Finite Group Databasebeta

← #30641655

Subgroups — #30641655

Order 16256.

660 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×C4×C22⋊D127C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic, complemented
3#22549766C212yesmaximal, complemented
4#22549757C212yesmaximal, complemented
5#22549755C212yesmaximal
6#22549741C211yesmaximal, characteristic, complemented
7#22549736C211yesmaximal, characteristic, complemented
8#22549733C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic127C211yesmaximal, characteristic, complemented
10C4×C22⋊Dic127C212yesmaximal, complemented
11C22×C4⋊C4×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14C4×C22⋊D127C418yescomplemented
15#16861167—24no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic
18C2×Dic127⋊C4C2211yescharacteristic
19C2×C22⋊Dic127C2212yes—
20C2×C22⋊Dic127C2211yescharacteristic
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2212yes—
23C2×(C2×D127)⋊C4C2211yescharacteristic
24C2×C4×Dic127C2212yes—
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2211yescharacteristic
27C127⋊#95C2212yes—
28C127⋊#95C2212yes—
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C23×C4×C127C2212yes—
33C23×C4×C127C2211yescharacteristic
34C2×C4⋊C4×C127—24no—
35C2×C4⋊C4×C127C2212yes—
36C2×C4⋊C4×C127C2211yescharacteristic
37C2×C4⋊C4×C127C2211yescharacteristic
38C23×D127C2311yescharacteristic
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C2×C22⋊D127C2×C414yes—
42C22×Dic127C2312yes—
43C22×Dic127C2312yes—
44C22×Dic127C2311yescharacteristic
45C22×Dic127C2311yescharacteristic
46C22×Dic127C2311yescharacteristic
47C2×C4×D127—22no—
48C2×C4×D127—22no—
49C2×C4×D127C2×C414yescomplemented
50C4⋊Dic127—22no—
51C4⋊Dic127—22no—
52Dic127⋊C4C2×C414yescomplemented
53C22⋊Dic127—24no—
54C22⋊Dic127C2×C414yes—
55(C2×D127)⋊C4—24no—
56(C2×D127)⋊C4—42no—
57(C2×D127)⋊C4C2×C414yescomplemented
58C4×Dic127—24no—
59C4×Dic127—42no—
60C4×Dic127C2×C414yescomplemented
61C24×C127C2311yescharacteristic
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127—24no—
65C22×C4×C127—24no—
66C22×C4×C127—42no—
67C22×C4×C127C2×C414yes—
68C22×C4×C127C2312yes—
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2312yes—
71C22×C4×C127C2311yescharacteristic
72C4⋊C4×C127—24no—
73C4⋊C4×C127—22no—
74C4⋊C4×C127D414yescomplemented
75C22×D127—21no—
76C22×D127—21no—
77C22×D127—21no—
78C22×D127—21no—
79C22×D127—21no—
80C22×D127—21no—
81C22×D127C22×C412yes—
82C22⋊D127—28no—
83C2×Dic127—21no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—21no—
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×Dic127C22×C412yes—
96C2×Dic127C22×C412yes—
97C2×Dic127C22×C412yes—
98C4×D127—28no—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127—21no—
103C23×C127C22×C412yes—
104C23×C127#4112yes—
105C23×C127#4112yes—
106C23×C127C2411yescharacteristic
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—28no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—48no—
116C2×C4×C127C22×C412yes—
117C2×C4×C127C22×C412yes—
118C2×C4×C127C2×D414yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127#4112yes—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—22no—
127C22×C127#11714yescomplemented
128C22×C127C23×C411yescharacteristic
129C22×C127C22×D411yescharacteristic
130C22×C127C2×#4111yescharacteristic
131C22×C127C2×#4111yescharacteristic
132C22×C127C2×#4111yescharacteristic
133C22×C127H(5,𝔽2)11yescharacteristic
134C22×C1272−1+411yescharacteristic
135C2×D127—22no—
136C2×D127—22no—
137C2×D127—22no—
138C2×D127—41no—
139C2×D127—41no—
140C2×D127—41no—
141C2×D127—41no—
142C4×C127—22no—
143C4×C127—24no—
144C4×C127—42no—
145C4×C127C4×D414yes—
146Dic127—22no—
147Dic127—22no—
148Dic127—22no—
149Dic127—42no—
150Dic127—42no—
151C2×C127—24no—
152C2×C127C2×#11711yescharacteristic
153C2×C127C2×C4×D411yescharacteristic
154C2×C127#52011yescharacteristic
155C2×C127#54611yescharacteristic
156C2×C127#54811yescharacteristic
157C2×C127#55211yescharacteristic
158C2×C127#55311yescharacteristic
159D127—42no—
160#2352—1271nomaximal, Sylow
161C127#235211yescharacteristic, complemented, Sylow
162C2×C4×D4—1271no—
163C2×#117—1271no—
164C22×C4⋊C4D12711yescharacteristic, complemented
165#388—1272no—
166#387—1272no—
167#386—1272no—
168#382—1271no—
169#380—1271no—
170#380—1271no—
171C4×C4⋊C4—1271no—
172C4×C22⋊C4—1272no—
173C22×D4—1271no—
174C23×C4—1271no—
175C23×C4C2×D12712yes—
176C23×C4C2×D12711yescharacteristic
177C4×D4—1278no—
178#117—2544no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—1271no—
181C2×C4⋊C4—1271no—
182C2×C4⋊C4—1271no—
183C2×C4⋊C4C2×D12711yescharacteristic
184C2×C4⋊C4C2×D12712yes—
185C2×C4⋊C4C2×D12711yescharacteristic
186C2×C22⋊C4—1271no—
187C2×C22⋊C4—1271no—
188C2×C22⋊C4—1272no—
189C2×C22⋊C4—1272no—
190C2×C22⋊C4—1272no—
191C2×C42—1271no—
192C2×C42—1272no—
193C2×C42—1272no—
194#95—1272no—
195#95—1272no—
196#95—1272no—
197#95—1272no—
198#95—1272no—
199C24—1271no—
200C24C22×D12711yescharacteristic
201C2×D4—1274no—
202C2×D4—2542no—
203C2×D4—2542no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—42no—
209C22×C4—1271no—
210C22×C4—1271no—
211C22×C4—1271no—
212C22×C4—1274no—
213C22×C4—1272no—
214C22×C4—1272no—
215C22×C4—2542no—
216C22×C4—2542no—
217C22×C4C4×D12714yes—
218C22×C4C22×D12712yes—
219C22×C4C22×D12711yescharacteristic
220C22×C4C22×D12712yes—
221C22×C4C22×D12711yescharacteristic
222C4⋊C4—24no—
223C4⋊C4—22no—
224C4⋊C4—1274no—
225C4⋊C4—2542no—
226C4⋊C4—2542no—
227C4⋊C4C22⋊D12714yescomplemented
228C22⋊C4—1274no—
229C22⋊C4—1274no—
230C22⋊C4—5082no—
231C22⋊C4—2544no—
232C22⋊C4—2544no—
233C42—1274no—
234C42—5082no—
235C42—2544no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—1272no—
241C23—2541no—
242C23—2541no—
243C23—2541no—
244C23—2541no—
245C23—2541no—
246C23—2541no—
247C23C2×C4×D12712yes—
248C23#1452324712yes—
249C23#1452324912yes—
250C23C23×D12711yescharacteristic
251D4—2548no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—28no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—48no—
261C2×C4—1272no—
262C2×C4—1272no—
263C2×C4—1272no—
264C2×C4—2542no—
265C2×C4—2542no—
266C2×C4—2542no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4—2542no—
270C2×C4—2541no—
271C2×C4—2541no—
272C2×C4—2541no—
273C2×C4—2541no—
274C2×C4—2541no—
275C2×C4—2541no—
276C2×C4—2548no—
277C2×C4C2×C4×D12712yes—
278C2×C4C2×C4×D12712yes—
279C2×C4C2×C22⋊D12714yes—
280C2×C4C2×C22⋊D12712yes—
281C2×C4#1452324812yes—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—5081no—
289C22—5081no—
290C22—5081no—
291C22—5081no—
292C22—2542no—
293C22—2542no—
294C22—2542no—
295C22#1686116714yescomplemented
296C22C22×C4×D12711yescharacteristic
297C22C22×C22⋊D12711yescharacteristic
298C22C2×#1452324711yescharacteristic
299C22C2×#1452324811yescharacteristic
300C22C2×#1452324911yescharacteristic
301C22#1686125411yescharacteristic
302C22#1686125711yescharacteristic
303C4—22no—
304C4—24no—
305C4—42no—
306C4—5082no—
307C4—5082no—
308C4—2542no—
309C4—2542no—
310C4—2542no—
311C4C4×C22⋊D12714yes—
312C2—24no—
313C2—5082no—
314C2C2×#1686116711yescharacteristic
315C2C2×C4×C22⋊D12711yescharacteristic
316C2#2255035711yescharacteristic
317C2#2255052011yescharacteristic
318C2#2255053711yescharacteristic
319C2#2255056111yescharacteristic
320C2#2255057411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC22×C4⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC22×C22⋊D127C22
hypercenterC22×C4⋊C4D127
hypocenterC127#2352
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16861167C2semidirect
3. #22549766C2semidirect
4. #22549757C2semidirect
6. #22549741C2semidirect
7. #22549736C2semidirect
8. #22549733C2semidirect
9. C4⋊C4×Dic127C2semidirect
10. C4×C22⋊Dic127C2semidirect
11. C22×C4⋊C4×C127C2semidirect
14. C4×C22⋊D127C4semidirect
49. C2×C4×D127C2×C4semidirect
52. Dic127⋊C4C2×C4semidirect
57. (C2×D127)⋊C4C2×C4semidirect
60. C4×Dic127C2×C4semidirect
74. C4⋊C4×C127D4semidirect
127. C22×C127#117semidirect
161. C127#2352semidirect
164. C22×C4⋊C4D127semidirect
227. C4⋊C4C22⋊D127semidirect
295. C22#16861167semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1272
C2×#168611672
#225497662
#225497572
#225497552
#225497412
#225497362
#225497332
C4⋊C4×Dic1272
C4×C22⋊Dic1272
C22×C4⋊C4×C1272
#2352127