Finite Group Databasebeta

← #30641402

Subgroups — #30641402

Order 16256.

632 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic127C211yesmaximal, characteristic
2C22×C4⋊Dic127C211yesmaximal, characteristic, complemented
3#22549775C212yesmaximal, complemented
4#22549769C212yesmaximal, complemented
5#22549762C212yesmaximal, complemented
6#22549756C211yesmaximal, characteristic
7#22549751C211yesmaximal, characteristic, complemented
8#22549751C212yesmaximal, complemented
9#22549739C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
11C2×#123×C127C211yesmaximal, characteristic
12C23×Dic127C2211yescharacteristic
13C23×Dic127C2211yescharacteristic
14C23×Dic127C2211yescharacteristic
15D4×Dic127—24no—
16C2×C4⋊Dic127—24no—
17C2×C4⋊Dic127C2212yes—
18C2×C4⋊Dic127C2212yes—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4⋊Dic127C2212yes—
21C2×C22⋊Dic127C2211yescharacteristic
22C2×C22⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2212yes—
25C2×C4×Dic127C2211yescharacteristic
26C2×C4×Dic127C2212yes—
27C127⋊#95C2212yes—
28C127⋊#95C2212yes—
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C22×D4×C127C2211yescharacteristic
33C23×C4×C127C2211yescharacteristic
34#123×C127C418yescomplemented
35C2×C4⋊C4×C127C2212yes—
36C2×C22⋊C4×C127C2212yes—
37C2×C22⋊C4×C127C2211yescharacteristic
38C22×Dic127—22no—
39C22×Dic127—22no—
40C22×Dic127—42no—
41C22×Dic127—42no—
42C22×Dic127—42no—
43C22×Dic127D414yes—
44C22×Dic127C2311yescharacteristic
45C22×Dic127C2312yes—
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2312yes—
48C22×Dic127C2311yescharacteristic
49C22×Dic127C2311yescharacteristic
50C4⋊Dic127—24no—
51C4⋊Dic127—24no—
52C4⋊Dic127—22no—
53C4⋊Dic127—42no—
54C22⋊Dic127—22no—
55C22⋊Dic127—22no—
56C22⋊Dic127—22no—
57C22⋊Dic127—42no—
58C4×Dic127—22no—
59C4×Dic127—42no—
60C24×C127C2311yescharacteristic
61C24×C127C2311yescharacteristic
62C2×D4×C127—22no—
63C2×D4×C127—22no—
64C2×D4×C127C2×C414yes—
65C22×C4×C127—22no—
66C22×C4×C127—22no—
67C22×C4×C127C2×C414yescomplemented
68C22×C4×C127C2312yes—
69C22×C4×C127C2312yes—
70C22×C4×C127C2311yescharacteristic
71C4⋊C4×C127C2×C418yescomplemented
72C22⋊C4×C127C2×C414yescomplemented
73C22⋊C4×C127C2×C418yescomplemented
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—44no—
87C2×Dic127—44no—
88C2×Dic127—44no—
89C2×Dic127C2×D412yes—
90C2×Dic127C2×D412yes—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—21no—
94C23×C127—21no—
95C23×C127—22no—
96C23×C127—22no—
97C23×C127C22×C412yes—
98C23×C127C22×C412yes—
99C23×C127C2×D412yes—
100C23×C127#4112yes—
101C23×C127C2411yescharacteristic
102D4×C127—28no—
103C2×C4×C127—28no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127C22×C414yes—
111C2×C4×C127C22×C414yes—
112C2×C4×C127C22×C412yes—
113C22×C127—24no—
114C22×C127—22no—
115C22×C127—22no—
116C22×C127—24no—
117C22×C127—22no—
118C22×C127—24no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127—42no—
122C22×C127C4×D414yes—
123C22×C127C23×C411yescharacteristic
124C22×C127C22×D411yescharacteristic
125C22×C127C2×#4111yescharacteristic
126C22×C127H(5,𝔽2)12yes—
127C22×C1272−1+412yes—
128C4×C127—24no—
129C4×C127—22no—
130C4×C127—24no—
131Dic127—22no—
132Dic127—22no—
133Dic127—41no—
134Dic127—42no—
135Dic127—41no—
136Dic127—42no—
137C2×C127—24no—
138C2×C127—42no—
139C2×C127C2×C4×D411yescharacteristic
140C2×C127#52012yes—
141C2×C127#53411yescharacteristic
142C2×C127#53611yescharacteristic
143C2×C127#53912yes—
144#2322—1271nomaximal, Sylow
145C127#232211yescharacteristic, complemented, Sylow
146C2×#123D12711yescharacteristic
147C2×C4×D4—1271no—
148C22×C4⋊C4—1271no—
149#389—1272no—
150#388—1272no—
151#387—1272no—
152#386—1271no—
153#385—1272no—
154#385—1271no—
155#381—1271no—
156C4×C22⋊C4—1271no—
157C22×D4C2×D12711yescharacteristic
158C23×C4—1271no—
159C23×C4—1271no—
160C23×C4—1271no—
161C23×C4C2×D12711yescharacteristic
162#123Dic12718yescomplemented
163C4×D4—2544no—
164C2×C4⋊C4—1272no—
165C2×C4⋊C4—1272no—
166C2×C4⋊C4—1272no—
167C2×C4⋊C4—1271no—
168C2×C4⋊C4—2544no—
169C2×C4⋊C4C2×D12712yes—
170C2×C22⋊C4—1272no—
171C2×C22⋊C4—1271no—
172C2×C22⋊C4—1271no—
173C2×C22⋊C4—1271no—
174C2×C22⋊C4C2×D12711yescharacteristic
175C2×C22⋊C4C2×D12712yes—
176C2×C42—1272no—
177C2×C42—1271no—
178#95—1272no—
179#95—1272no—
180#95—1272no—
181#95—1272no—
182#95—1272no—
183C24C22×D12711yescharacteristic
184C24C22×D12711yescharacteristic
185C2×D4—22no—
186C2×D4—22no—
187C2×D4C2×Dic12714yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—1274no—
191C22×C4—1271no—
192C22×C4—1271no—
193C22×C4—1271no—
194C22×C4—1272no—
195C22×C4—1271no—
196C22×C4—1272no—
197C22×C4—5082no—
198C22×C4—5082no—
199C22×C4—5082no—
200C22×C4—2542no—
201C22×C4—2542no—
202C22×C4C2×Dic12714yescomplemented
203C22×C4C22×D12712yes—
204C22×C4C22×D12711yescharacteristic
205C22×C4C22×D12712yes—
206C4⋊C4—5082no—
207C4⋊C4—2544no—
208C4⋊C4—2544no—
209C4⋊C4—2542no—
210C4⋊C4C2×Dic12718yescomplemented
211C22⋊C4—5082no—
212C22⋊C4—2542no—
213C22⋊C4—2542no—
214C22⋊C4—2542no—
215C22⋊C4C2×Dic12714yescomplemented
216C22⋊C4C2×Dic12718yescomplemented
217C42—5082no—
218C42—2542no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23—22no—
225C23C22×Dic12712yes—
226C23C22×Dic12712yes—
227C23D4×D12712yes—
228C23#1452324712yes—
229C23C23×D12711yescharacteristic
230D4—28no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—28no—
238C2×C4—1272no—
239C2×C4—1272no—
240C2×C4—5084no—
241C2×C4—5084no—
242C2×C4—5084no—
243C2×C4—2541no—
244C2×C4—2541no—
245C2×C4—2542no—
246C2×C4—2542no—
247C2×C4—2542no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4—2542no—
251C2×C4—2542no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4C22×Dic12714yes—
256C2×C4C22×Dic12714yes—
257C2×C4C22×Dic12712yes—
258C22—24no—
259C22—22no—
260C22—22no—
261C22—24no—
262C22—22no—
263C22—24no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22D4×Dic12714yes—
268C22C23×Dic12711yescharacteristic
269C22C2×D4×D12711yescharacteristic
270C22C2×#1452324711yescharacteristic
271C22#1686125512yes—
272C22#1686125612yes—
273C4—24no—
274C4—22no—
275C4—24no—
276C4—5082no—
277C4—5081no—
278C4—5081no—
279C4—5082no—
280C4—2542no—
281C4—2542no—
282C2—24no—
283C2—42no—
284C2C2×D4×Dic12711yescharacteristic
285C2#2255035912yes—
286C2#2255043711yescharacteristic
287C2#2255044811yescharacteristic
288C2#2255047012yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×#123×C127C2
socle, cosocleC23×C127C24
prime radicalC22×D4×C127C22
hypercenterC2×#123D127
hypocenterC127#2322
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic127C2semidirect
3. #22549775C2semidirect
4. #22549769C2semidirect
5. #22549762C2semidirect
7. #22549751C2semidirect
8. #22549751C2semidirect
9. #22549739C2semidirect
10. C22⋊C4×Dic127C2semidirect
34. #123×C127C4semidirect
67. C22×C4×C127C2×C4semidirect
71. C4⋊C4×C127C2×C4semidirect
72. C22⋊C4×C127C2×C4semidirect
73. C22⋊C4×C127C2×C4semidirect
145. C127#2322semidirect
162. #123Dic127semidirect
202. C22×C4C2×Dic127semidirect
210. C4⋊C4C2×Dic127semidirect
215. C22⋊C4C2×Dic127semidirect
216. C22⋊C4C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
C22×C4⋊Dic1272
#225497752
#225497692
#225497622
#225497562
#225497512
#225497392
C22⋊C4×Dic1272
C2×#123×C1272
#2322127