Finite Group Databasebeta

← #30641581

Subgroups — #30641581

Order 16256.

792 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861165C211yesmaximal, characteristic, complemented
2C22×C22⋊Dic127C211yesmaximal, characteristic
3C22×C22⋊Dic127C211yesmaximal, characteristic
4#22549770C212yesmaximal, complemented
5#22549764C212yesmaximal, complemented
6#22549748C212yesmaximal, complemented
7#22549737C211yesmaximal, characteristic, complemented
8#22549734C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic127C212yesmaximal, complemented
10C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
11#379×C127C211yesmaximal, characteristic
12C23×Dic127C2212yes—
13C23×Dic127C2211yescharacteristic
14C23×Dic127C2211yescharacteristic
15#16861165—24no—
16C2×Dic127⋊C4C2212yescomplemented
17C2×Dic127⋊C4C2212yescomplemented
18C2×C22⋊Dic127—24no—
19C2×C22⋊Dic127C418yescomplemented
20C2×C22⋊Dic127C2212yes—
21C2×C22⋊Dic127C2211yescharacteristic
22C2×C22⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C4×Dic127C2212yescomplemented
28C2×C4×Dic127C2212yescomplemented
29C127⋊#95C2212yescomplemented
30C127⋊#95C2212yescomplemented
31C127⋊#95C2212yescomplemented
32C127⋊#95C2212yescomplemented
33C25×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2211yescharacteristic
36C2×C22⋊C4×C127C2212yes—
37C2×C22⋊C4×C127C2212yes—
38C22×Dic127—24no—
39C22×Dic127—22no—
40C22×Dic127—22no—
41C22×Dic127—24no—
42C22×Dic127—42no—
43C22×Dic127—42no—
44C22×Dic127C2×C414yescomplemented
45C22×Dic127C2×C414yescomplemented
46C22×Dic127C2312yes—
47C22×Dic127C2311yescharacteristic
48C22×Dic127C2311yescharacteristic
49C22×Dic127C2311yescharacteristic
50C22×Dic127C2311yescharacteristic
51C22×Dic127C2312yes—
52Dic127⋊C4—24no—
53Dic127⋊C4—42no—
54C22⋊Dic127—24no—
55C22⋊Dic127—22no—
56C22⋊Dic127C2×C414yes—
57C22⋊Dic127C2×C414yes—
58C22⋊Dic127C2×C414yescomplemented
59C22⋊Dic127C2×C414yescomplemented
60C22⋊Dic127D414yes—
61C4×Dic127—24no—
62C4×Dic127—42no—
63C24×C127—22no—
64C24×C127—24no—
65C24×C127—22no—
66C24×C127—42no—
67C24×C127C2×C414yes—
68C24×C127C2311yescharacteristic
69C24×C127C2311yescharacteristic
70C24×C127C2311yescharacteristic
71C22×C4×C127C2312yes—
72C22×C4×C127C2312yes—
73C22⋊C4×C127—22no—
74C22⋊C4×C127—24no—
75C22⋊C4×C127—22no—
76C22⋊C4×C127—42no—
77C2×Dic127—28no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—44no—
90C2×Dic127—48no—
91C2×Dic127C22×C412yes—
92C2×Dic127C22×C412yes—
93C2×Dic127C22×C412yes—
94C2×Dic127C22×C412yes—
95C2×Dic127C2×D414yes—
96C23×C127—21no—
97C23×C127—22no—
98C23×C127—21no—
99C23×C127—22no—
100C23×C127—22no—
101C23×C127—28no—
102C23×C127—28no—
103C23×C127—21no—
104C23×C127—22no—
105C23×C127—21no—
106C23×C127—22no—
107C23×C127—44no—
108C23×C127—48no—
109C23×C127—48no—
110C23×C127C22×C412yes—
111C23×C127C22×C412yes—
112C23×C127C22×C412yes—
113C23×C127C2×D412yes—
114C23×C127#4114yes—
115C23×C127#4112yes—
116C23×C127C2411yescharacteristic
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—22no—
127C22×C127—24no—
128C22×C127—24no—
129C22×C127—24no—
130C22×C127—24no—
131C22×C127—22no—
132C22×C127—22no—
133C22×C127—22no—
134C22×C127—416no—
135C22×C127—42no—
136C22×C127—42no—
137C22×C127#11714yescomplemented
138C22×C127C4×D414yes—
139C22×C127C23×C411yescharacteristic
140C22×C127C22×D411yescharacteristic
141C22×C127C2×#4112yes—
142C22×C127C2×#4111yescharacteristic
143C22×C127H(5,𝔽2)12yes—
144C4×C127—42no—
145C4×C127—42no—
146Dic127—22no—
147Dic127—22no—
148Dic127—22no—
149Dic127—22no—
150Dic127—24no—
151Dic127—42no—
152C2×C127—24no—
153C2×C127—24no—
154C2×C127—42no—
155C2×C127C2×#11711yescharacteristic
156C2×C127C2×C4×D411yescharacteristic
157C2×C127#51811yescharacteristic
158C2×C127#54612yes—
159C2×C127#55112yes—
160#2342—1271nomaximal, Sylow
161C127#234211yescharacteristic, complemented, Sylow
162C2×#117—1271no—
163C22×C22⋊C4—1271no—
164C22×C22⋊C4—1271no—
165#388—1272no—
166#388—1272no—
167#385—1272no—
168#381—1271no—
169#380—1271no—
170#379D12711yescharacteristic
171C4×C22⋊C4—1272no—
172C4×C22⋊C4—1271no—
173C25C2×D12711yescharacteristic
174C23×C4—1271no—
175C23×C4—1271no—
176C23×C4—1272no—
177#117—2544no—
178C2×C4⋊C4—1272no—
179C2×C4⋊C4—1272no—
180C2×C22⋊C4—1278no—
181C2×C22⋊C4—1271no—
182C2×C22⋊C4—1271no—
183C2×C22⋊C4—1271no—
184C2×C22⋊C4—1271no—
185C2×C22⋊C4—1271no—
186C2×C22⋊C4—1271no—
187C2×C22⋊C4—1272no—
188C2×C22⋊C4—2544no—
189C2×C22⋊C4C2×D12711yescharacteristic
190C2×C22⋊C4C2×D12711yescharacteristic
191C2×C22⋊C4C2×D12712yes—
192C2×C22⋊C4C2×D12712yes—
193C2×C42—1272no—
194C2×C42—1272no—
195#95—1272no—
196#95—1272no—
197#95—1272no—
198#95—1272no—
199C24—22no—
200C24—24no—
201C24—22no—
202C24—42no—
203C24C4×D12714yes—
204C24C22×D12711yescharacteristic
205C24C22×D12711yescharacteristic
206C24C22×D12711yescharacteristic
207C22×C4—1274no—
208C22×C4—1274no—
209C22×C4—1271no—
210C22×C4—1271no—
211C22×C4—1271no—
212C22×C4—1271no—
213C22×C4—1272no—
214C22×C4—1272no—
215C22×C4—5082no—
216C22×C4—5082no—
217C22×C4—2544no—
218C22×C4—2544no—
219C22×C4—2542no—
220C22×C4—2542no—
221C22×C4C22×D12712yes—
222C22×C4C22×D12712yes—
223C4⋊C4—5082no—
224C4⋊C4—2544no—
225C22⋊C4—22no—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—42no—
229C22⋊C4—1274no—
230C22⋊C4—1274no—
231C22⋊C4—1274no—
232C22⋊C4—1274no—
233C22⋊C4—1274no—
234C22⋊C4—2544no—
235C22⋊C4—2542no—
236C42—5082no—
237C42—2544no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—22no—
247C23—28no—
248C23—28no—
249C23—44no—
250C23—48no—
251C23—48no—
252C23C2×C4×D12712yes—
253C23C2×C4×D12712yes—
254C23C2×C4×D12712yes—
255C23D4×D12712yes—
256C23#1452324714yes—
257C23#1452324712yes—
258C23C23×D12711yescharacteristic
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—1272no—
266C2×C4—1272no—
267C2×C4—1272no—
268C2×C4—1272no—
269C2×C4—1274no—
270C2×C4—5088no—
271C2×C4—5084no—
272C2×C4—2542no—
273C2×C4—2542no—
274C2×C4—2542no—
275C2×C4—2541no—
276C2×C4—2541no—
277C2×C4—2541no—
278C2×C4—2541no—
279C2×C4—2542no—
280C2×C4—2542no—
281C2×C4—2542no—
282C2×C4—2542no—
283C2×C4—2548no—
284C22—24no—
285C22—24no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—24no—
291C22—24no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—416no—
296C22—42no—
297C22—42no—
298C22#1686116514yescomplemented
299C22(C22⋊D127)⋊C414yes—
300C22C22×C4×D12711yescharacteristic
301C22C2×D4×D12711yescharacteristic
302C22C2×#1452324711yescharacteristic
303C22C2×#1452324712yes—
304C22#1686125412yes—
305C4—42no—
306C4—42no—
307C4—5082no—
308C4—2542no—
309C4—2542no—
310C4—2542no—
311C4—2542no—
312C4—2544no—
313C2—24no—
314C2—24no—
315C2—42no—
316C2C2×#1686116511yescharacteristic
317C2C2×(C22⋊D127)⋊C411yescharacteristic
318C2#2255034911yescharacteristic
319C2#2255051312yes—
320C2#2255055712yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#379×C127C2
socle, cosocleC23×C127C24
prime radicalC25×C127C22
hypercenter#379D127
hypocenterC127#2342
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#16861165C2semidirect
4. #22549770C2semidirect
5. #22549764C2semidirect
6. #22549748C2semidirect
7. #22549737C2semidirect
8. #22549734C2semidirect
9. C22⋊C4×Dic127C2semidirect
10. C22⋊C4×Dic127C2semidirect
16. C2×Dic127⋊C4C22semidirect
17. C2×Dic127⋊C4C22semidirect
19. C2×C22⋊Dic127C4semidirect
27. C2×C4×Dic127C22semidirect
28. C2×C4×Dic127C22semidirect
29. C127⋊#95C22semidirect
30. C127⋊#95C22semidirect
31. C127⋊#95C22semidirect
32. C127⋊#95C22semidirect
44. C22×Dic127C2×C4semidirect
45. C22×Dic127C2×C4semidirect
58. C22⋊Dic127C2×C4semidirect
59. C22⋊Dic127C2×C4semidirect
137. C22×C127#117semidirect
161. C127#2342semidirect
298. C22#16861165semidirect

Maximal subgroups

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

subgroupindex
C2×#168611652
C22×C22⋊Dic1272
#225497702
#225497642
#225497482
#225497372
#225497342
C22⋊C4×Dic1272
#379×C1272
#2342127