Finite Group Databasebeta

← #397×D127 #30641953

Subgroups — #397×D127 #30641953

Order 16256.

820 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D127C212yesmaximal
3C2×C22⋊C4×D127C211yesmaximal, characteristic
4#22549805C212yesmaximal, complemented
5#22549804C212yesmaximal, complemented
6#22549803C211yesmaximal, characteristic, complemented
7#22549802C211yesmaximal, characteristic, complemented
8#22549801C211yesmaximal, characteristic, complemented
9#95×D127C212yesmaximal, complemented
10#95×D127C211yesmaximal, characteristic, complemented
11#397×C127C211yesmaximal, characteristic, complemented
12C24×D127C2211yescharacteristic
13C22×C4×D127C2212yes—
14C22×C4×D127C2212yes—
15C22×C4×D127C2211yescharacteristic
16C22×C4×D127C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic, complemented
18C2×Dic127⋊C4C2212yescomplemented
19C4⋊C4×D127—24no—
20C2×C22⋊Dic127C2212yes—
21C2×C22⋊Dic127C2211yescharacteristic
22C2×(C2×D127)⋊C4C2211yescharacteristic
23C2×(C2×D127)⋊C4C2212yes—
24C2×(C2×D127)⋊C4C2212yes—
25C2×(C2×D127)⋊C4C2211yescharacteristic
26C22⋊C4×D127—28no—
27C22⋊C4×D127—24no—
28C127⋊#95C2211yescharacteristic, complemented
29C127⋊#95C2212yescomplemented
30C127⋊#95C2212yescomplemented
31C127⋊#95C2212yescomplemented
32C127⋊#95C2212yescomplemented
33C2×C4⋊C4×C127C2211yescharacteristic, complemented
34C2×C22⋊C4×C127C2211yescharacteristic
35C2×C22⋊C4×C127C2212yes—
36#95×C127C2211yescharacteristic, complemented
37#95×C127C2212yescomplemented
38C23×D127—22no—
39C23×D127—22no—
40C23×D127—24no—
41C23×D127—24no—
42C23×D127D414yes—
43C23×D127C2311yescharacteristic
44C23×D127C2311yescharacteristic
45C22×Dic127C2312yes—
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2312yes—
48C22×Dic127C2311yescharacteristic
49C2×C4×D127—24no—
50C2×C4×D127—24no—
51C2×C4×D127—24no—
52C2×C4×D127—22no—
53C2×C4×D127—24no—
54C2×C4×D127—24no—
55C2×C4×D127—24no—
56C2×C4×D127—24no—
57C2×C4×D127—24no—
58C2×C4×D127D414yes—
59C4⋊Dic127—22no—
60Dic127⋊C4—24no—
61C22⋊Dic127—22no—
62C22⋊Dic127—24no—
63(C2×D127)⋊C4—24no—
64(C2×D127)⋊C4—24no—
65(C2×D127)⋊C4—22no—
66(C2×D127)⋊C4—22no—
67C24×C127C2311yescharacteristic
68C22×C4×C127C2312yes—
69C22×C4×C127C2311yescharacteristic
70C22×C4×C127C2312yes—
71C22×C4×C127C2311yescharacteristic
72C4⋊C4×C127—22no—
73C22⋊C4×C127—24no—
74C22⋊C4×C127—22no—
75C22×D127—21no—
76C22×D127—21no—
77C22×D127—22no—
78C22×D127—22no—
79C22×D127—28no—
80C22×D127—216no—
81C22×D127—44no—
82C22×D127—44no—
83C22×D127—48no—
84C22×D127C2×D412yes—
85C22×D127C2×D412yes—
86C22×D127C2×D412yes—
87C22×D127#4112yes—
88C22×D127#4114yes—
89C22×D127#4114yes—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—22no—
95C2×Dic127—22no—
96C2×Dic127—22no—
97C2×Dic127—22no—
98C2×Dic127—22no—
99C2×Dic127C2×D412yes—
100C4×D127—28no—
101C4×D127—48no—
102C4×D127—48no—
103C4×D127—44no—
104C23×C127—21no—
105C23×C127—21no—
106C23×C127—22no—
107C23×C127—22no—
108C23×C127C2×D412yes—
109C23×C127C2411yescharacteristic
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127C2×D412yes—
120C22×C127—22no—
121C22×C127—24no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—42no—
125C22×C127C22×D411yescharacteristic
126C22×C127C22×D411yescharacteristic
127C22×C127C2×#4111yescharacteristic
128C22×C127C2×#4112yes—
129C22×C127C2×#4112yes—
130C2×D127—22no—
131C2×D127—24no—
132C2×D127—41no—
133C2×D127—41no—
134C2×D127—42no—
135C2×D127—416no—
136C2×D127#12114yes—
137C2×D127#12314yes—
138C2×D127#12318yes—
139C2×D127#12414yes—
140C2×D127#12618yes—
141C4×C127—22no—
142C4×C127—42no—
143C4×C127—42no—
144C4×C127—41no—
145Dic127—22no—
146Dic127—42no—
147Dic127—42no—
148Dic127—41no—
149C2×C127—42no—
150C2×C127C2×#12111yescharacteristic
151C2×C127C2×#12311yescharacteristic
152C2×C127C2×#12312yes—
153C2×C127C2×#12411yescharacteristic
154C2×C127C2×#12612yes—
155D127—42no—
156D127#39718yesdirect product factor
157C2×#397—1271nomaximal, Sylow
158C127C2×#39711yescharacteristic, complemented, Sylow
159C22×C4⋊C4—1271no—
160C22×C22⋊C4—1271no—
161C22×C22⋊C4—1272no—
162#397—1271no—
163#397—1271no—
164#397—1272no—
165#397—1271no—
166#397—1272no—
167#397D12711yescharacteristic, direct product factor
168C2×#95—1271no—
169C2×#95—1272no—
170C25—1271no—
171C23×C4—1271no—
172C23×C4—1272no—
173C23×C4—1271no—
174C23×C4—1272no—
175C2×C4⋊C4—1272no—
176C2×C4⋊C4—1271no—
177C2×C4⋊C4—2544no—
178C2×C4⋊C4C2×D12711yescharacteristic, complemented
179C2×C22⋊C4—1271no—
180C2×C22⋊C4—1272no—
181C2×C22⋊C4—1271no—
182C2×C22⋊C4—1272no—
183C2×C22⋊C4—1272no—
184C2×C22⋊C4—1271no—
185C2×C22⋊C4—2544no—
186C2×C22⋊C4—2548no—
187C2×C22⋊C4C2×D12712yes—
188C2×C22⋊C4C2×D12711yescharacteristic
189#95—1272no—
190#95—1272no—
191#95—1271no—
192#95—1272no—
193#95—1272no—
194#95C2×D12712yescomplemented
195#95C2×D12711yescharacteristic, complemented
196C24—1271no—
197C24—1274no—
198C24—1271no—
199C24—2542no—
200C24—2542no—
201C24—2544no—
202C24—2544no—
203C24C22×D12711yescharacteristic
204C22×C4—1271no—
205C22×C4—1274no—
206C22×C4—1272no—
207C22×C4—1271no—
208C22×C4—1272no—
209C22×C4—2542no—
210C22×C4—2544no—
211C22×C4—2544no—
212C22×C4—2544no—
213C22×C4—2544no—
214C22×C4—2544no—
215C22×C4—2544no—
216C22×C4—2544no—
217C22×C4—2544no—
218C22×C4C22×D12712yes—
219C22×C4C22×D12711yescharacteristic
220C22×C4C22×D12712yes—
221C22×C4C22×D12711yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—2544no—
224C4⋊C4—2542no—
225C22⋊C4—24no—
226C22⋊C4—22no—
227C22⋊C4—2542no—
228C22⋊C4—2544no—
229C22⋊C4—2544no—
230C22⋊C4—2542no—
231C22⋊C4—2544no—
232C22⋊C4—2542no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—1272no—
238C23—1272no—
239C23—1272no—
240C23—1274no—
241C23—1274no—
242C23—1272no—
243C23—5084no—
244C23—5084no—
245C23—5088no—
246C23—2548no—
247C23—25416no—
248C23—2541no—
249C23—2542no—
250C23—2542no—
251C23—2541no—
252C23D4×D12712yes—
253C23C23×D12711yescharacteristic
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—1272no—
264C2×C4—5088no—
265C2×C4—5088no—
266C2×C4—5084no—
267C2×C4—2548no—
268C2×C4—2542no—
269C2×C4—2542no—
270C2×C4—2541no—
271C2×C4—2542no—
272C2×C4—2542no—
273C2×C4—2542no—
274C2×C4—2542no—
275C2×C4—2542no—
276C2×C4—2542no—
277C2×C4D4×D12712yes—
278C22—22no—
279C22—24no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—1274no—
284C22—1278no—
285C22—1274no—
286C22—1274no—
287C22—1278no—
288C22—50816no—
289C22—5081no—
290C22—5081no—
291C22—5082no—
292C22—2542no—
293C22—2544no—
294C22C2×D4×D12711yescharacteristic
295C22C2×D4×D12711yescharacteristic
296C22#41×D12711yescharacteristic
297C22#41×D12712yes—
298C22#41×D12712yes—
299C4—22no—
300C4—42no—
301C4—42no—
302C4—41no—
303C4—5082no—
304C4—5082no—
305C4—5081no—
306C4—2542no—
307C2—42no—
308C2—1278no—
309C2—5082no—
310C2#121×D12711yescharacteristic
311C2#123×D12711yescharacteristic
312C2#123×D12712yes—
313C2#124×D12711yescharacteristic
314C2#126×D12712yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#397×C127C2
prime radicalC24×D127C22
hypercenter#397D127
hypocenterC127C2×#397
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D127C2semidirect
4. #22549805C2semidirect
5. #22549804C2semidirect
6. #22549803C2semidirect
7. #22549802C2semidirect
8. #22549801C2semidirect
9. #95×D127C2semidirect
10. #95×D127C2semidirect
11. #397×C127C2semidirect
17. C2×C4⋊Dic127C22semidirect
18. C2×Dic127⋊C4C22semidirect
28. C127⋊#95C22semidirect
29. C127⋊#95C22semidirect
30. C127⋊#95C22semidirect
31. C127⋊#95C22semidirect
32. C127⋊#95C22semidirect
33. C2×C4⋊C4×C127C22semidirect
36. #95×C127C22semidirect
37. #95×C127C22semidirect
156. D127#397direct (N × H)
158. C127C2×#397semidirect
167. #397D127direct (N × H)
178. C2×C4⋊C4C2×D127semidirect
194. #95C2×D127semidirect
195. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1272
C2×C22⋊C4×D1272
#225498052
#225498042
#225498032
#225498022
#225498012
#95×D1272
#397×C1272
C2×#397127