Finite Group Databasebeta

← #30641385

Subgroups — #30641385

Order 16256.

808 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861178C211yesmaximal, characteristic
2C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic127C211yesmaximal, characteristic, complemented
4#22549776C212yesmaximal, complemented
5#22549770C212yesmaximal, complemented
6#22549764C212yesmaximal, complemented
7#22549753C211yesmaximal, characteristic, complemented
8#22549752C212yesmaximal, complemented
9#22549748C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
11#379×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C22×C22⋊D127C2212yes—
14C23×Dic127C2211yescharacteristic
15C23×Dic127C2211yescharacteristic
16C22×C4×D127C2211yescharacteristic, complemented
17#16861178C418yescomplemented
18(C22⋊D127)⋊C4—24no—
19C2×Dic127⋊C4C2211yescharacteristic, complemented
20C2×Dic127⋊C4C2212yescomplemented
21C2×C22⋊Dic127—24no—
22C2×C22⋊Dic127C2212yes—
23C2×C22⋊Dic127C2212yes—
24C2×C22⋊Dic127C2212yes—
25C2×C22⋊Dic127C2211yescharacteristic
26C2×(C2×D127)⋊C4C2211yescharacteristic, complemented
27C2×(C2×D127)⋊C4C2212yescomplemented
28C2×C4×Dic127C2211yescharacteristic, complemented
29C2×C4×Dic127C2212yescomplemented
30C127⋊#95C2212yescomplemented
31C127⋊#95C2212yescomplemented
32C127⋊#95C2212yescomplemented
33C25×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2211yescharacteristic
36C2×C22⋊C4×C127C2212yes—
37C2×C22⋊C4×C127C2212yes—
38C23×D127C2311yescharacteristic
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C2×C22⋊D127—42no—
42C2×C22⋊D127—42no—
43C2×C22⋊D127C2×C414yes—
44C2×C22⋊D127C2×C418yescomplemented
45C22×Dic127—22no—
46C22×Dic127—42no—
47C22×Dic127—42no—
48C22×Dic127D414yescomplemented
49C22×Dic127C2312yes—
50C22×Dic127C2311yescharacteristic
51C22×Dic127C2312yes—
52C22×Dic127C2311yescharacteristic
53C22×Dic127C2311yescharacteristic
54C2×C4×D127—22no—
55C2×C4×D127—42no—
56Dic127⋊C4—22no—
57Dic127⋊C4—42no—
58C22⋊Dic127—24no—
59C22⋊Dic127—24no—
60C22⋊Dic127C2×C418yescomplemented
61C22⋊Dic127C2×C414yescomplemented
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—42no—
64C4×Dic127—22no—
65C4×Dic127—42no—
66C24×C127—22no—
67C24×C127—22no—
68C24×C127—42no—
69C24×C127—42no—
70C24×C127C2×C414yescomplemented
71C24×C127C2312yes—
72C24×C127C2311yescharacteristic
73C22×C4×C127C2311yescharacteristic
74C22×C4×C127C2312yes—
75C22×C4×C127C2311yescharacteristic
76C22⋊C4×C127—22no—
77C22⋊C4×C127—22no—
78C22⋊C4×C127—42no—
79C22⋊C4×C127—42no—
80C22×D127—21no—
81C22×D127—21no—
82C22×D127—22no—
83C22×D127—22no—
84C22×D127C22×C412yes—
85C22⋊D127—28no—
86C22⋊D127—48no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127—22no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C2×Dic127—22no—
96C2×Dic127—44no—
97C2×Dic127—44no—
98C2×Dic127C22×C412yes—
99C2×Dic127C22×C414yes—
100C2×Dic127C2×D412yes—
101C2×Dic127C2×D412yes—
102C4×D127—44no—
103C23×C127—22no—
104C23×C127—22no—
105C23×C127—28no—
106C23×C127—22no—
107C23×C127—22no—
108C23×C127—22no—
109C23×C127—22no—
110C23×C127—22no—
111C23×C127—22no—
112C23×C127—48no—
113C23×C127—44no—
114C23×C127—44no—
115C23×C127—48no—
116C23×C127C22×C414yes—
117C23×C127C22×C412yes—
118C23×C127C2×D412yes—
119C23×C127#4112yes—
120C23×C127C2411yescharacteristic
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—22no—
124C2×C4×C127—22no—
125C2×C4×C127—22no—
126C2×C4×C127—22no—
127C2×C4×C127—22no—
128C22×C127—24no—
129C22×C127—22no—
130C22×C127—22no—
131C22×C127—24no—
132C22×C127—24no—
133C22×C127—24no—
134C22×C127—24no—
135C22×C127—416no—
136C22×C127—42no—
137C22×C127—42no—
138C22×C127—42no—
139C22×C127—42no—
140C22×C127C4×D414yes—
141C22×C127C23×C411yescharacteristic
142C22×C127C22×D411yescharacteristic
143C22×C127C2×#4111yescharacteristic
144C22×C127H(5,𝔽2)12yes—
145C22×C127H(5,𝔽2)12yes—
146C2×D127—22no—
147C2×D127—24no—
148C2×D127—41no—
149C2×D127—41no—
150C2×D127—42no—
151C4×C127—41no—
152C4×C127—41no—
153C4×C127—42no—
154Dic127—22no—
155Dic127—24no—
156Dic127—22no—
157Dic127—22no—
158Dic127—42no—
159C2×C127—24no—
160C2×C127—44no—
161C2×C127C2×C4×D411yescharacteristic
162C2×C127#51812yes—
163C2×C127#53411yescharacteristic
164C2×C127#53511yescharacteristic
165C2×C127#53812yes—
166D127—42no—
167#2321—1271nomaximal, Sylow
168C127#232111yescharacteristic, complemented, Sylow
169C2×#120—1271no—
170C2×C4×D4—1271no—
171C22×C22⋊C4—1271no—
172#390—1272no—
173#388—1272no—
174#388—1272no—
175#386—1272no—
176#386—1271no—
177#385—1271no—
178#379D12711yescharacteristic, complemented
179C4×C22⋊C4—1271no—
180C25C2×D12711yescharacteristic
181C22×D4—1272no—
182C22×D4—1271no—
183C23×C4—1271no—
184C23×C4—1271no—
185C23×C4—1271no—
186#120—1278no—
187C4×D4—2544no—
188C2×C4⋊C4—1272no—
189C2×C4⋊C4—1271no—
190C2×C22⋊C4—1271no—
191C2×C22⋊C4—1272no—
192C2×C22⋊C4—1272no—
193C2×C22⋊C4—1272no—
194C2×C22⋊C4—1272no—
195C2×C22⋊C4—1271no—
196C2×C22⋊C4—2544no—
197C2×C22⋊C4C2×D12711yescharacteristic
198C2×C22⋊C4C2×D12711yescharacteristic
199C2×C22⋊C4C2×D12712yes—
200C2×C22⋊C4C2×D12712yes—
201C2×C42—1272no—
202C2×C42—1271no—
203#95—1272no—
204#95—1272no—
205#95—1272no—
206C24—22no—
207C24—22no—
208C24—42no—
209C24—42no—
210C24—1271no—
211C24C4×D12714yescomplemented
212C24C22×D12712yes—
213C24C22×D12711yescharacteristic
214C2×D4—1278no—
215C2×D4—1274no—
216C2×D4—5082no—
217C2×D4—5082no—
218C2×D4—2542no—
219C2×D4—2542no—
220C22×C4—1274no—
221C22×C4—1271no—
222C22×C4—1272no—
223C22×C4—1271no—
224C22×C4—1271no—
225C22×C4—1272no—
226C22×C4—5082no—
227C22×C4—5082no—
228C22×C4—5082no—
229C22×C4—2542no—
230C22×C4—2542no—
231C22×C4C22×D12712yes—
232C22×C4C22×D12711yescharacteristic
233C22×C4C22×D12711yescharacteristic
234C4⋊C4—5082no—
235C4⋊C4—2542no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—1278no—
241C22⋊C4—1274no—
242C22⋊C4—5082no—
243C22⋊C4—2542no—
244C22⋊C4—2544no—
245C22⋊C4—2544no—
246C42—5082no—
247C42—2542no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—28no—
257C23—48no—
258C23—44no—
259C23—44no—
260C23—48no—
261C23—1272no—
262C23—2541no—
263C23—2541no—
264C23—2542no—
265C23—2542no—
266C23C2×C4×D12714yes—
267C23C2×C4×D12712yes—
268C23D4×D12712yes—
269C23#1452324712yes—
270C23C23×D12711yescharacteristic
271D4—5088no—
272D4—2548no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—1272no—
281C2×C4—1274no—
282C2×C4—1272no—
283C2×C4—1272no—
284C2×C4—5084no—
285C2×C4—5084no—
286C2×C4—5084no—
287C2×C4—2542no—
288C2×C4—2542no—
289C2×C4—2542no—
290C2×C4—2541no—
291C2×C4—2541no—
292C2×C4—2542no—
293C2×C4—2542no—
294C2×C4—2542no—
295C2×C4—2542no—
296C22—24no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—416no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—5081no—
309C22—5081no—
310C22—5082no—
311C22—2542no—
312C22—2544no—
313C22(C22⋊D127)⋊C414yes—
314C22C22×C4×D12711yescharacteristic
315C22C2×D4×D12711yescharacteristic
316C22C2×#1452324711yescharacteristic
317C22#1686125412yes—
318C22#1686125412yes—
319C4—41no—
320C4—41no—
321C4—42no—
322C4—5082no—
323C4—2542no—
324C4—2544no—
325C4—2542no—
326C4—2542no—
327C2—24no—
328C2—44no—
329C2—5082no—
330C2C2×(C22⋊D127)⋊C411yescharacteristic
331C2#2255034912yes—
332C2#2255043611yescharacteristic
333C2#2255044111yescharacteristic
334C2#2255046112yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#379×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861178C2
hypercenter#379D127
hypocenterC127#2321
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D127)⋊C4C2semidirect
3. C22×C22⋊Dic127C2semidirect
4. #22549776C2semidirect
5. #22549770C2semidirect
6. #22549764C2semidirect
7. #22549753C2semidirect
8. #22549752C2semidirect
9. #22549748C2semidirect
10. C22⋊C4×Dic127C2semidirect
11. #379×C127C2semidirect
16. C22×C4×D127C22semidirect
17. #16861178C4semidirect
19. C2×Dic127⋊C4C22semidirect
20. C2×Dic127⋊C4C22semidirect
26. C2×(C2×D127)⋊C4C22semidirect
27. C2×(C2×D127)⋊C4C22semidirect
28. C2×C4×Dic127C22semidirect
29. C2×C4×Dic127C22semidirect
30. C127⋊#95C22semidirect
31. C127⋊#95C22semidirect
32. C127⋊#95C22semidirect
44. C2×C22⋊D127C2×C4semidirect
48. C22×Dic127D4semidirect
60. C22⋊Dic127C2×C4semidirect
61. C22⋊Dic127C2×C4semidirect
70. C24×C127C2×C4semidirect
168. C127#2321semidirect
178. #379D127semidirect
211. C24C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611782
C2×(C22⋊D127)⋊C42
C22×C22⋊Dic1272
#225497762
#225497702
#225497642
#225497532
#225497522
#225497482
C22⋊C4×Dic1272
#379×C1272
#2321127