861205C2214yescomplemented
21#16861196—22no—
22#16861196C2214yescomplemented
23C2×C22⋊Dic127—22no—
24C2×C4×Dic127C2211yescharacteristic
25#16861119C2214yescomplemented
26C22×D4×C127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28C2×D8×C127—24no—
29C2×D8×C127C2214yescomplemented
30C22×C8×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—
40C2×C22⋊D127—28no—
41C2×C22⋊D127—48no—
42C22×Dic127—21no—
43C22×Dic127D412yes—
44#14523237—28no—
45#14523236—28no—
46#14523233—28no—
47C2×C127⋊C8D414yescomplemented
48C2×C127⋊C8C2312yes—
49C22⋊Dic127—28no—
50C4×Dic127D412yes—
51C4×Dic127C2312yes—
52C24×C127—21no—
53C24×C127—21no—
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127—22no—
57C2×D4×C127—22no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127C2312yes—
61C2×D4×C127C2312yes—
62C22×C4×C127C2311yescharacteristic
63D8×C127—28no—
64C2×C8×C127D414yescomplemented
65C2×C8×C127C2312yes—
66C22×D127—22no—
67C22×D127—42no—
68C22×D127—44no—
69Dic254—22no—
70Dic254—22no—
71Dic254—42no—
72D508—22no—
73D508—22no—
74D508—42no—
75C22⋊D127—416no—
76C2×Dic127—24no—
77C2×Dic127—22no—
78C2×Dic127—42no—
79C2×Dic127C2×D414yes—
80C127⋊C8C2×D414yescomplemented
81C23×C127—22no—
82C23×C127—22no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127—44no—
86C23×C127—44no—
87C23×C127C2×D411yescharacteristic
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—42no—
93D4×C127—42no—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D412yes—
97C2×C4×C127C2411yescharacteristic
98C8×C127C2×D414yescomplemented
99C22×C127—42no—
100C22×C127—42no—
101C22×C127—44no—
102C22×C127—44no—
103C22×C127—84no—
104C22×C127—84no—
105C22×C127#12712yes—
106C22×C127AGL(1,ℤ/8)14yes—
107C22×C127C22×D411yescharacteristic
108C2×D127—42no—
109C2×D127—44no—
110C2×D127—84no—
111C4×C127#12712yes—
112C4×C127C22×D411yescharacteristic
113C4×C127C22×D411yescharacteristic
114Dic127—24no—
115Dic127—42no—
116C2×C127—82no—
117C2×C127—82no—
118C2×C127#49614yescomplemented
119C2×C127C2×#12711yescharacteristic
120C2×C127C2×AGL(1,ℤ/8)12yes—
121D127—82no—
122C2×#496—1271nomaximal, Sylow
123C127C2×#49611yescharacteristic, complemented, Sylow
124C22×SD16—1271no—
125C22×SD16—1271no—
126C22×D8—1271no—
127C22×D8D12711yescharacteristic, complemented
128C2×#127—1271no—
129C2×#124—1271no—
130#496—1278no—
131C2×#97—1271no—
132C22×Q8—1271no—
133C22×D4—1271no—
134C22×D4—2542no—
135C22×D4C2×D12711yescharacteristic
136C22×D4C2×D12711yescharacteristic
137C2×SD16—1274no—
138C2×SD16—1274no—
139C2×SD16—2544no—
140C2×SD16—2544no—
141C2×D8—24no—
142C2×D8—1274no—
143C2×D8—2544no—
144C2×D8C2×D12714yescomplemented
145C22×C8—1271no—
146C22×C8C2×D12711yescharacteristic
147#127—1274no—
148#127—2542no—
149#124—1274no—
150#124—2542no—
151C2×C22⋊C4—2542no—
152C2×C42—1271no—
153#97—1274no—
154C24—21no—
155C24—21no—
156C24—2541no—
157C2×Q8—1272no—
158C2×Q8—2542no—
159C2×Q8—2542no—
160C2×Q8—2541no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—1272no—
168C2×D4—5088no—
169C2×D4—2548no—
170C2×D4—2541no—
171C2×D4—2542no—
172C2×D4—2542no—
173C2×D4C22×D12712yes—
174C2×D4C22×D12712yes—
175C22×C4—1272no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178SD16—2548no—
179SD16—2548no—
180D8—28no—
181D8—2548no—
182C2×C8—1274no—
183C2×C8—1272no—
184C2×C8C22⋊D12714yescomplemented
185C2×C8C22×D12712yes—
186C22⋊C4—2548no—
187C42—1272no—
188C42—1272no—
189C23—22no—
190C23—22no—
191C23—42no—
192C23—42no—
193C23—44no—
194C23—44no—
195C23—5082no—
196C23—5084no—
197C23—2542no—
198C23D4×D12711yescharacteristic
199Q8—5082no—
200Q8—2542no—
201Q8—2542no—
202D4—22no—
203D4—22no—
204D4—22no—
205D4—22no—
206D4—42no—
207D4—42no—
208D4—50816no—
209D4—5082no—
210D4—2542no—
211D4—2542no—
212C2×C4—1274no—
213C2×C4—5082no—
214C2×C4—2542no—
215C2×C4—2544no—
216C2×C4D4×D12711yescharacteristic
217C2×C4D4×D12712yes—
218C2×C4C2×C22⋊D12712yes—
219C2×C4C23×D12711yescharacteristic
220C8—1274no—
221C8C2×C22⋊D12714yescomplemented
222C22—44no—
223C22—44no—
224C22—42no—
225C22—42no—
226C22—84no—
227C22—84no—
228C22—10164no—
229C22—5084no—
230C22—5082no—
231C22#1686120512yes—
232C22#1686123314yes—
233C22C2×D4×D12711yescharacteristic
234C4—5082no—
235C4—2544no—
236C4#1686120512yes—
237C4C2×D4×D12711yescharacteristic
238C4C22×C22⋊D12711yescharacteristic
239C2—82no—
240C2—82no—
241C2—10162no—
242C2#2255026414yesdirect product factor
243C2C2×#1686120511yescharacteristic
244C2C2×#1686123312yes—
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC23×C127C2×D4
hypercenterC22×D8D127
hypocenterC127C2×#496
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14523237C2semidirect
2. C22×#14523236C2semidirect
3. C22×#14523233C2semidirect
4. C2×#16861205C2semidirect
5. C2×#16861196C2semidirect
6. #22550264C2direct (N × H)
7. C2×#16861119C2semidirect
8. C22×D8×C127C2semidirect
13. C2×#14523237C22semidirect
15. C2×#14523236C22semidirect
17. C2×#14523233C22semidirect
20. #16861205C22semidirect
22. #16861196C22semidirect
25. #16861119C22semidirect
29. C2×D8×C127C22semidirect
47. C2×C127⋊C8D4semidirect
64. C2×C8×C127D4semidirect
80. C127⋊C8C2×D4semidirect
98. C8×C127C2×D4semidirect
118. C2×C127#496semidirect
123. C127C2×#496semidirect
127. C22×D8D127semidirect
144. C2×D8C2×D127semidirect
184. C2×C8C22⋊D127semidirect
221. C8C2×C22⋊D127semidirect
242. C2#22550264direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232372
C22×#145232362
C22×#145232332
C2×#168612052
C2×#168611962
#225502642
C2×#168611192
C22×D8×C1272
C2×#496127