Finite Group Databasebeta

← #30642049

Subgroups — #30642049

Order 16256.

1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14523248C211yesmaximal, characteristic, complemented
2C2×#16861178C212yesmaximal, complemented
3#22549788C212yesmaximal, complemented
4#22549785C212yesmaximal, complemented
5#22549755C212yesmaximal, complemented
6#22549752C214yesmaximal, complemented
7#22549739C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—28no—
10C2×#14523248—24no—
11C22×Dic254C2211yescharacteristic, complemented
12C22×D508C2211yescharacteristic, complemented
13C22×C22⋊D127C2214yes—
14C22×C22⋊D127C2212yes—
15C22×C4×D127C2212yescomplemented
16#16861178—28no—
17C2×C22⋊Dic127C2214yes—
18C2×C22⋊Dic127C2212yes—
19C2×(C2×D127)⋊C4C2214yescomplemented
20C127⋊#95C2212yescomplemented
21C127⋊#95C2214yescomplemented
22C127⋊#95C2212yescomplemented
23C25×C127C2211yescharacteristic
24C23×C4×C127C2211yescharacteristic
25C23×C4×C127C2211yescharacteristic
26C2×C22⋊C4×C127—24no—
27C2×C22⋊C4×C127C2212yes—
28C2×C22⋊C4×C127C2212yes—
29C23×D127C2312yes—
30#14523248—232no—
31C2×Dic254—24no—
32C2×Dic254—22no—
33C2×D508—24no—
34C2×D508—22no—
35C2×C22⋊D127—28no—
36C2×C22⋊D127—24no—
37C2×C22⋊D127—24no—
38C2×C22⋊D127—28no—
39C2×C22⋊D127—44no—
40C2×C22⋊D127D418yescomplemented
41C22×Dic127C2314yes—
42C22×Dic127C2312yes—
43C2×C4×D127—24no—
44C2×C4×D127—24no—
45C2×C4×D127—24no—
46C22⋊Dic127—28no—
47C22⋊Dic127—24no—
48(C2×D127)⋊C4—44no—
49C24×C127—22no—
50C24×C127—24no—
51C24×C127—24no—
52C24×C127—42no—
53C24×C127C2312yes—
54C24×C127C2311yescharacteristic
55C22×C4×C127—24no—
56C22×C4×C127—22no—
57C22×C4×C127—42no—
58C22×C4×C127D414yescomplemented
59C22×C4×C127C2312yes—
60C22×C4×C127C2311yescharacteristic
61C22×C4×C127C2311yescharacteristic
62C22⋊C4×C127—24no—
63C22⋊C4×C127—24no—
64C22×D127—22no—
65C22×D127—22no—
66C22×D127—24no—
67C22×D127—24no—
68C22×D127C2×D414yes—
69Dic254—28no—
70D508—28no—
71C22⋊D127—216no—
72C22⋊D127—416no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—24no—
76C2×Dic127—24no—
77C2×Dic127—24no—
78C2×Dic127C2×D414yes—
79C4×D127—216no—
80C23×C127—22no—
81C23×C127—22no—
82C23×C127—28no—
83C23×C127—22no—
84C23×C127—24no—
85C23×C127—22no—
86C23×C127—24no—
87C23×C127—416no—
88C23×C127—44no—
89C23×C127—44no—
90C23×C127C2×D414yes—
91C23×C127C2×D412yes—
92C23×C127#4114yescomplemented
93C23×C127C2411yescharacteristic
94C2×C4×C127—28no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—24no—
99C2×C4×C127—44no—
100C2×C4×C127C2×D412yes—
101C2×C4×C127C2×D412yes—
102C22×C127—22no—
103C22×C127—22no—
104C22×C127—28no—
105C22×C127—24no—
106C22×C127—28no—
107C22×C127—416no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127—44no—
111C22×C127#12014yes—
112C22×C127C22×D412yes—
113C22×C127C22×D411yescharacteristic
114C22×C127C2×#4112yes—
115C22×C127H(5,𝔽2)12yes—
116C2×D127—24no—
117C2×D127—28no—
118C2×D127—42no—
119C2×D127—42no—
120C2×D127—44no—
121C4×C127—22no—
122C4×C127—22no—
123C4×C127—42no—
124Dic127—24no—
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×#41—1271no—
136C2×#120—1272no—
137C22×C22⋊C4D12711yescharacteristic, complemented
138#393—1272no—
139#392—1272no—
140#386—1272no—
141#386—1274no—
142#381—1271no—
143C25C2×D12711yescharacteristic
144C2×#41—2548no—
145C2×#41—2544no—
146C22×Q8—1271no—
147C22×D4—1272no—
148C22×D4—1274no—
149C22×D4—1271no—
150C23×C4—1272no—
151C23×C4C2×D12711yescharacteristic
152C23×C4C2×D12711yescharacteristic
153#120—2548no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—1274no—
156C2×C22⋊C4—1272no—
157C2×C22⋊C4—1274no—
158C2×C22⋊C4C2×D12712yes—
159C2×C22⋊C4C2×D12712yes—
160#95—1272no—
161#95—1272no—
162#95—1274no—
163C24—22no—
164C24—24no—
165C24—24no—
166C24—42no—
167C24—1272no—
168C24C22×D12712yes—
169C24C22×D12711yescharacteristic
170#41—25432no—
171C2×Q8—2544no—
172C2×Q8—2542no—
173C2×D4—1278no—
174C2×D4—5084no—
175C2×D4—2544no—
176C2×D4—2548no—
177C2×D4—2548no—
178C2×D4—2544no—
179C2×D4—2544no—
180C2×D4—2542no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—1272no—
185C22×C4—1274no—
186C22×C4—2544no—
187C22×C4—2544no—
188C22×C4—2544no—
189C22×C4C22⋊D12714yescomplemented
190C22×C4C22×D12712yes—
191C22×C4C22×D12711yescharacteristic
192C22×C4C22×D12711yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—5084no—
196C22⋊C4—2548no—
197C22⋊C4—2544no—
198C23—22no—
199C23—22no—
200C23—28no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—416no—
206C23—44no—
207C23—44no—
208C23—1274no—
209C23—2542no—
210C23—2542no—
211C23—2544no—
212C23—2544no—
213C23D4×D12714yes—
214C23C2×C22⋊D12712yes—
215C23#1452324814yescomplemented
216C23C23×D12711yescharacteristic
217Q8—2548no—
218D4—50816no—
219D4—2548no—
220D4—25416no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—28no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—1274no—
228C2×C4—25416no—
229C2×C4—2544no—
230C2×C4—2544no—
231C2×C4—2544no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4C2×C22⋊D12712yes—
235C2×C4C2×C22⋊D12712yes—
236C22—22no—
237C22—22no—
238C22—28no—
239C22—24no—
240C22—28no—
241C22—416no—
242C22—42no—
243C22—42no—
244C22—44no—
245C22—5082no—
246C22—5082no—
247C22—5084no—
248C22—2544no—
249C22—2548no—
250C22#1686117714yes—
251C22C2×D4×D12712yes—
252C22C22×C22⋊D12711yescharacteristic
253C22C2×#1452324812yes—
254C22#1686125412yes—
255C4—22no—
256C4—22no—
257C4—42no—
258C4—5084no—
259C4—2544no—
260C2—24no—
261C2—44no—
262C2—5084no—
263C2C2×#1686117711yescharacteristic
264C2#2255038711yescharacteristic
265C2#2255044311yescharacteristic
266C2#2255051614yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC22×C22⋊C4×C127C2
hypercenterC22×C22⋊C4D127
hypocenterC127#2404
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#14523248C2semidirect
2. C2×#16861178C2semidirect
3. #22549788C2semidirect
4. #22549785C2semidirect
5. #22549755C2semidirect
6. #22549752C2semidirect
7. #22549739C2semidirect
8. C22×C22⋊C4×C127C2semidirect
11. C22×Dic254C22semidirect
12. C22×D508C22semidirect
15. C22×C4×D127C22semidirect
19. C2×(C2×D127)⋊C4C22semidirect
20. C127⋊#95C22semidirect
21. C127⋊#95C22semidirect
22. C127⋊#95C22semidirect
40. C2×C22⋊D127D4semidirect
58. C22×C4×C127D4semidirect
92. C23×C127#41semidirect
134. C127#2404semidirect
137. C22×C22⋊C4D127semidirect
189. C22×C4C22⋊D127semidirect
215. C23#14523248semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168611782
#225497882
#225497852
#225497552
#225497522
#225497392
C22×C22⋊C4×C1272
#2404127