Finite Group Databasebeta

← #30648725

Subgroups — #30648725

Order 16256.

492 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550421C211yesmaximal, characteristic, complemented
2C2×D4×Dic127C211yesmaximal, characteristic, complemented
3#22550190C211yesmaximal, characteristic, complemented
4#22550187C211yesmaximal, characteristic, complemented
5#22550154C212yesmaximal, complemented
6#22550069C212yesmaximal, complemented
7#22550010C211yesmaximal, characteristic, complemented
8D8×Dic127C212yesmaximal, complemented
9#22549956C212yesmaximal, complemented
10#22549926C211yesmaximal, characteristic, complemented
11#447×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C23×Dic127—21no—
14C2×#14523233C2212yes—
15#16861208C2211yescharacteristic, complemented
16#16861205C2211yescharacteristic, complemented
17#16861195—21no—
18#16861183C2211yescharacteristic
19#16861181—21no—
20D4×Dic127—21no—
21D4×Dic127—21no—
22D4×Dic127C2212yes—
23D4×Dic127C2212yes—
24D4×Dic127C2211yescharacteristic, complemented
25#16861171C2211yescharacteristic, complemented
26(C22⋊D127)⋊C4—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32Dic127⋊C8C2212yescomplemented
33#16861131C2212yescomplemented
34#16861130C2212yes—
35#16861130C2212yescomplemented
36C22⋊(C127⋊C8)C2211yescharacteristic
37C8×Dic127C2212yescomplemented
38C22×D4×C127C2211yescharacteristic
39C2×D8×C127C2212yes—
40#121×C127C2211yescharacteristic
41#102×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43#14523247—44no—
44C2×Dic254—21no—
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—21no—
47C2×C22⋊D127—21no—
48C2×C22⋊D127—21no—
49C22×Dic127—22no—
50C22×Dic127—22no—
51C22×Dic127—21no—
52C22×Dic127—21no—
53C22×Dic127—44no—
54C22×Dic127D411yescharacteristic
55C22×Dic127D411yescharacteristic
56C2×C4×D127—21no—
57#14523233—24no—
58C2×C127⋊C8C2312yes—
59C4⋊Dic127—21no—
60C4⋊Dic127C2311yescharacteristic
61C4⋊Dic127C2312yes—
62Dic127⋊C4—21no—
63C22⋊Dic127—21no—
64C22⋊Dic127—22no—
65C22⋊Dic127—22no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68C4×Dic127D411yescharacteristic
69C4×Dic127D411yescharacteristic
70C4×Dic127C2311yescharacteristic
71C4×Dic127C2311yescharacteristic
72C24×C127—21no—
73C2×D4×C127—24no—
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127C2311yescharacteristic
77C2×D4×C127C2312yes—
78C22×C4×C127C2311yescharacteristic
79D8×C127—24no—
80C2×C8×C127C2312yes—
81C4⋊C4×C127C2311yescharacteristic
82C22⋊C4×C127—21no—
83C22×D127—21no—
84Dic254—42no—
85D508—22no—
86C22⋊D127—42no—
87C22⋊D127—42no—
88C22⋊D127—42no—
89C2×Dic127—21no—
90C2×Dic127—22no—
91C2×Dic127—41no—
92C2×Dic127—42no—
93C2×Dic127—44no—
94C2×Dic127—44no—
95C2×Dic127—42no—
96C2×Dic127D814yescomplemented
97C2×Dic127C2×D411yescharacteristic
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D411yescharacteristic
100C2×Dic127C2×D411yescharacteristic
101C4×D127—42no—
102C127⋊C8—22no—
103C23×C127—24no—
104C23×C127—22no—
105C23×C127—21no—
106C23×C127—42no—
107C23×C127—42no—
108C23×C127C2×D411yescharacteristic
109D4×C127—24no—
110D4×C127—22no—
111D4×C127—22no—
112D4×C127—42no—
113D4×C127#4114yescomplemented
114C2×C4×C127—22no—
115C2×C4×C127—21no—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2411yescharacteristic
118C8×C127—22no—
119C22×C127—21no—
120C22×C127—21no—
121C22×C127—24no—
122C22×C127—48no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127—41no—
128C22×C127C2×D812yes—
129C22×C127C22×D411yescharacteristic
130C2×D127—41no—
131C2×D127—41no—
132C2×D127—41no—
133C4×C127—21no—
134C4×C127—41no—
135C4×C127C2×#4112yes—
136Dic127—21no—
137Dic127—21no—
138Dic127—42no—
139Dic127—41no—
140Dic127C2×D812yes—
141Dic127C2×D812yes—
142C2×C127—22no—
143C2×C127—44no—
144C2×C127—81no—
145C2×C127#52511yescharacteristic
146C2×C127C22×D811yescharacteristic
147C2×C127#57511yescharacteristic
148D127—81no—
149#3100—1271nomaximal, Sylow
150C127#310011yescharacteristic, complemented, Sylow
151#532—1271no—
152C2×C4×D4—1271no—
153#480—1271no—
154#480—1271no—
155#474—1272no—
156#459—1272no—
157#447—1271no—
158#447D12711yescharacteristic, complemented
159C4×D8—1272no—
160C4×D8—1272no—
161#431—1271no—
162C2×#41—2541no—
163C22×D4C2×D12711yescharacteristic
164C23×C4—2541no—
165C2×D8—1272no—
166C2×D8C2×D12712yes—
167C4⋊Q8—1271no—
168#127—1271no—
169#124—2541no—
170#121—1271no—
171#121—2541no—
172#121C2×D12711yescharacteristic
173C4×D4—1272no—
174C4×D4—1272no—
175C4×D4—1271no—
176C4×D4—1271no—
177C4×D4—2541no—
178C4×D4—2541no—
179C4×D4—2541no—
180C2×C4⋊C4—1271no—
181C2×C22⋊C4—2541no—
182C2×C42—1271no—
183#107—1272no—
184#107—1272no—
185C4⋊C8—1272no—
186#102—1272no—
187#102—1272no—
188#102—1272no—
189#102C2×D12712yes—
190C22⋊C8—1271no—
191C22⋊C8C2×D12711yescharacteristic
192C4×C8—1272no—
193C24—21no—
194#41—5084no—
195C2×Q8—2541no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—1271no—
200C2×D4—2541no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4C22×D12711yescharacteristic
204C2×D4C22×D12712yes—
205C22×C4—1271no—
206C22×C4—1271no—
207C22×C4—5084no—
208C22×C4—2541no—
209C22×C4—2542no—
210C22×C4—2542no—
211C22×C4—2541no—
212C22×C4—2541no—
213C22×C4C22×D12711yescharacteristic
214D8—24no—
215D8—2544no—
216C2×C8—1272no—
217C2×C8C22×D12712yes—
218C4⋊C4—1272no—
219C4⋊C4—1271no—
220C4⋊C4—2541no—
221C4⋊C4—2541no—
222C4⋊C4C22×D12711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2541no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C22⋊C4—2541no—
228C22⋊C4—2541no—
229C42—1271no—
230C42—1271no—
231C42—1271no—
232C42—1271no—
233C23—24no—
234C23—22no—
235C23—21no—
236C23—42no—
237C23—42no—
238C23—2541no—
239C23D4×D12711yescharacteristic
240Q8—5082no—
241D4—24no—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—5082no—
246D4—5082no—
247D4—5082no—
248D4—2542no—
249D4#1452324714yescomplemented
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—1274no—
253C2×C4—1271no—
254C2×C4—1271no—
255C2×C4—1271no—
256C2×C4—1271no—
257C2×C4—5082no—
258C2×C4—5084no—
259C2×C4—5084no—
260C2×C4—5082no—
261C2×C4—5082no—
262C2×C4—5081no—
263C2×C4—2542no—
264C2×C4—2541no—
265C2×C4D4×D12711yescharacteristic
266C2×C4C23×D12711yescharacteristic
267C8—22no—
268C8—2542no—
269C22—21no—
270C22—21no—
271C22—24no—
272C22—48no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—5081no—
279C22—5081no—
280C22—5081no—
281C22D8×D12712yes—
282C22C2×D4×D12711yescharacteristic
283C4—21no—
284C4—41no—
285C4—1272no—
286C4—1272no—
287C4—5082no—
288C4—5081no—
289C4—2541no—
290C4—2541no—
291C4C2×#1452324712yes—
292C2—22no—
293C2—44no—
294C2—81no—
295C2—10161no—
296C2#2255038211yescharacteristic
297C2C2×D8×D12711yescharacteristic
298C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#447×C127C2
socleC22×C127C22×D4
hypercenter#447D127
hypocenterC127#3100
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22550421C2semidirect
2. C2×D4×Dic127C2semidirect
3. #22550190C2semidirect
4. #22550187C2semidirect
5. #22550154C2semidirect
6. #22550069C2semidirect
7. #22550010C2semidirect
8. D8×Dic127C2semidirect
9. #22549956C2semidirect
10. #22549926C2semidirect
11. #447×C127C2semidirect
15. #16861208C22semidirect
16. #16861205C22semidirect
24. D4×Dic127C22semidirect
25. #16861171C22semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. Dic127⋊C8C22semidirect
33. #16861131C22semidirect
35. #16861130C22semidirect
37. C8×Dic127C22semidirect
96. C2×Dic127D8semidirect
113. D4×C127#41semidirect
150. C127#3100semidirect
158. #447D127semidirect
249. D4#14523247semidirect

Maximal subgroups

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

subgroupindex
#225504212
C2×D4×Dic1272
#225501902
#225501872
#225501542
#225500692
#225500102
D8×Dic1272
#225499562
#225499262
#447×C1272
#3100127