Finite Group Databasebeta

← #30647188

Subgroups — #30647188

Order 16256.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#16861254C211yesmaximal, characteristic
2#22550349C212yesmaximal, complemented
3#22549848C214yesmaximal, complemented
4C2×#16861124C211yesmaximal, characteristic, complemented
5C2×#16861123C212yesmaximal, complemented
6#99×D127C214yesmaximal, complemented
7C2×#99×C127C211yesmaximal, characteristic, complemented
8#16861254—24no—
9#16861254—24no—
10C2×#14523248C412yescomplemented
11C2×#14523247C412yescomplemented
12C2×#14523247C2212yes—
13C22×C22⋊D127C412yescomplemented
14C22×C22⋊D127C2211yescharacteristic
15C22×C22⋊D127C2211yescharacteristic
16C2×D4×D127C412yescomplemented
17C2×D4×D127C2212yes—
18C4×C22⋊D127—24no—
19(C22⋊D127)⋊C4—24no—
20#16861165C2214yescomplemented
21C2×C22⋊Dic127C2212yescomplemented
22C22⋊C4×D127C2214yescomplemented
23#16861124C2214yescomplemented
24#16861123C2218yescomplemented
25C22×D4×C127C2211yescharacteristic
26C2×C22⋊C4×C127C2212yescomplemented
27#99×C127C2214yescomplemented
28C23×D127C2×C411yescharacteristic, complemented
29C23×D127C2×C412yescomplemented
30C23×D127C2311yescharacteristic
31#14523248—24no—
32#14523248—24no—
33#14523247—28no—
34#14523247—28no—
35C2×Dic254C2×C412yescomplemented
36C2×D508C2×C412yescomplemented
37C2×C22⋊D127—24no—
38C2×C22⋊D127—24no—
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C2×C22⋊D127C2×C412yes—
42C2×C22⋊D127C2×C412yes—
43C2×C22⋊D127C2×C412yescomplemented
44C2×C22⋊D127C2×C412yescomplemented
45C2×C22⋊D127C2×C412yescomplemented
46C2×C22⋊D127D414yes—
47C2×C22⋊D127D414yes—
48C2×C22⋊D127C2312yes—
49D4×D127—28no—
50D4×D127—28no—
51C22×Dic127—22no—
52C22×Dic127C2×C412yescomplemented
53C22×Dic127C2×C411yescharacteristic, complemented
54C22×Dic127C2311yescharacteristic
55C2×C4×D127—24no—
56C2×C4×D127C2×C412yescomplemented
57C2×C4×D127C2×C412yescomplemented
58Dic127⋊C4—24no—
59C22⋊Dic127—22no—
60C22⋊Dic127C2314yes—
61(C2×D127)⋊C4—24no—
62C4×Dic127—24no—
63C24×C127C2×C411yescharacteristic, complemented
64C24×C127C2311yescharacteristic
65C2×D4×C127—22no—
66C2×D4×C127—22no—
67C2×D4×C127C2×C412yescomplemented
68C2×D4×C127C2312yes—
69C22×C4×C127—22no—
70C22×C4×C127C2×C411yescharacteristic, complemented
71C22⋊C4×C127—22no—
72C22⋊C4×C127C2314yes—
73C22×D127—22no—
74C22×D127—24no—
75C22×D127—81no—
76C22×D127—82no—
77C22×D127—81no—
78C22×D127C22×C412yes—
79C22×D127C22×C412yes—
80C22×D127C2×D414yes—
81Dic254—24no—
82D508—24no—
83C22⋊D127—22no—
84C22⋊D127—22no—
85C22⋊D127—24no—
86C22⋊D127—24no—
87C22⋊D127—24no—
88C22⋊D127C22⋊C418yes—
89C22⋊D127C22⋊C414yes—
90C22⋊D127C22⋊C414yes—
91C2×Dic127—22no—
92C2×Dic127—24no—
93C2×Dic127—24no—
94C2×Dic127—82no—
95C2×Dic127C22×C412yes—
96C2×Dic127C22×C412yes—
97C2×Dic127C2×D414yes—
98C4×D127—24no—
99C4×D127—24no—
100C4×D127—84no—
101C23×C127—21no—
102C23×C127—21no—
103C23×C127—81no—
104C23×C127—81no—
105C23×C127C22×C412yes—
106C23×C127C22×C411yescharacteristic
107C23×C127C22×C411yescharacteristic
108C23×C127C2×D412yes—
109C23×C127C2×D412yes—
110C23×C127C2411yescharacteristic
111D4×C127—24no—
112D4×C127—24no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—24no—
116C2×C4×C127—82no—
117C2×C4×C127C22×C412yes—
118C22×C127—22no—
119C22×C127—42no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127—42no—
123C22×C127—41no—
124C22×C127—42no—
125C22×C127—81no—
126C22×C127—81no—
127C22×C127C2×C22⋊C411yescharacteristic
128C22×C127C2×C22⋊C411yescharacteristic
129C22×C127C2×C22⋊C412yes—
130C22×C127C23×C411yescharacteristic
131C22×C127C22×D412yes—
132C2×D127—22no—
133C2×D127—44no—
134C2×D127—42no—
135C2×D127—42no—
136C2×D127—84no—
137C2×D127—82no—
138C2×D127C2×C22⋊C414yes—
139C4×C127—22no—
140C4×C127—44no—
141Dic127—22no—
142Dic127—44no—
143Dic127C2×C22⋊C414yes—
144C2×C127—22no—
145C2×C127—21no—
146C2×C127—22no—
147C2×C127—21no—
148C2×C127—21no—
149C2×C127—42no—
150C2×C127C22×C22⋊C411yescharacteristic
151D127—24no—
152D127—42no—
153#2883—1271nomaximal, Sylow
154C127#288311yescharacteristic, complemented, Sylow
155C2×H(5,𝔽2)—1271no—
156#518—1272no—
157#410—1274no—
158C2×#99—1271no—
159C2×#99—1272no—
160C2×#99—1274no—
161C2×#99D12711yescharacteristic, complemented
162H(5,𝔽2)—2544no—
163H(5,𝔽2)—2544no—
164C2×#41—1272no—
165C2×#41—1272no—
166C2×#41—1272no—
167C22×D4—1271no—
168C22×D4—1272no—
169C22×D4—1271no—
170C22×D4—1272no—
171C22×D4—1272no—
172C22×D4C2×D12711yescharacteristic
173C4×D4—2544no—
174C4×D4—2544no—
175#117—1274no—
176C2×C22⋊C4—1272no—
177C2×C22⋊C4—1274no—
178C2×C22⋊C4C2×D12712yescomplemented
179#99—1274no—
180#99—1278no—
181#99C2×D12714yescomplemented
182C24—1271no—
183C24—1272no—
184C24—1271no—
185C24C4×D12711yescharacteristic, complemented
186C24C22×D12711yescharacteristic
187#41—2544no—
188#41—2544no—
189#41—2548no—
190#41—2548no—
191C2×Q8—1272no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—1272no—
195C2×D4—1272no—
196C2×D4—1274no—
197C2×D4—1274no—
198C2×D4—1272no—
199C2×D4—1272no—
200C2×D4—1272no—
201C2×D4—1272no—
202C2×D4—1272no—
203C2×D4—2548no—
204C2×D4—2548no—
205C2×D4—2544no—
206C2×D4—2544no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4C4×D12712yescomplemented
210C2×D4C22×D12712yes—
211C22×C4—22no—
212C22×C4—1271no—
213C22×C4—1272no—
214C22×C4—1271no—
215C22×C4—1272no—
216C22×C4—1272no—
217C22×C4—2544no—
218C22×C4—2542no—
219C22×C4C4×D12711yescharacteristic, complemented
220C4⋊C4—2544no—
221C22⋊C4—22no—
222C22⋊C4—1274no—
223C22⋊C4—2544no—
224C22⋊C4—2542no—
225C22⋊C4C22×D12714yes—
226C42—2544no—
227C23—21no—
228C23—21no—
229C23—81no—
230C23—81no—
231C23—1272no—
232C23—1274no—
233C23—1272no—
234C23—10161no—
235C23—10162no—
236C23—10161no—
237C23—2544no—
238C23—2542no—
239C23C2×C4×D12712yes—
240C23C2×C4×D12711yescharacteristic
241C23C2×C4×D12711yescharacteristic
242C23D4×D12712yes—
243C23D4×D12712yes—
244C23C23×D12711yescharacteristic
245Q8—2544no—
246D4—24no—
247D4—24no—
248D4—1274no—
249D4—1274no—
250D4—1278no—
251D4—2544no—
252D4—2544no—
253D4—2544no—
254D4—2544no—
255D4—2542no—
256D4—2542no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—82no—
261C2×C4—1272no—
262C2×C4—1274no—
263C2×C4—1272no—
264C2×C4—10164no—
265C2×C4—10162no—
266C2×C4—2544no—
267C2×C4—2544no—
268C2×C4—2544no—
269C2×C4—2544no—
270C2×C4—2542no—
271C2×C4C2×C4×D12712yes—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—1274no—
282C22—10162no—
283C22—10164no—
284C22—5082no—
285C22—5084no—
286C22—5082no—
287C22—2542no—
288C22C22⋊C4×D12711yescharacteristic
289C22C22⋊C4×D12711yescharacteristic
290C22C22⋊C4×D12712yes—
291C22C22×C4×D12711yescharacteristic
292C22C2×D4×D12712yes—
293C4—22no—
294C4—44no—
295C4—1274no—
296C4—5084no—
297C4—2542no—
298C2—22no—
299C2—21no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—42no—
304C2—5082no—
305C2—2544no—
306C2C2×C22⋊C4×D12711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D127
commutatorC22×C127C23×C4
FrattiniC23C23×D127
FittingC2×#99×C127C2
socleC2×C127C22×C22⋊C4
prime radicalC2×#16861254C2
cosocleC23×C127C24
hypercenterC2×#99D127
hypocenterC127#2883
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #22550349C2semidirect
3. #22549848C2semidirect
4. C2×#16861124C2semidirect
5. C2×#16861123C2semidirect
6. #99×D127C2semidirect
7. C2×#99×C127C2semidirect
10. C2×#14523248C4semidirect
11. C2×#14523247C4semidirect
13. C22×C22⋊D127C4semidirect
16. C2×D4×D127C4semidirect
20. #16861165C22semidirect
21. C2×C22⋊Dic127C22semidirect
22. C22⋊C4×D127C22semidirect
23. #16861124C22semidirect
24. #16861123C22semidirect
26. C2×C22⋊C4×C127C22semidirect
27. #99×C127C22semidirect
28. C23×D127C2×C4semidirect
29. C23×D127C2×C4semidirect
35. C2×Dic254C2×C4semidirect
36. C2×D508C2×C4semidirect
43. C2×C22⋊D127C2×C4semidirect
44. C2×C22⋊D127C2×C4semidirect
45. C2×C22⋊D127C2×C4semidirect
52. C22×Dic127C2×C4semidirect
53. C22×Dic127C2×C4semidirect
56. C2×C4×D127C2×C4semidirect
57. C2×C4×D127C2×C4semidirect
63. C24×C127C2×C4semidirect
67. C2×D4×C127C2×C4semidirect
70. C22×C4×C127C2×C4semidirect
154. C127#2883semidirect
161. C2×#99D127semidirect
178. C2×C22⋊C4C2×D127semidirect
181. #99C2×D127semidirect
185. C24C4×D127semidirect
209. C2×D4C4×D127semidirect
219. C22×C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
#225503492
#225498482
C2×#168611242
C2×#168611232
#99×D1272
C2×#99×C1272
#2883127