Finite Group Databasebeta

← #30647074

Subgroups — #30647074

Order 16256.

924 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 189 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×#16861184C211yesmaximal, characteristic, complemented
3C2×#16861182C214yesmaximal, complemented
4C2×#16861180C211yesmaximal, characteristic, complemented
5C2×C42×D127C211yesmaximal, characteristic, complemented
6#22549778C211yesmaximal, characteristic, complemented
7#22549776C214yesmaximal, complemented
8#22549745C211yesmaximal, characteristic, complemented
9#390×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×D508C2212yes—
12C22×C22⋊D127C2214yes—
13C22×C22⋊D127C2212yes—
14C22×C4×D127C2211yescharacteristic
15C22×C4×D127C2212yes—
16#16861205—24no—
17#16861184—24no—
18#16861182—216no—
19#16861180—24no—
20C2×C4⋊Dic127C2211yescharacteristic
21C2×Dic127⋊C4C2214yes—
22C2×C22⋊Dic127C2212yes—
23C2×(C2×D127)⋊C4C2212yes—
24C2×(C2×D127)⋊C4C2214yes—
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2211yescharacteristic
27C42×D127—24no—
28C22×D4×C127C2211yescharacteristic
29C2×C4⋊C4×C127C2211yescharacteristic
30C2×C22⋊C4×C127C2214yes—
31C2×C42×C127C2211yescharacteristic
32C23×D127C2312yes—
33C23×D127C2311yescharacteristic
34C2×D508—24no—
35C2×D508—28no—
36C2×D508—22no—
37C2×D508—22no—
38C2×D508—22no—
39C2×C22⋊D127—24no—
40C2×C22⋊D127—28no—
41C2×C22⋊D127—28no—
42C2×C22⋊D127—28no—
43C2×C22⋊D127—28no—
44C22×Dic127C2312yes—
45C22×Dic127C2311yescharacteristic
46C22×Dic127C2312yes—
47C2×C4×D127—24no—
48C2×C4×D127—24no—
49C2×C4×D127—22no—
50C2×C4×D127—22no—
51C2×C4×D127D414yescomplemented
52C2×C4×D127D418yescomplemented
53C4⋊Dic127—22no—
54Dic127⋊C4—28no—
55C22⋊Dic127—24no—
56(C2×D127)⋊C4—24no—
57(C2×D127)⋊C4—28no—
58C4×Dic127—24no—
59C4×Dic127D414yescomplemented
60C24×C127C2312yes—
61C2×D4×C127—22no—
62C2×D4×C127—22no—
63C2×D4×C127—22no—
64C22×C4×C127C2312yes—
65C22×C4×C127C2312yes—
66C22×C4×C127C2311yescharacteristic
67C4⋊C4×C127—22no—
68C22⋊C4×C127—28no—
69C42×C127—22no—
70C22×D127—21no—
71C22×D127—22no—
72C22×D127—21no—
73C22×D127—24no—
74C22×D127—22no—
75C22×D127—22no—
76C22×D127—24no—
77C22×D127—22no—
78C22×D127C2×D412yes—
79C22×D127C2×D414yes—
80D508—48no—
81D508—44no—
82C22⋊D127—48no—
83C22⋊D127—416no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—24no—
87C2×Dic127—22no—
88C2×Dic127C2×D412yes—
89C2×Dic127C2×D414yes—
90C2×Dic127C2×D414yes—
91C2×Dic127#4114yes—
92C4×D127—28no—
93C4×D127—216no—
94C23×C127—24no—
95C23×C127—22no—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—24no—
99C23×C127C2411yescharacteristic
100D4×C127—44no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—24no—
104C2×C4×C127—22no—
105C2×C4×C127C2×D414yes—
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D412yes—
108C2×C4×C127#4112yes—
109C22×C127—42no—
110C22×C127—44no—
111C22×C127—42no—
112C22×C127—44no—
113C22×C127—42no—
114C22×C127C22×D411yescharacteristic
115C22×C127C22×D412yes—
116C22×C127C22×D411yescharacteristic
117C22×C127C2×#4112yes—
118C22×C127C2×#4111yescharacteristic
119C2×D127—24no—
120C2×D127—22no—
121C2×D127—24no—
122C2×D127—22no—
123C2×D127—44no—
124C2×D127—42no—
125C2×D127—42no—
126C2×D127—44no—
127C2×D127—42no—
128C2×D127#12714yescomplemented
129C4×C127—24no—
130C4×C127—42no—
131C4×C127#12114yescomplemented
132Dic127—22no—
133Dic127—42no—
134Dic127#12118yescomplemented
135C2×C127—82no—
136C2×C127C2×#12112yes—
137C2×C127C2×#12111yescharacteristic
138C2×C127C2×#12711yescharacteristic
139C2×C127#53212yes—
140C2×C127#53211yescharacteristic
141D127—24no—
142D127—82no—
143#2856—1271nomaximal, Sylow
144C127#285611yescharacteristic, complemented, Sylow
145C2×#127—1271no—
146C2×#121—1271no—
147C2×#121—1271no—
148C2×#121—1274no—
149C22×C42—1271no—
150#390—1274no—
151#390—1271no—
152#390D12711yescharacteristic, complemented
153#384—1271no—
154C22×D4—1272no—
155C22×D4—1274no—
156C22×D4—1272no—
157C22×D4—1271no—
158C22×D4C2×D12711yescharacteristic
159C23×C4—1272no—
160C23×C4—1271no—
161#127—2544no—
162#121—25416no—
163#121—2544no—
164#121—2544no—
165C2×C4⋊C4—1274no—
166C2×C4⋊C4—1271no—
167C2×C4⋊C4C2×D12711yescharacteristic
168C2×C22⋊C4—1272no—
169C2×C22⋊C4—1272no—
170C2×C22⋊C4—1274no—
171C2×C22⋊C4C2×D12714yes—
172C2×C42—1271no—
173C2×C42—1272no—
174C2×C42—2544no—
175C2×C42C2×D12711yescharacteristic
176C24—1271no—
177C24—1272no—
178C24C22×D12712yes—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—2548no—
183C2×D4—2544no—
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—1272no—
193C22×C4—1271no—
194C22×C4—1278no—
195C22×C4—1274no—
196C22×C4—1272no—
197C22×C4—2544no—
198C22×C4—2544no—
199C22×C4—2542no—
200C22×C4—2542no—
201C22×C4C22×D12712yes—
202C22×C4C22×D12712yes—
203C22×C4C22×D12711yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—2548no—
206C4⋊C4—2542no—
207C22⋊C4—28no—
208C22⋊C4—2544no—
209C22⋊C4—2544no—
210C22⋊C4—2548no—
211C42—22no—
212C42—1274no—
213C42—2544no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—24no—
218C23—22no—
219C23—1274no—
220C23—1272no—
221C23—2544no—
222C23—2542no—
223C23—2542no—
224C23—2544no—
225C23—2542no—
226C23—2541no—
227C23—2542no—
228C23—2541no—
229C23C23×D12711yescharacteristic
230D4—44no—
231D4—5088no—
232D4—50816no—
233D4—5088no—
234D4—5084no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—1274no—
240C2×C4—1274no—
241C2×C4—1274no—
242C2×C4—1272no—
243C2×C4—25416no—
244C2×C4—2544no—
245C2×C4—2542no—
246C2×C4—2541no—
247C2×C4—2541no—
248C2×C4—2548no—
249C2×C4D4×D12714yes—
250C2×C4D4×D12712yes—
251C2×C4D4×D12712yes—
252C2×C4#1452324812yes—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—1274no—
259C22—5084no—
260C22—5082no—
261C22—5082no—
262C22—5084no—
263C22—5082no—
264C22—2544no—
265C22—2542no—
266C22—2544no—
267C22—2542no—
268C22C2×D4×D12711yescharacteristic
269C22C2×D4×D12712yes—
270C22C2×D4×D12711yescharacteristic
271C22#41×D12712yes—
272C22C2×#1452324811yescharacteristic
273C4—24no—
274C4—42no—
275C4—1278no—
276C4—5082no—
277C4—2542no—
278C4#1686118214yescomplemented
279C2—82no—
280C2—10162no—
281C2—2544no—
282C2#121×D12712yes—
283C2C2×#1686118211yescharacteristic
284C2#127×D12711yescharacteristic
285C2#2255042312yes—
286C2#2255042511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#390×C127C2
hypercenter#390D127
hypocenterC127#2856
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16861205C2semidirect
2. C2×#16861184C2semidirect
3. C2×#16861182C2semidirect
4. C2×#16861180C2semidirect
5. C2×C42×D127C2semidirect
6. #22549778C2semidirect
7. #22549776C2semidirect
8. #22549745C2semidirect
9. #390×C127C2semidirect
51. C2×C4×D127D4semidirect
52. C2×C4×D127D4semidirect
59. C4×Dic127D4semidirect
128. C2×D127#127semidirect
131. C4×C127#121semidirect
134. Dic127#121semidirect
144. C127#2856semidirect
152. #390D127semidirect
278. C4#16861182semidirect

Maximal subgroups

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

subgroupindex
C2×#168612052
C2×#168611842
C2×#168611822
C2×#168611802
C2×C42×D1272
#225497782
#225497762
#225497452
#390×C1272
#2856127