Finite Group Databasebeta

← #30641294

Subgroups — #30641294

Order 16256.

924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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
2C2×D4×Dic127C211yesmaximal, characteristic, complemented
3C2×#16861171C211yesmaximal, characteristic, complemented
4C2×(C22⋊D127)⋊C4C214yesmaximal, complemented
5#22549778C211yesmaximal, characteristic, complemented
6#22549776C214yesmaximal, complemented
7#22549745C211yesmaximal, characteristic, complemented
8C42×Dic127C211yesmaximal, characteristic, complemented
9#390×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C22×C22⋊D127C2214yes—
12C23×Dic127C2212yes—
13C22×C4×D127C2212yes—
14#16861205C418yescomplemented
15D4×Dic127—24no—
16#16861171—24no—
17(C22⋊D127)⋊C4—216no—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×Dic127⋊C4C2214yes—
20C2×C22⋊Dic127C2212yes—
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2214yes—
23C2×C4×Dic127C2212yes—
24C2×C4×Dic127C2211yescharacteristic
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28C2×C4⋊C4×C127C2211yescharacteristic
29C2×C22⋊C4×C127C2214yes—
30C2×C42×C127C2211yescharacteristic
31C23×D127C2312yes—
32C2×D508—22no—
33C2×D508—22no—
34C2×D508C2×C414yescomplemented
35C2×C22⋊D127—28no—
36C2×C22⋊D127—28no—
37C2×C22⋊D127C2×C4116yescomplemented
38C22×Dic127—28no—
39C22×Dic127—24no—
40C22×Dic127C2312yes—
41C22×Dic127C2311yescharacteristic
42C22×Dic127C2312yes—
43C22×Dic127C2311yescharacteristic
44C2×C4×D127—28no—
45C2×C4×D127—24no—
46C4⋊Dic127—22no—
47Dic127⋊C4—28no—
48C22⋊Dic127—24no—
49(C2×D127)⋊C4—24no—
50(C2×D127)⋊C4—28no—
51C4×Dic127—24no—
52C4×Dic127C2×C414yescomplemented
53C4×Dic127D418yescomplemented
54C4×Dic127D414yescomplemented
55C24×C127C2312yes—
56C2×D4×C127—22no—
57C2×D4×C127—22no—
58C2×D4×C127C2×C414yescomplemented
59C22×C4×C127C2312yes—
60C22×C4×C127C2312yes—
61C22×C4×C127C2311yescharacteristic
62C4⋊C4×C127—22no—
63C22⋊C4×C127—28no—
64C42×C127—22no—
65C22×D127—24no—
66C22×D127—22no—
67C22×D127—24no—
68C22×D127—22no—
69C22×D127C22×C414yes—
70D508—28no—
71C22⋊D127—232no—
72C2×Dic127—22no—
73C2×Dic127—24no—
74C2×Dic127—22no—
75C2×Dic127—48no—
76C2×Dic127C22×C414yes—
77C2×Dic127C2×D412yes—
78C2×Dic127C2×D414yes—
79C2×Dic127C2×D414yes—
80C2×Dic127C2×D412yes—
81C2×Dic127#4114yes—
82C4×D127—48no—
83C23×C127—24no—
84C23×C127—22no—
85C23×C127—22no—
86C23×C127—24no—
87C23×C127C22×C414yes—
88C23×C127C2411yescharacteristic
89D4×C127—28no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—24no—
93C2×C4×C127—22no—
94C2×C4×C127C22×C412yes—
95C2×C4×C127C2×D414yes—
96C2×C4×C127C2×D412yes—
97C2×C4×C127#4112yes—
98C22×C127—24no—
99C22×C127—28no—
100C22×C127—42no—
101C22×C127—44no—
102C22×C127—42no—
103C22×C127C23×C411yescharacteristic
104C22×C127C22×D412yes—
105C22×C127C22×D411yescharacteristic
106C22×C127C2×#4112yes—
107C22×C127C2×#4111yescharacteristic
108C2×D127—28no—
109C2×D127—24no—
110C2×D127—42no—
111C2×D127—44no—
112C2×D127—42no—
113C4×C127—24no—
114C4×C127—42no—
115C4×C127C4×D414yescomplemented
116Dic127—22no—
117Dic127—42no—
118Dic127C4×D418yescomplemented
119Dic127#12714yescomplemented
120C2×C127—44no—
121C2×C127C2×C4×D412yes—
122C2×C127C2×C4×D411yescharacteristic
123C2×C127C2×#12711yescharacteristic
124C2×C127#53212yes—
125C2×C127#53211yescharacteristic
126D127—44no—
127C4×#127—1271nomaximal, Sylow
128C127C4×#12711yescharacteristic, complemented, Sylow
129C2×#127—1271no—
130C2×C4×D4—1271no—
131C2×C4×D4—1271no—
132C2×C4×D4—1274no—
133#390—1274no—
134#390—1271no—
135#390D12711yescharacteristic, complemented
136#384—1271no—
137C43—1271no—
138C22×D4—1274no—
139C22×D4—1271no—
140C22×D4C2×D12711yescharacteristic
141C23×C4—1272no—
142C23×C4—1272no—
143#127—1278no—
144C4×D4—25416no—
145C4×D4—2544no—
146C4×D4—2544no—
147C2×C4⋊C4—1274no—
148C2×C4⋊C4—1271no—
149C2×C4⋊C4C2×D12711yescharacteristic
150C2×C22⋊C4—1272no—
151C2×C22⋊C4—1272no—
152C2×C22⋊C4—1274no—
153C2×C22⋊C4C2×D12714yes—
154C2×C42—1272no—
155C2×C42—1271no—
156C2×C42—1271no—
157C2×C42—1272no—
158C2×C42C2×D12711yescharacteristic
159C24—1272no—
160C24C22×D12712yes—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—12716no—
164C2×D4—1274no—
165C2×D4—2548no—
166C2×D4—2548no—
167C2×D4—2542no—
168C2×D4—2542no—
169C2×D4C4×D12714yescomplemented
170C22×C4—1272no—
171C22×C4—1271no—
172C22×C4—1271no—
173C22×C4—1272no—
174C22×C4—2548no—
175C22×C4—2544no—
176C22×C4—2548no—
177C22×C4—2544no—
178C22×C4C22×D12712yes—
179C22×C4C22×D12712yes—
180C22×C4C22×D12711yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—2548no—
183C4⋊C4—2542no—
184C22⋊C4—28no—
185C22⋊C4—2544no—
186C22⋊C4—2544no—
187C22⋊C4—2548no—
188C42—22no—
189C42—1274no—
190C42—1278no—
191C42—1274no—
192C42—2544no—
193C23—24no—
194C23—22no—
195C23—24no—
196C23—22no—
197C23—1274no—
198C23—2544no—
199C23—2542no—
200C23—2544no—
201C23—2542no—
202C23C2×C4×D12714yes—
203C23C23×D12711yescharacteristic
204D4—28no—
205D4—25432no—
206D4—2548no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—1274no—
212C2×C4—1274no—
213C2×C4—1274no—
214C2×C4—1272no—
215C2×C4—1274no—
216C2×C4—1272no—
217C2×C4—5088no—
218C2×C4—5088no—
219C2×C4—2544no—
220C2×C4—2542no—
221C2×C4—2542no—
222C2×C4C2×C4×D12712yes—
223C2×C4D4×D12714yes—
224C2×C4D4×D12712yes—
225C2×C4#1452324712yes—
226C22—28no—
227C22—24no—
228C22—42no—
229C22—44no—
230C22—42no—
231C22—5082no—
232C22—5084no—
233C22—5082no—
234C22—2548no—
235C22—2544no—
236C22C22×C4×D12711yescharacteristic
237C22C2×D4×D12712yes—
238C22C2×D4×D12711yescharacteristic
239C22#41×D12712yes—
240C22C2×#1452324711yescharacteristic
241C4—24no—
242C4—42no—
243C4—1278no—
244C4—1274no—
245C4—5082no—
246C4—2542no—
247C4(C22⋊D127)⋊C414yescomplemented
248C2—44no—
249C2—5084no—
250C2C4×D4×D12712yes—
251C2C2×(C22⋊D127)⋊C411yescharacteristic
252C2#127×D12711yescharacteristic
253C2#2255042112yes—
254C2#2255042511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#390×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861205C2
hypercenter#390D127
hypocenterC127C4×#127
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic127C2semidirect
3. C2×#16861171C2semidirect
4. C2×(C22⋊D127)⋊C4C2semidirect
5. #22549778C2semidirect
6. #22549776C2semidirect
7. #22549745C2semidirect
8. C42×Dic127C2semidirect
9. #390×C127C2semidirect
14. #16861205C4semidirect
34. C2×D508C2×C4semidirect
37. C2×C22⋊D127C2×C4semidirect
52. C4×Dic127C2×C4semidirect
53. C4×Dic127D4semidirect
54. C4×Dic127D4semidirect
58. C2×D4×C127C2×C4semidirect
115. C4×C127C4×D4semidirect
118. Dic127C4×D4semidirect
119. Dic127#127semidirect
128. C127C4×#127semidirect
135. #390D127semidirect
169. C2×D4C4×D127semidirect
247. C4(C22⋊D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168612052
C2×D4×Dic1272
C2×#168611712
C2×(C22⋊D127)⋊C42
#225497782
#225497762
#225497452
C42×Dic1272
#390×C1272
C4×#127127