Finite Group Databasebeta

← #30647073

Subgroups — #30647073

Order 16256.

924 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 201 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×#16861205C211yesmaximal, characteristic, complemented
2C2×#16861183C211yesmaximal, characteristic, complemented
3C2×#16861181C214yesmaximal, complemented
4C22×C4×Dic127C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic, complemented
6#22549777C211yesmaximal, characteristic, complemented
7#22549776C214yesmaximal, complemented
8#22549745C211yesmaximal, characteristic, complemented
9C2×#121×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C22⋊D127C2214yes—
12C22×C22⋊D127C2212yes—
13C23×Dic127C2212yes—
14#16861205—24no—
15#16861183—24no—
16#16861181—216no—
17C2×C4⋊Dic127C2211yescharacteristic
18C2×Dic127⋊C4C2214yes—
19C2×C22⋊Dic127C2214yes—
20C2×(C2×D127)⋊C4C2212yes—
21C2×(C2×D127)⋊C4C2214yes—
22C2×C4×Dic127—24no—
23C2×C4×Dic127C2212yes—
24C2×C4×Dic127C2211yescharacteristic
25C2×C4×Dic127C2211yescharacteristic
26C22×D4×C127C2211yescharacteristic
27C22×D4×C127C2212yes—
28C23×C4×C127C2211yescharacteristic
29#121×C127—24no—
30C2×C4⋊C4×C127C2211yescharacteristic
31C2×C22⋊C4×C127C2212yes—
32C23×D127C2312yes—
33C2×D508—22no—
34C2×D508—22no—
35C2×D508—22no—
36C2×C22⋊D127—28no—
37C2×C22⋊D127—28no—
38C2×C22⋊D127—28no—
39C2×C22⋊D127—28no—
40C2×C22⋊D127—24no—
41C22×Dic127—24no—
42C22×Dic127—24no—
43C22×Dic127D418yescomplemented
44C22×Dic127C2312yes—
45C22×Dic127C2312yes—
46C22×Dic127C2312yes—
47C4⋊Dic127—22no—
48Dic127⋊C4—28no—
49C22⋊Dic127—28no—
50(C2×D127)⋊C4—28no—
51(C2×D127)⋊C4—24no—
52C4×Dic127—24no—
53C4×Dic127—22no—
54C4×Dic127D414yescomplemented
55C24×C127C2312yes—
56C24×C127C2311yescharacteristic
57C2×D4×C127—24no—
58C2×D4×C127—28no—
59C2×D4×C127—22no—
60C2×D4×C127—22no—
61C2×D4×C127—22no—
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127D414yescomplemented
65C22×C4×C127C2312yes—
66C22×C4×C127C2311yescharacteristic
67C22×C4×C127C2311yescharacteristic
68C4⋊C4×C127—22no—
69C22⋊C4×C127—24no—
70C22×D127—24no—
71C22×D127—22no—
72C22×D127—22no—
73C22×D127—24no—
74C22×D127—22no—
75D508—44no—
76C22⋊D127—48no—
77C22⋊D127—416no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—216no—
81C2×Dic127—22no—
82C2×Dic127—24no—
83C2×Dic127C2×D414yes—
84C2×Dic127C2×D414yes—
85C2×Dic127C2×D414yes—
86C2×Dic127#4114yes—
87C23×C127—21no—
88C23×C127—21no—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—24no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—24no—
95C23×C127C2×D414yes—
96C23×C127C2×D412yes—
97C23×C127C2411yescharacteristic
98D4×C127—48no—
99D4×C127—44no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—28no—
103C2×C4×C127—22no—
104C2×C4×C127—24no—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D412yes—
108C2×C4×C127#4112yes—
109C22×C127—22no—
110C22×C127—24no—
111C22×C127—24no—
112C22×C127—22no—
113C22×C127—42no—
114C22×C127—44no—
115C22×C127—42no—
116C22×C127—44no—
117C22×C127—42no—
118C22×C127#12714yescomplemented
119C22×C127C22×D411yescharacteristic
120C22×C127C22×D412yes—
121C22×C127C22×D411yescharacteristic
122C22×C127C2×#4112yes—
123C22×C127C2×#4111yescharacteristic
124C2×D127—44no—
125C2×D127—42no—
126C2×D127—42no—
127C2×D127—44no—
128C2×D127—42no—
129C4×C127—22no—
130C4×C127—42no—
131C4×C127#12114yescomplemented
132Dic127—24no—
133Dic127—42no—
134Dic127#12118yescomplemented
135C2×C127—24no—
136C2×C127—82no—
137C2×C127C2×#12112yes—
138C2×C127C2×#12111yescharacteristic
139C2×C127C2×#12711yescharacteristic
140C2×C127#53212yes—
141C2×C127#53211yescharacteristic
142D127—82no—
143#2856—1271nomaximal, Sylow
144C127#285611yescharacteristic, complemented, Sylow
145C2×#127—1271no—
146C2×#121—1274no—
147C2×#121—1271no—
148C2×#121D12711yescharacteristic, complemented
149C22×C42—1271no—
150#390—1271no—
151#390—1271no—
152#390—1274no—
153#384—1271no—
154C22×D4—1272no—
155C22×D4—1274no—
156C22×D4—1271no—
157C22×D4C2×D12712yes—
158C22×D4C2×D12711yescharacteristic
159C23×C4—1272no—
160C23×C4C2×D12711yescharacteristic
161#127—2544no—
162#121—24no—
163#121—25416no—
164#121—2544no—
165C2×C4⋊C4—1274no—
166C2×C4⋊C4—1271no—
167C2×C4⋊C4C2×D12711yescharacteristic
168C2×C22⋊C4—1274no—
169C2×C22⋊C4—1274no—
170C2×C22⋊C4—1272no—
171C2×C22⋊C4C2×D12712yes—
172C2×C42—1271no—
173C2×C42—1272no—
174C2×C42—1271no—
175C2×C42—2544no—
176C24—1272no—
177C24C22×D12712yes—
178C24C22×D12711yescharacteristic
179C2×D4—28no—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—2548no—
185C2×D4—2548no—
186C2×D4—2548no—
187C2×D4—2548no—
188C2×D4—2544no—
189C2×D4—2542no—
190C2×D4—2542no—
191C2×D4—2542no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—1272no—
195C22×C4—1278no—
196C22×C4—1272no—
197C22×C4—1272no—
198C22×C4—2544no—
199C22×C4—2544no—
200C22×C4C22⋊D12714yescomplemented
201C22×C4C22×D12712yes—
202C22×C4C22×D12711yescharacteristic
203C22×C4C22×D12711yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—2548no—
206C4⋊C4—2542no—
207C22⋊C4—24no—
208C22⋊C4—2548no—
209C22⋊C4—2548no—
210C22⋊C4—2544no—
211C42—1274no—
212C42—2544no—
213C42—2542no—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—24no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—2544no—
223C23—2542no—
224C23—2542no—
225C23—2544no—
226C23—2542no—
227C23D4×D12714yes—
228C23C2×C22⋊D12712yes—
229C23C23×D12711yescharacteristic
230D4—48no—
231D4—44no—
232D4—50816no—
233D4—5088no—
234D4—5084no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—28no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—1274no—
241C2×C4—1274no—
242C2×C4—1274no—
243C2×C4—1274no—
244C2×C4—25416no—
245C2×C4—2544no—
246C2×C4—2542no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4D4×D12712yes—
250C2×C4C2×C22⋊D12712yes—
251C2×C4C2×C22⋊D12712yes—
252C2×C4#1452324712yes—
253C22—24no—
254C22—22no—
255C22—24no—
256C22—22no—
257C22—44no—
258C22—42no—
259C22—42no—
260C22—44no—
261C22—42no—
262C22—5084no—
263C22—5082no—
264C22—5082no—
265C22—5084no—
266C22—5082no—
267C22#1686120514yescomplemented
268C22C2×D4×D12711yescharacteristic
269C22C2×D4×D12712yes—
270C22C22×C22⋊D12711yescharacteristic
271C22#41×D12712yes—
272C22C2×#1452324711yescharacteristic
273C4—22no—
274C4—42no—
275C4—1278no—
276C4—5082no—
277C4—2544no—
278C4#1686118114yescomplemented
279C2—24no—
280C2—82no—
281C2—10162no—
282C2#121×D12712yes—
283C2C2×#1686118111yescharacteristic
284C2C2×#1686120511yescharacteristic
285C2#2255042112yes—
286C2#2255042211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×#121×C127C2
hypercenterC2×#121D127
hypocenterC127#2856
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16861205C2semidirect
2. C2×#16861183C2semidirect
3. C2×#16861181C2semidirect
4. C22×C4×Dic127C2semidirect
5. #22549779C2semidirect
6. #22549777C2semidirect
7. #22549776C2semidirect
8. #22549745C2semidirect
9. C2×#121×C127C2semidirect
43. C22×Dic127D4semidirect
54. C4×Dic127D4semidirect
64. C22×C4×C127D4semidirect
118. C22×C127#127semidirect
131. C4×C127#121semidirect
134. Dic127#121semidirect
144. C127#2856semidirect
148. C2×#121D127semidirect
200. C22×C4C22⋊D127semidirect
267. C22#16861205semidirect
278. C4#16861181semidirect

Maximal subgroups

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

subgroupindex
C2×#168612052
C2×#168611832
C2×#168611812
C22×C4×Dic1272
#225497792
#225497772
#225497762
#225497452
C2×#121×C1272
#2856127