Finite Group Databasebeta

← C2×#22550069 #30648140

Subgroups — C2×#22550069 #30648140

Order 16256.

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

normal subgroups only

This group has 648 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×#14523233C211yesmaximal, characteristic, complemented
2C2×#16861205C211yesmaximal, characteristic, complemented
3C2×D4×Dic127C211yesmaximal, characteristic, complemented
4#22550069C218yesmaximal, direct product factor
5C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
6C2×#16861131C211yesmaximal, characteristic, complemented
7C2×#16861130C211yesmaximal, characteristic, complemented
8C22×D8×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C22×C22⋊D127—21no—
12C23×Dic127—21no—
13C2×#14523233—24no—
14C2×#14523233C2214yescomplemented
15C22×C127⋊C8C2211yescharacteristic
16#16861205—22no—
17#16861205C2214yescomplemented
18D4×Dic127—22no—
19D4×Dic127C2214yescomplemented
20C2×C4⋊Dic127C2211yescharacteristic
21C2×C22⋊Dic127—21no—
22C2×C4×Dic127C2211yescharacteristic
23Dic127⋊C8C2214yescomplemented
24#16861131C2214yescomplemented
25#16861130C2214yescomplemented
26C22×D4×C127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28C2×D8×C127—24no—
29C2×D8×C127C2214yescomplemented
30C22×C8×C127C2211yescharacteristic
31C23×D127—21no—
32C2×D508—22no—
33C2×D508—21no—
34C2×D508—22no—
35C2×D508C2312yes—
36C2×C22⋊D127—24no—
37C2×C22⋊D127—24no—
38C2×C22⋊D127—44no—
39C2×C22⋊D127—44no—
40C22×Dic127—24no—
41C22×Dic127—21no—
42C22×Dic127—44no—
43C22×Dic127D411yescharacteristic
44C22×Dic127D411yescharacteristic
45#14523233—28no—
46C2×C127⋊C8—22no—
47C2×C127⋊C8C2312yes—
48C4⋊Dic127—21no—
49C4⋊Dic127C2312yes—
50C22⋊Dic127—24no—
51C4×Dic127D412yes—
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—22no—
68C22×D127—42no—
69C22×D127—42no—
70C22×D127—42no—
71D508—22no—
72D508—22no—
73D508—42no—
74C22⋊D127—48no—
75C22⋊D127—48no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—42no—
79C2×Dic127—48no—
80C2×Dic127D814yescomplemented
81C2×Dic127C2×D412yes—
82C2×Dic127C2×D412yes—
83C127⋊C8—22no—
84C23×C127—24no—
85C23×C127—22no—
86C23×C127—22no—
87C23×C127—42no—
88C23×C127—42no—
89C23×C127—42no—
90C23×C127—42no—
91C23×C127—42no—
92C23×C127C2×D411yescharacteristic
93D4×C127—24no—
94D4×C127—22no—
95D4×C127—22no—
96D4×C127—22no—
97D4×C127—42no—
98D4×C127C2×D414yescomplemented
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D412yes—
101C2×C4×C127#4112yes—
102C2×C4×C127C2411yescharacteristic
103C8×C127—22no—
104C22×C127—24no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—48no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127—42no—
111C22×C127—84no—
112C22×C127#12112yes—
113C22×C127C2×D812yes—
114C22×C127AGL(1,ℤ/8)12yes—
115C22×C127C22×D411yescharacteristic
116C2×D127—42no—
117C2×D127—42no—
118C2×D127—42no—
119C2×D127—84no—
120C4×C127#12112yes—
121C4×C127C22×D411yescharacteristic
122C4×C127C2×#4111yescharacteristic
123Dic127—22no—
124Dic127—42no—
125Dic127C2×D814yescomplemented
126C2×C127—44no—
127C2×C127—82no—
128C2×C127#45914yescomplemented
129C2×C127C2×#12111yescharacteristic
130C2×C127C22×D811yescharacteristic
131C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
132D127—82no—
133C2×#459—1271nomaximal, Sylow
134C127C2×#45911yescharacteristic, complemented, Sylow
135C22×D8—1271no—
136C22×D8D12711yescharacteristic, complemented
137C2×#127—1271no—
138C2×C4×D4—1271no—
139#459—1278no—
140C2×C4⋊C8—1271no—
141C2×#102—1271no—
142C2×#102—1271no—
143C22×D4—1271no—
144C22×D4—2541no—
145C22×D4—2541no—
146C22×D4C2×D12711yescharacteristic
147C22×D4C2×D12711yescharacteristic
148C23×C4—2541no—
149C2×D8—24no—
150C2×D8—1274no—
151C2×D8—2544no—
152C2×D8C2×D12714yescomplemented
153C22×C8—1271no—
154C22×C8C2×D12711yescharacteristic
155#127—1274no—
156#127—2542no—
157C4×D4—1274no—
158C4×D4—2542no—
159C2×C4⋊C4—1271no—
160C2×C22⋊C4—2541no—
161C2×C42—1271no—
162C4⋊C8—1274no—
163#102—1274no—
164#102—1274no—
165C24—21no—
166C24—21no—
167C24—2541no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—1272no—
174C2×D4—5084no—
175C2×D4—5084no—
176C2×D4—2544no—
177C2×D4—2544no—
178C2×D4—2541no—
179C2×D4—2542no—
180C2×D4—2542no—
181C2×D4C22⋊D12714yescomplemented
182C2×D4C22×D12712yes—
183C2×D4C22×D12712yes—
184C22×C4—1271no—
185C22×C4—1271no—
186C22×C4—5084no—
187C22×C4—2544no—
188C22×C4—2541no—
189C22×C4C22×D12711yescharacteristic
190D8—28no—
191D8—2548no—
192C2×C8—22no—
193C2×C8—1272no—
194C2×C8—2542no—
195C2×C8C22×D12712yes—
196C4⋊C4—1272no—
197C4⋊C4—2541no—
198C22⋊C4—2544no—
199C42—1272no—
200C42—1272no—
201C23—22no—
202C23—24no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—5082no—
210C23—5082no—
211C23—5082no—
212C23—2542no—
213C23D4×D12711yescharacteristic
214D4—22no—
215D4—22no—
216D4—22no—
217D4—24no—
218D4—42no—
219D4—5088no—
220D4—5088no—
221D4—5082no—
222D4—2542no—
223D4—2542no—
224D4C2×C22⋊D12714yescomplemented
225C2×C4—1274no—
226C2×C4—1272no—
227C2×C4—1272no—
228C2×C4—5088no—
229C2×C4—5082no—
230C2×C4—2542no—
231C2×C4—2542no—
232C2×C4D4×D12711yescharacteristic
233C2×C4C2×C22⋊D12712yes—
234C2×C4#1452324712yes—
235C2×C4C23×D12711yescharacteristic
236C8—22no—
237C8—2542no—
238C22—24no—
239C22—48no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—84no—
246C22—10164no—
247C22—5082no—
248C22—5082no—
249C22—5082no—
250C22#1686118112yes—
251C22D8×D12712yes—
252C22#1686123312yes—
253C22C2×D4×D12711yescharacteristic
254C4—1274no—
255C4—5082no—
256C4—2542no—
257C4#1686118112yes—
258C4C22×C22⋊D12711yescharacteristic
259C4C2×#1452324711yescharacteristic
260C2—44no—
261C2—82no—
262C2—10162no—
263C2#2255006914yesdirect product factor
264C2C2×#1686118111yescharacteristic
265C2C2×D8×D12711yescharacteristic
266C2C2×#1686123311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC23×C127C2×D4
hypercenterC22×D8D127
hypocenterC127C2×#459
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×#14523233C2semidirect
2. C2×#16861205C2semidirect
3. C2×D4×Dic127C2semidirect
4. #22550069C2direct (N × H)
5. C2×Dic127⋊C8C2semidirect
6. C2×#16861131C2semidirect
7. C2×#16861130C2semidirect
8. C22×D8×C127C2semidirect
14. C2×#14523233C22semidirect
17. #16861205C22semidirect
19. D4×Dic127C22semidirect
23. Dic127⋊C8C22semidirect
24. #16861131C22semidirect
25. #16861130C22semidirect
29. C2×D8×C127C22semidirect
60. C2×D4×C127D4semidirect
80. C2×Dic127D8semidirect
98. D4×C127C2×D4semidirect
125. Dic127C2×D8semidirect
128. C2×C127#459semidirect
134. C127C2×#459semidirect
136. C22×D8D127semidirect
152. C2×D8C2×D127semidirect
181. C2×D4C22⋊D127semidirect
224. D4C2×C22⋊D127semidirect
263. C2#22550069direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168612052
C2×D4×Dic1272
#225500692
C2×Dic127⋊C82
C2×#168611312
C2×#168611302
C22×D8×C1272
C2×#459127