Finite Group Databasebeta

← #450×D127 #30647909

Subgroups — #450×D127 #30647909

Order 16256.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×D4×D127C211yesmaximal, characteristic
2C2×SD16×D127C212yesmaximal, complemented
3C22⋊Q8×D127C211yesmaximal, characteristic, complemented
4#22550029C211yesmaximal, characteristic, complemented
5#22550028C212yesmaximal, complemented
6#22550027C211yesmaximal, characteristic, complemented
7#22550026C211yesmaximal, characteristic, complemented
8#22550025C212yesmaximal, complemented
9#102×D127C212yesmaximal, complemented
10C22⋊C8×D127C211yesmaximal, characteristic, complemented
11#450×C127C211yesmaximal, characteristic, complemented
12C24×D127—21no—
13C2×Q8×D127C2211yescharacteristic, complemented
14C22×D508C2211yescharacteristic
15C22×C22⋊D127—21no—
16C2×D4×D127—28no—
17C2×D4×D127—21no—
18C2×D4×D127—21no—
19C2×D4×D127C2212yes—
20C2×D4×D127C2212yes—
21C22×C4×D127C2211yescharacteristic
22C2×#14523237C2212yescomplemented
23C2×#14523236C2212yescomplemented
24C2×#14523235C2212yescomplemented
25SD16×D127—28no—
26C2×C8×D127C2212yescomplemented
27#16861189C2211yescharacteristic, complemented
28#16861188C2211yescharacteristic, complemented
29#16861187—21no—
30#16861186—21no—
31#16861185C2211yescharacteristic, complemented
32C4⋊C4×D127—21no—
33C4⋊C4×D127C2211yescharacteristic, complemented
34C22⋊C4×D127—21no—
35#16861131C2212yescomplemented
36#16861130C2212yescomplemented
37#16861129C2212yescomplemented
38C22⋊(C127⋊C8)C2211yescharacteristic, complemented
39(C2×D127)⋊C8C2212yescomplemented
40C22×D4×C127C2211yescharacteristic
41C2×SD16×C127C2212yescomplemented
42C22⋊Q8×C127C2211yescharacteristic, complemented
43#102×C127C2212yescomplemented
44C22⋊C8×C127C2211yescharacteristic, complemented
45C23×D127—28no—
46C23×D127—22no—
47C23×D127—22no—
48C23×D127—21no—
49C23×D127—44no—
50C23×D127—44no—
51C23×D127D411yescharacteristic
52C2×Dic254—21no—
53C2×Dic254C2311yescharacteristic, complemented
54Q8×D127—24no—
55C2×D508—24no—
56C2×D508—21no—
57C2×D508C2312yes—
58C2×C22⋊D127—28no—
59C2×C22⋊D127—22no—
60C2×C22⋊D127—22no—
61D4×D127—28no—
62D4×D127—28no—
63D4×D127—24no—
64D4×D127—24no—
65D4×D127—24no—
66D4×D127D418yes—
67C22×Dic127D411yescharacteristic
68C2×C4×D127—24no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127D411yescharacteristic
72C2×C4×D127D411yescharacteristic
73C2×C4×D127C2311yescharacteristic
74C2×C4×D127C2311yescharacteristic
75#14523237—24no—
76#14523236—24no—
77#14523235—24no—
78C2×C127⋊C8C2312yescomplemented
79C8×D127—24no—
80C4⋊Dic127—21no—
81C4⋊Dic127C2311yescharacteristic, complemented
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85C22⋊Dic127—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88C24×C127—21no—
89C2×Q8×C127C2311yescharacteristic, complemented
90C2×D4×C127—24no—
91C2×D4×C127—21no—
92C2×D4×C127C2312yes—
93C22×C4×C127C2311yescharacteristic
94SD16×C127—24no—
95C2×C8×C127C2312yescomplemented
96C4⋊C4×C127—21no—
97C4⋊C4×C127C2311yescharacteristic, complemented
98C22⋊C4×C127—21no—
99C22×D127—22no—
100C22×D127—22no—
101C22×D127—28no—
102C22×D127—24no—
103C22×D127—22no—
104C22×D127—28no—
105C22×D127—44no—
106C22×D127—432no—
107C22×D127—42no—
108C22×D127—42no—
109C22×D127—44no—
110C22×D127—44no—
111C22×D127—44no—
112C22×D127SD1614yescomplemented
113C22×D127C2×D411yescharacteristic
114C22×D127C2×D411yescharacteristic
115Dic254—22no—
116Dic254—42no—
117D508—24no—
118D508—22no—
119D508C2×D414yes—
120C22⋊D127—28no—
121C22⋊D127—28no—
122C2×Dic127—22no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C4×D127—22no—
128C4×D127—22no—
129C4×D127—22no—
130C4×D127—42no—
131C4×D127—42no—
132C4×D127C2×D414yes—
133C127⋊C8—22no—
134C23×C127—24no—
135C23×C127—22no—
136C23×C127—42no—
137C23×C127—42no—
138C23×C127C2×D411yescharacteristic
139Q8×C127—22no—
140D4×C127—24no—
141D4×C127—22no—
142D4×C127C2×D414yes—
143C2×C4×C127—22no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C8×C127—22no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—24no—
152C22×C127—48no—
153C22×C127—42no—
154C22×C127—42no—
155C22×C127C2×SD1612yescomplemented
156C22×C127C22×D411yescharacteristic
157C2×D127—21no—
158C2×D127—21no—
159C2×D127—28no—
160C2×D127—24no—
161C2×D127—416no—
162C2×D127—48no—
163C2×D127—42no—
164C2×D127—42no—
165C2×D127—416no—
166C2×D127#12012yes—
167C2×D127C2×SD1612yes—
168C2×D127C2×SD1612yescomplemented
169C2×D127AGL(1,ℤ/8)12yes—
170C4×C127—21no—
171C4×C127—41no—
172C4×C127—41no—
173C4×C127C22×D412yes—
174Dic127—21no—
175Dic127—41no—
176Dic127—41no—
177Dic127#12012yes—
178C2×C127—22no—
179C2×C127—44no—
180C2×C127C2×#12011yescharacteristic
181C2×C127C22×SD1611yescharacteristic
182C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
183D127—22no—
184D127—44no—
185D127#45014yesdirect product factor
186C2×#450—1271nomaximal, Sylow
187C127C2×#45011yescharacteristic, complemented, Sylow
188C23×D4—1271no—
189C22×SD16—1272no—
190C2×C22⋊Q8—1271no—
191#450—1271no—
192#450—1272no—
193#450—1271no—
194#450—1272no—
195#450—1271no—
196#450D12711yescharacteristic, direct product factor
197C2×#102—1272no—
198C2×C22⋊C8—1271no—
199C25—2541no—
200C22×Q8—1271no—
201C22×D4—1272no—
202C22×D4—1272no—
203C22×D4—1271no—
204C22×D4—2541no—
205C22×D4—2541no—
206C22×D4—2541no—
207C22×D4—2548no—
208C22×D4C2×D12711yescharacteristic
209C23×C4—1271no—
210C2×SD16—1272no—
211C2×SD16—1272no—
212C2×SD16—1272no—
213C2×SD16—2548no—
214C2×SD16C2×D12712yescomplemented
215C22×C8—1272no—
216C22⋊Q8—1271no—
217C22⋊Q8—1271no—
218C22⋊Q8—1271no—
219C22⋊Q8—2541no—
220C22⋊Q8—2541no—
221C22⋊Q8C2×D12711yescharacteristic, complemented
222C2×C4⋊C4—1271no—
223C2×C4⋊C4—2541no—
224C2×C22⋊C4—2541no—
225#102—1272no—
226#102—1272no—
227#102—1272no—
228#102C2×D12712yescomplemented
229C22⋊C8—1271no—
230C22⋊C8—1272no—
231C22⋊C8C2×D12711yescharacteristic, complemented
232C24—21no—
233C24—1271no—
234C24—5084no—
235C24—5084no—
236C24—2542no—
237C24—2542no—
238C24—2548no—
239C24—2541no—
240C2×Q8—1271no—
241C2×Q8—2541no—
242C2×Q8—2544no—
243C2×Q8C22×D12711yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—1278no—
247C2×D4—1272no—
248C2×D4—2544no—
249C2×D4—2544no—
250C2×D4—2542no—
251C2×D4—2542no—
252C2×D4—2544no—
253C2×D4—2548no—
254C2×D4—2548no—
255C2×D4—2548no—
256C2×D4—2544no—
257C2×D4—2541no—
258C2×D4C22×D12712yes—
259C22×C4—1271no—
260C22×C4—1271no—
261C22×C4—1271no—
262C22×C4—1271no—
263C22×C4—1271no—
264C22×C4—2544no—
265C22×C4—2541no—
266C22×C4—2541no—
267C22×C4C22×D12711yescharacteristic
268SD16—24no—
269SD16—2544no—
270SD16—2544no—
271SD16—2544no—
272C2×C8—1272no—
273C2×C8—2544no—
274C2×C8C22×D12712yescomplemented
275C4⋊C4—21no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4C22×D12711yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—1271no—
291C23—1274no—
292C23—1271no—
293C23—5082no—
294C23—5084no—
295C23—5084no—
296C23—5084no—
297C23—50832no—
298C23—5084no—
299C23—5082no—
300C23—2548no—
301C23—2542no—
302C23—2542no—
303C23—2544no—
304C23—2548no—
305C23—2542no—
306C23D4×D12711yescharacteristic
307Q8—22no—
308Q8—5082no—
309Q8—2542no—
310D4—24no—
311D4—22no—
312D4—1274no—
313D4—2544no—
314D4—2548no—
315D4—2548no—
316D4—2542no—
317D4D4×D12714yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—1271no—
322C2×C4—1274no—
323C2×C4—1271no—
324C2×C4—5082no—
325C2×C4—5082no—
326C2×C4—2542no—
327C2×C4—2542no—
328C2×C4—2542no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2542no—
332C2×C4D4×D12711yescharacteristic
333C2×C4C23×D12711yescharacteristic
334C8—22no—
335C8—2542no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—1272no—
343C22—1272no—
344C22—1272no—
345C22—1272no—
346C22—5088no—
347C22—50816no—
348C22—50816no—
349C22—5082no—
350C22—5082no—
351C22—2548no—
352C22—2544no—
353C22—2541no—
354C22—2541no—
355C22SD16×D12712yescomplemented
356C22C2×D4×D12711yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—1272no—
361C4—5081no—
362C4—5081no—
363C4—2541no—
364C4C2×D4×D12712yes—
365C2—22no—
366C2—44no—
367C2—1274no—
368C2—5084no—
369C2—2542no—
370C2#120×D12711yescharacteristic
371C2C2×SD16×D12711yescharacteristic
372C2AGL(1,ℤ/8)×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×D127C2
hypercenter#450D127
hypocenterC127C2×#450
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D127C2semidirect
3. C22⋊Q8×D127C2semidirect
4. #22550029C2semidirect
5. #22550028C2semidirect
6. #22550027C2semidirect
7. #22550026C2semidirect
8. #22550025C2semidirect
9. #102×D127C2semidirect
10. C22⋊C8×D127C2semidirect
11. #450×C127C2semidirect
13. C2×Q8×D127C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523236C22semidirect
24. C2×#14523235C22semidirect
26. C2×C8×D127C22semidirect
27. #16861189C22semidirect
28. #16861188C22semidirect
31. #16861185C22semidirect
33. C4⋊C4×D127C22semidirect
35. #16861131C22semidirect
36. #16861130C22semidirect
37. #16861129C22semidirect
38. C22⋊(C127⋊C8)C22semidirect
39. (C2×D127)⋊C8C22semidirect
41. C2×SD16×C127C22semidirect
42. C22⋊Q8×C127C22semidirect
43. #102×C127C22semidirect
44. C22⋊C8×C127C22semidirect
53. C2×Dic254C23semidirect
78. C2×C127⋊C8C23semidirect
81. C4⋊Dic127C23semidirect
89. C2×Q8×C127C23semidirect
95. C2×C8×C127C23semidirect
97. C4⋊C4×C127C23semidirect
112. C22×D127SD16semidirect
155. C22×C127C2×SD16semidirect
168. C2×D127C2×SD16semidirect
185. D127#450direct (N × H)
187. C127C2×#450semidirect
196. #450D127direct (N × H)
214. C2×SD16C2×D127semidirect
221. C22⋊Q8C2×D127semidirect
228. #102C2×D127semidirect
231. C22⋊C8C2×D127semidirect
243. C2×Q8C22×D127semidirect
274. C2×C8C22×D127semidirect
281. C4⋊C4C22×D127semidirect
355. C22SD16×D127semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×SD16×D1272
C22⋊Q8×D1272
#225500292
#225500282
#225500272
#225500262
#225500252
#102×D1272
C22⋊C8×D1272
#450×C1272
C2×#450127