Finite Group Databasebeta

← #30642045

Subgroups — #30642045

Order 16256.

1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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×#16861177C212yesmaximal, complemented
2C22×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
3#22549787C212yesmaximal, complemented
4#22549784C212yesmaximal, complemented
5#22549756C212yesmaximal, complemented
6#22549755C214yesmaximal, complemented
7#22549739C211yesmaximal, characteristic, complemented
8C22×#41×C127C211yesmaximal, characteristic, complemented
9C24×D127C2211yescharacteristic
10C22×C22⋊D127C2214yes—
11C23×Dic127C2211yescharacteristic
12#16861177—28no—
13C2×C22⋊Dic127C2212yes—
14C2×C22⋊Dic127C2214yescomplemented
15C2×(C2×D127)⋊C4—24no—
16C2×(C2×D127)⋊C4C2212yes—
17C2×(C2×D127)⋊C4C2214yes—
18C2×(C2×D127)⋊C4C2212yes—
19C127⋊#95C2212yescomplemented
20C127⋊#95C2214yescomplemented
21C127⋊#95C2212yescomplemented
22C2×#41×C127—28no—
23C2×#41×C127—24no—
24C22×Q8×C127C2211yescharacteristic, complemented
25C22×D4×C127C2212yes—
26C22×D4×C127C2211yescharacteristic, complemented
27C23×C4×C127C2211yescharacteristic
28C23×C4×C127C2212yescomplemented
29C23×D127—24no—
30C23×D127—22no—
31C23×D127—24no—
32C23×D127—42no—
33C23×D127C2312yes—
34C2×C22⋊D127—28no—
35C2×C22⋊D127—28no—
36C2×C22⋊D127—44no—
37C22×Dic127—22no—
38C22×Dic127—42no—
39C22×Dic127C2312yes—
40C22×Dic127C2314yes—
41C22⋊Dic127—24no—
42C22⋊Dic127—44no—
43(C2×D127)⋊C4—24no—
44(C2×D127)⋊C4—24no—
45(C2×D127)⋊C4—28no—
46C24×C127C2312yes—
47C24×C127C2311yescharacteristic
48#41×C127—232no—
49C2×Q8×C127—24no—
50C2×Q8×C127—22no—
51C2×D4×C127—24no—
52C2×D4×C127—24no—
53C2×D4×C127—24no—
54C2×D4×C127—22no—
55C2×D4×C127D418yescomplemented
56C22×C4×C127—24no—
57C22×C4×C127—24no—
58C22×C4×C127—24no—
59C22×C4×C127—24no—
60C22×C4×C127D414yescomplemented
61C22×C4×C127C2311yescharacteristic
62C22×C4×C127C2311yescharacteristic
63C22×C4×C127C2312yes—
64C22×D127—28no—
65C22×D127—22no—
66C22×D127—24no—
67C22×D127—22no—
68C22×D127—24no—
69C22×D127—416no—
70C22×D127—44no—
71C22×D127—44no—
72C22×D127#4114yescomplemented
73C22⋊D127—416no—
74C2×Dic127—22no—
75C2×Dic127—24no—
76C2×Dic127—24no—
77C2×Dic127—24no—
78C2×Dic127—24no—
79C2×Dic127—44no—
80C23×C127—22no—
81C23×C127—22no—
82C23×C127—22no—
83C23×C127—22no—
84C23×C127—24no—
85C23×C127—24no—
86C23×C127C2×D414yes—
87C23×C127C2×D414yes—
88C23×C127C2×D412yes—
89C23×C127C2411yescharacteristic
90Q8×C127—28no—
91D4×C127—28no—
92D4×C127—216no—
93C2×C4×C127—216no—
94C2×C4×C127—28no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127C2×D412yes—
100C2×C4×C127C2×D414yes—
101C2×C4×C127C2×D412yes—
102C22×C127—22no—
103C22×C127—22no—
104C22×C127—28no—
105C22×C127—24no—
106C22×C127—28no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—44no—
110C22×C127#12014yes—
111C22×C127C22×D411yescharacteristic
112C22×C127C22×D412yes—
113C22×C127C2×#4112yes—
114C22×C127H(5,𝔽2)12yes—
115C2×D127—24no—
116C2×D127—28no—
117C2×D127—416no—
118C2×D127—42no—
119C2×D127—42no—
120C2×D127—44no—
121C4×C127—22no—
122C4×C127—24no—
123C4×C127—22no—
124Dic127—42no—
125Dic127—44no—
126C2×C127—24no—
127C2×C127—44no—
128C2×C127C2×#12011yescharacteristic
129C2×C127#52511yescharacteristic
130C2×C127#53511yescharacteristic
131C2×C127#54614yes—
132D127—44no—
133#2404—1271nomaximal, Sylow
134C127#240411yescharacteristic, complemented, Sylow
135C22×#41D12711yescharacteristic, complemented
136C2×#120—1272no—
137C22×C22⋊C4—1271no—
138#393—1272no—
139#392—1272no—
140#386—1272no—
141#386—1274no—
142#381—1271no—
143C25—1271no—
144C2×#41—28no—
145C2×#41—24no—
146C22×Q8C2×D12711yescharacteristic, complemented
147C22×D4—1274no—
148C22×D4C2×D12712yes—
149C22×D4C2×D12711yescharacteristic, complemented
150C23×C4—1271no—
151C23×C4C2×D12711yescharacteristic
152C23×C4C2×D12712yescomplemented
153#120—2548no—
154C2×C22⋊C4—1274no—
155C2×C22⋊C4—1272no—
156C2×C22⋊C4—1272no—
157C2×C22⋊C4—1272no—
158C2×C22⋊C4—1274no—
159C2×C22⋊C4—2544no—
160#95—1272no—
161#95—1272no—
162#95—1274no—
163C24—1272no—
164C24—5082no—
165C24—2544no—
166C24—2544no—
167C24—2542no—
168C24C22×D12712yes—
169C24C22×D12711yescharacteristic
170#41—232no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—5084no—
178C2×D4—2548no—
179C2×D4—2548no—
180C2×D4C22⋊D12718yescomplemented
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—1274no—
186C22×C4—1272no—
187C22×C4—5082no—
188C22×C4—2542no—
189C22×C4C22⋊D12714yescomplemented
190C22×C4C22×D12711yescharacteristic
191C22×C4C22×D12711yescharacteristic
192C22×C4C22×D12712yes—
193C22⋊C4—5084no—
194C22⋊C4—2544no—
195C22⋊C4—2544no—
196C22⋊C4—2544no—
197C22⋊C4—2548no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—1274no—
205C23—5084no—
206C23—50816no—
207C23—5084no—
208C23—2548no—
209C23—2542no—
210C23—2544no—
211C23—2542no—
212C23—2544no—
213C23C2×C22⋊D12714yes—
214C23C2×C22⋊D12714yes—
215C23C2×C22⋊D12712yes—
216C23C23×D12711yescharacteristic
217Q8—28no—
218D4—28no—
219D4—216no—
220D4—50816no—
221C2×C4—216no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—28no—
227C2×C4—5084no—
228C2×C4—2542no—
229C2×C4—2544no—
230C2×C4—2544no—
231C2×C4—2544no—
232C2×C4—2544no—
233C2×C4C2×C22⋊D12712yes—
234C2×C4C2×C22⋊D12714yes—
235C2×C4C2×C22⋊D12712yes—
236C22—22no—
237C22—22no—
238C22—28no—
239C22—24no—
240C22—28no—
241C22—42no—
242C22—42no—
243C22—44no—
244C22—50816no—
245C22—5082no—
246C22—5082no—
247C22—5084no—
248C22—2544no—
249C22—2548no—
250C22#1686117814yes—
251C22C22×C22⋊D12711yescharacteristic
252C22C22×C22⋊D12712yes—
253C22#41×D12712yes—
254C22#1686125512yes—
255C4—22no—
256C4—24no—
257C4—22no—
258C4—5082no—
259C4—5084no—
260C2—24no—
261C2—44no—
262C2—5084no—
263C2C2×#1686117811yescharacteristic
264C2#2255038311yescharacteristic
265C2#2255044011yescharacteristic
266C2#2255051414yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC22×#41×C127C2
hypercenterC22×#41D127
hypocenterC127#2404
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861177C2semidirect
2. C22×(C2×D127)⋊C4C2semidirect
3. #22549787C2semidirect
4. #22549784C2semidirect
5. #22549756C2semidirect
6. #22549755C2semidirect
7. #22549739C2semidirect
8. C22×#41×C127C2semidirect
14. C2×C22⋊Dic127C22semidirect
19. C127⋊#95C22semidirect
20. C127⋊#95C22semidirect
21. C127⋊#95C22semidirect
24. C22×Q8×C127C22semidirect
26. C22×D4×C127C22semidirect
28. C23×C4×C127C22semidirect
55. C2×D4×C127D4semidirect
60. C22×C4×C127D4semidirect
72. C22×D127#41semidirect
134. C127#2404semidirect
135. C22×#41D127semidirect
146. C22×Q8C2×D127semidirect
149. C22×D4C2×D127semidirect
152. C23×C4C2×D127semidirect
180. C2×D4C22⋊D127semidirect
189. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611772
C22×(C2×D127)⋊C42
#225497872
#225497842
#225497562
#225497552
#225497392
C22×#41×C1272
#2404127