Finite Group Databasebeta

← #30641401

Subgroups — #30641401

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×#16861193C211yesmaximal, characteristic
2C2×C4×C22⋊D127C211yesmaximal, characteristic
3C22×Dic127⋊C4C211yesmaximal, characteristic, complemented
4#22549774C212yesmaximal, complemented
5#22549767C212yesmaximal, complemented
6#22549761C212yesmaximal, complemented
7#22549753C211yesmaximal, characteristic
8#22549751C211yesmaximal, characteristic, complemented
9#22549750C212yesmaximal, complemented
10#22549737C211yesmaximal, characteristic, complemented
11C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C23×Dic127C2211yescharacteristic
14C23×Dic127C2211yescharacteristic
15C22×C4×D127C2211yescharacteristic
16#16861193C418yescomplemented
17C4×C22⋊D127—24no—
18C2×C4⋊Dic127C2212yes—
19C2×C4⋊Dic127C2212yes—
20C2×Dic127⋊C4—24no—
21C2×Dic127⋊C4C2212yes—
22C2×Dic127⋊C4C2212yes—
23C2×Dic127⋊C4C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×(C2×D127)⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2212yes—
27C2×(C2×D127)⋊C4C2212yes—
28C2×C4×Dic127C2211yescharacteristic
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C127⋊#95C2212yes—
33C23×C4×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2211yescharacteristic
36C2×C42×C127C2212yes—
37#95×C127C2212yes—
38C23×D127C2311yescharacteristic
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C2×C22⋊D127C2×C414yes—
42C22×Dic127—22no—
43C22×Dic127—22no—
44C22×Dic127—22no—
45C22×Dic127—42no—
46C22×Dic127C2×C414yescomplemented
47C22×Dic127C2311yescharacteristic
48C22×Dic127C2312yes—
49C22×Dic127C2311yescharacteristic
50C22×Dic127C2312yes—
51C22×Dic127C2311yescharacteristic
52C2×C4×D127—22no—
53C2×C4×D127—42no—
54C4⋊Dic127—42no—
55C4⋊Dic127C2×C418yescomplemented
56Dic127⋊C4—24no—
57Dic127⋊C4—24no—
58Dic127⋊C4—22no—
59C22⋊Dic127—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—42no—
62(C2×D127)⋊C4C2×C418yescomplemented
63C4×Dic127—22no—
64C24×C127C2311yescharacteristic
65C22×C4×C127—42no—
66C22×C4×C127D414yes—
67C22×C4×C127C2312yes—
68C22×C4×C127C2312yes—
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C22⋊C4×C127—22no—
72C22⋊C4×C127C2×C414yescomplemented
73C42×C127—42no—
74C22×D127—21no—
75C22×D127—21no—
76C22×D127—22no—
77C22×D127—22no—
78C22×D127C22×C412yes—
79C22⋊D127—28no—
80C2×Dic127—28no—
81C2×Dic127—21no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—21no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127—44no—
93C2×Dic127C22×C414yes—
94C2×Dic127C22×C412yes—
95C4×D127—44no—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127C22×C412yes—
99C23×C127C2×D412yes—
100C23×C127#4112yes—
101C23×C127C2411yescharacteristic
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—44no—
110C2×C4×C127C22×C414yes—
111C2×C4×C127C2×D412yes—
112C2×C4×C127C2×D412yes—
113C22×C127—24no—
114C22×C127—22no—
115C22×C127—22no—
116C22×C127—24no—
117C22×C127C4×D414yes—
118C22×C127C23×C411yescharacteristic
119C22×C127C22×D411yescharacteristic
120C22×C127C2×#4111yescharacteristic
121C22×C127H(5,𝔽2)12yes—
122C22×C1272−1+412yes—
123C2×D127—22no—
124C2×D127—24no—
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—42no—
128C4×C127—24no—
129C4×C127—22no—
130C4×C127—22no—
131C4×C127—42no—
132Dic127—24no—
133Dic127—22no—
134Dic127—41no—
135Dic127—42no—
136Dic127—41no—
137C2×C127—24no—
138C2×C127C2×C4×D411yescharacteristic
139C2×C127#52012yes—
140C2×C127#53411yescharacteristic
141C2×C127#53611yescharacteristic
142C2×C127#53912yes—
143D127—42no—
144#2322—1271nomaximal, Sylow
145C127#232211yescharacteristic, complemented, Sylow
146C2×#123—1271no—
147C2×C4×D4—1271no—
148C22×C4⋊C4—1271no—
149#389—1272no—
150#388—1272no—
151#387—1272no—
152#386—1271no—
153#385—1271no—
154#385—1272no—
155#381—1271no—
156C4×C22⋊C4D12711yescharacteristic, complemented
157C22×D4—1271no—
158C23×C4—1271no—
159C23×C4—1271no—
160C23×C4—1271no—
161C23×C4C2×D12711yescharacteristic
162#123—1278no—
163C4×D4—2544no—
164C2×C4⋊C4—1272no—
165C2×C4⋊C4—1272no—
166C2×C4⋊C4—1272no—
167C2×C4⋊C4—1272no—
168C2×C4⋊C4—1271no—
169C2×C4⋊C4—2544no—
170C2×C22⋊C4—1272no—
171C2×C22⋊C4—1272no—
172C2×C22⋊C4—1271no—
173C2×C22⋊C4—1271no—
174C2×C22⋊C4C2×D12711yescharacteristic
175C2×C22⋊C4C2×D12711yescharacteristic
176C2×C42—1271no—
177C2×C42C2×D12712yes—
178#95—1272no—
179#95—1272no—
180#95—1272no—
181#95—1272no—
182#95C2×D12712yes—
183C24—1271no—
184C24C22×D12711yescharacteristic
185C2×D4—1274no—
186C2×D4—2542no—
187C2×D4—2542no—
188C22×C4—42no—
189C22×C4—1274no—
190C22×C4—1271no—
191C22×C4—1272no—
192C22×C4—1271no—
193C22×C4—1271no—
194C22×C4—1272no—
195C22×C4—5082no—
196C22×C4—5082no—
197C22×C4—2542no—
198C22×C4—2542no—
199C22×C4—2542no—
200C22×C4—2542no—
201C22×C4D50814yes—
202C22×C4C22×D12712yes—
203C22×C4C22×D12712yes—
204C22×C4C22×D12711yescharacteristic
205C22×C4C22×D12711yescharacteristic
206C4⋊C4—1278no—
207C4⋊C4—5082no—
208C4⋊C4—2544no—
209C4⋊C4—2544no—
210C4⋊C4—2542no—
211C22⋊C4—22no—
212C22⋊C4—1278no—
213C22⋊C4—5082no—
214C22⋊C4—2542no—
215C22⋊C4—2542no—
216C22⋊C4C4×D12714yescomplemented
217C42—42no—
218C42—2542no—
219C23—22no—
220C23—22no—
221C23—1272no—
222C23—2541no—
223C23—2541no—
224C23—2542no—
225C23—2542no—
226C23C2×C4×D12712yes—
227C23C2×D50812yes—
228C23#1452324812yes—
229C23C23×D12711yescharacteristic
230D4—2548no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—44no—
239C2×C4—1274no—
240C2×C4—1272no—
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—2541no—
251C2×C4—2542no—
252C2×C4—2542no—
253C2×C4—2541no—
254C2×C4—2548no—
255C2×C4C2×C4×D12714yes—
256C2×C4C2×D50812yes—
257C2×C4C2×D50812yes—
258C22—24no—
259C22—22no—
260C22—22no—
261C22—24no—
262C22—5081no—
263C22—5081no—
264C22—5082no—
265C22—2542no—
266C22—2544no—
267C22C4×D50814yes—
268C22C22×C4×D12711yescharacteristic
269C22C22×D50811yescharacteristic
270C22C2×#1452324811yescharacteristic
271C22#1686125412yes—
272C22#1686125612yes—
273C4—24no—
274C4—22no—
275C4—22no—
276C4—42no—
277C4—5081no—
278C4—5082no—
279C4—5081no—
280C4—2544no—
281C4—2542no—
282C2—24no—
283C2—5082no—
284C2C2×C4×D50811yescharacteristic
285C2#2255035512yes—
286C2#2255043811yescharacteristic
287C2#2255045211yescharacteristic
288C2#2255046812yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC4×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC22×C22⋊D127C22
hypercenterC4×C22⋊C4D127
hypocenterC127#2322
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×Dic127⋊C4C2semidirect
4. #22549774C2semidirect
5. #22549767C2semidirect
6. #22549761C2semidirect
8. #22549751C2semidirect
9. #22549750C2semidirect
10. #22549737C2semidirect
11. C4×C22⋊C4×C127C2semidirect
16. #16861193C4semidirect
46. C22×Dic127C2×C4semidirect
55. C4⋊Dic127C2×C4semidirect
62. (C2×D127)⋊C4C2×C4semidirect
72. C22⋊C4×C127C2×C4semidirect
145. C127#2322semidirect
156. C4×C22⋊C4D127semidirect
216. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611932
C2×C4×C22⋊D1272
C22×Dic127⋊C42
#225497742
#225497672
#225497612
#225497532
#225497512
#225497502
#225497372
C4×C22⋊C4×C1272
#2322127