Finite Group Databasebeta

← #30648085

Subgroups — #30648085

Order 16256.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16861255C211yesmaximal, characteristic, complemented
2#22550674C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550052C214yesmaximal, complemented
5#22550042C214yesmaximal, complemented
6#22549893C211yesmaximal, characteristic, complemented
7#22549892C211yesmaximal, characteristic, complemented
8C2×#16861125C211yesmaximal, characteristic, complemented
9#575×C127C211yesmaximal, characteristic, complemented
10#16861255—22no—
11#16861255—22no—
12#16861255—22no—
13#16861255C2214yes—
14C2×#14523249—21no—
15C2×#14523248—21no—
16C2×#14523248C2211yescharacteristic
17C22×D508—21no—
18C22×D508C2211yescharacteristic
19C2×D4×D127—22no—
20C2×D4×D127—21no—
21#16861240C2212yescomplemented
22#16861239—21no—
23#16861235C2212yescomplemented
24#16861234—21no—
25#16861229—22no—
26C2×#14523232C2211yescharacteristic, complemented
27#16861205C2212yescomplemented
28#16861198C2212yescomplemented
29#16861183—22no—
30#16861177—22no—
31#16861168C2211yescharacteristic, complemented
32#16861136C2214yescomplemented
33#16861135C2214yescomplemented
34#16861125C2214yescomplemented
35C2×#41×C127C2211yescharacteristic
36C2×#41×C127C2211yescharacteristic, complemented
37#137×C127—21no—
38#137×C127C2212yescomplemented
39AGL(1,ℤ/8)×C127—21no—
40AGL(1,ℤ/8)×C127C2212yescomplemented
41#135×C127—22no—
42C2×M16×C127C2211yescharacteristic, complemented
43C23×D127—22no—
44C23×D127D412yes—
45#14523249—24no—
46#14523249—24no—
47#14523248—24no—
48#14523248—24no—
49#14523248—21no—
50#14523248—21no—
51#14523248D412yescomplemented
52#14523248C2312yes—
53C2×Dic254D411yescharacteristic, complemented
54C2×D508—22no—
55C2×D508—24no—
56C2×D508—24no—
57C2×D508—21no—
58C2×D508—21no—
59C2×D508—21no—
60C2×D508—21no—
61C2×D508D412yes—
62C2×D508D411yescharacteristic, complemented
63C2×D508C2312yes—
64C2×C22⋊D127—22no—
65C2×C22⋊D127—24no—
66C2×C22⋊D127—21no—
67D4×D127—28no—
68D4×D127—24no—
69D4×D127—28no—
70D4×D127—24no—
71C2×C4×D127—22no—
72C2×C4×D127—21no—
73#14523238—22no—
74#14523236—22no—
75#14523235—22no—
76#14523233—22no—
77#14523232—21no—
78#14523232C2312yescomplemented
79C2×C127⋊C8—21no—
80C4⋊Dic127—21no—
81C22⋊Dic127—21no—
82(C2×D127)⋊C4—24no—
83C4×Dic127C2312yescomplemented
84#41×C127—21no—
85#41×C127—21no—
86#41×C127—21no—
87#41×C127—21no—
88#41×C127—21no—
89#41×C127—21no—
90#41×C127D412yescomplemented
91#41×C127C2312yes—
92C2×Q8×C127D411yescharacteristic, complemented
93C2×Q8×C127C2311yescharacteristic, complemented
94C2×D4×C127—21no—
95C2×D4×C127—21no—
96C2×D4×C127D411yescharacteristic, complemented
97C2×D4×C127C2311yescharacteristic, complemented
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100C22×C4×C127C2311yescharacteristic
101Q16×C127—22no—
102SD16×C127—22no—
103SD16×C127—22no—
104D8×C127—22no—
105M16×C127—21no—
106M16×C127C2312yescomplemented
107C2×C8×C127—21no—
108C22×D127—22no—
109C22×D127—21no—
110C22×D127—28no—
111C22×D127—22no—
112C22×D127—82no—
113C22×D127—84no—
114C22×D127C2×D414yes—
115Dic254—21no—
116Dic254C2×D412yes—
117D508—28no—
118D508—22no—
119D508—22no—
120D508—21no—
121D508—22no—
122D508—22no—
123D508C2×D412yes—
124C22⋊D127—28no—
125C22⋊D127—22no—
126C22⋊D127—22no—
127C22⋊D127—84no—
128C2×Dic127—21no—
129C2×Dic127—22no—
130C4×D127—28no—
131C4×D127—22no—
132C4×D127—22no—
133C127⋊C8—22no—
134C23×C127—21no—
135C23×C127—21no—
136C23×C127C2×D411yescharacteristic
137Q8×C127—21no—
138Q8×C127—21no—
139Q8×C127—21no—
140Q8×C127C2×D412yes—
141D4×C127—22no—
142D4×C127—22no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127—21no—
146D4×C127—41no—
147D4×C127—41no—
148D4×C127C2×D412yes—
149C2×C4×C127—21no—
150C2×C4×C127—22no—
151C2×C4×C127—22no—
152C2×C4×C127—21no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127C2×D411yescharacteristic
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2×D411yescharacteristic
158C2×C4×C127C2×D411yescharacteristic
159C2×C4×C127C2×D411yescharacteristic
160C2×C4×C127C2411yescharacteristic
161C8×C127—22no—
162C22×C127—22no—
163C22×C127—41no—
164C22×C127—42no—
165C22×C127—42no—
166C22×C127—41no—
167C22×C127#12011yescharacteristic
168C22×C127#12011yescharacteristic
169C22×C127C22×D411yescharacteristic
170C2×D127—24no—
171C2×D127—22no—
172C2×D127—44no—
173C2×D127—42no—
174C2×D127—44no—
175C2×D127—82no—
176C2×D127—88no—
177C4×C127—22no—
178C4×C127—41no—
179C4×C127#12011yescharacteristic
180C4×C127#12011yescharacteristic
181C4×C127C22×D411yescharacteristic
182C4×C127C22×D411yescharacteristic
183Dic127—22no—
184Dic127—42no—
185C2×C127—21no—
186C2×C127—21no—
187C2×C127—21no—
188C2×C127—42no—
189C2×C127—81no—
190C2×C127C2×#12011yescharacteristic
191D127—44no—
192D127—42no—
193#3019—1271nomaximal, Sylow
194C127#301911yescharacteristic, complemented, Sylow
195C2×H(5,𝔽2)—1271no—
196#575—1271no—
197#575D12711yescharacteristic, complemented
198#535—1271no—
199#455—1274no—
200#453—1274no—
201#421—1271no—
202#421—1271no—
203C2×#100—1271no—
204H(5,𝔽2)—1274no—
205H(5,𝔽2)—2542no—
206H(5,𝔽2)—2542no—
207H(5,𝔽2)—2542no—
208C2×#41—1271no—
209C2×#41—2541no—
210C2×#41—2541no—
211C2×#41C2×D12711yescharacteristic
212C2×#41C2×D12711yescharacteristic, complemented
213C22×D4—1271no—
214C22×D4—2541no—
215C22×D4—2542no—
216C22×D4—2541no—
217#137—21no—
218#137—1272no—
219#137—2541no—
220#137C2×D12712yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—1272no—
223AGL(1,ℤ/8)—2541no—
224AGL(1,ℤ/8)C2×D12712yescomplemented
225#135—22no—
226#135—2542no—
227C2×M16—1271no—
228C2×M16C2×D12711yescharacteristic, complemented
229#127—1272no—
230#124—1272no—
231#121—2542no—
232#120—2542no—
233#117—1271no—
234#104—1274no—
235#104—1274no—
236#100—1274no—
237C24—1272no—
238C24—2542no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—1272no—
246#41—1272no—
247#41—2544no—
248#41—2544no—
249#41—2544no—
250#41—2544no—
251#41—2541no—
252#41—2541no—
253#41C22⋊D12712yescomplemented
254#41C22×D12712yes—
255C2×Q8—1271no—
256C2×Q8C22⋊D12711yescharacteristic, complemented
257C2×Q8C22×D12711yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—1271no—
261C2×D4—1272no—
262C2×D4—1272no—
263C2×D4—2544no—
264C2×D4—2542no—
265C2×D4—2544no—
266C2×D4—2548no—
267C2×D4—2548no—
268C2×D4—2544no—
269C2×D4—2544no—
270C2×D4—2542no—
271C2×D4—2541no—
272C2×D4—2544no—
273C2×D4—2541no—
274C2×D4—2541no—
275C2×D4—2541no—
276C2×D4—2541no—
277C2×D4C22⋊D12711yescharacteristic, complemented
278C2×D4C22×D12711yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2542no—
282C22×C4—2541no—
283C22×C4C22×D12711yescharacteristic
284Q16—22no—
285Q16—2542no—
286SD16—22no—
287SD16—22no—
288SD16—2542no—
289SD16—2542no—
290D8—22no—
291D8—2542no—
292M16—21no—
293M16—1272no—
294M16—2541no—
295M16C22×D12712yescomplemented
296C2×C8—21no—
297C2×C8—2541no—
298C4⋊C4—2541no—
299C22⋊C4—2544no—
300C22⋊C4—2541no—
301C42—1272no—
302C23—21no—
303C23—21no—
304C23—1274no—
305C23—10164no—
306C23—10162no—
307C23—2548no—
308C23—2542no—
309C23—2541no—
310C23—2542no—
311C23D4×D12711yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—1272no—
316Q8—2541no—
317Q8C2×C22⋊D12712yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—1272no—
326D4—10164no—
327D4—2548no—
328D4—2548no—
329D4—2542no—
330D4—2542no—
331D4—2542no—
332D4—2542no—
333D4—2541no—
334D4—2542no—
335D4—2542no—
336D4C2×C22⋊D12712yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—2548no—
344C2×C4—2542no—
345C2×C4—2541no—
346C2×C4—2542no—
347C2×C4—2542no—
348C2×C4D4×D12711yescharacteristic
349C2×C4D4×D12711yescharacteristic
350C2×C4D4×D12711yescharacteristic
351C2×C4C2×C22⋊D12711yescharacteristic
352C2×C4C2×C22⋊D12711yescharacteristic
353C2×C4C23×D12711yescharacteristic
354C8—22no—
355C8—2542no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—10162no—
362C22—10168no—
363C22—5084no—
364C22—5082no—
365C22—5084no—
366C22—2542no—
367C22—2544no—
368C22#1686117711yescharacteristic
369C22#1686117711yescharacteristic
370C22C2×D4×D12711yescharacteristic
371C4—22no—
372C4—41no—
373C4—5082no—
374C4—2542no—
375C4#1686117711yescharacteristic
376C4#1686117711yescharacteristic
377C4C2×D4×D12711yescharacteristic
378C4C22×C22⋊D12711yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—5082no—
385C2—5084no—
386C2C2×#1686117711yescharacteristic
subgroupG/H
centerC2C2×#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#575×C127C2
socleC2×C127C2×#120
hypercenter#575D127
hypocenterC127#3019
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. #22550674C2semidirect
3. #22550440C2semidirect
4. #22550052C2semidirect
5. #22550042C2semidirect
6. #22549893C2semidirect
7. #22549892C2semidirect
8. C2×#16861125C2semidirect
9. #575×C127C2semidirect
21. #16861240C22semidirect
23. #16861235C22semidirect
26. C2×#14523232C22semidirect
27. #16861205C22semidirect
28. #16861198C22semidirect
31. #16861168C22semidirect
32. #16861136C22semidirect
33. #16861135C22semidirect
34. #16861125C22semidirect
36. C2×#41×C127C22semidirect
38. #137×C127C22semidirect
40. AGL(1,ℤ/8)×C127C22semidirect
42. C2×M16×C127C22semidirect
51. #14523248D4semidirect
53. C2×Dic254D4semidirect
62. C2×D508D4semidirect
78. #14523232C23semidirect
83. C4×Dic127C23semidirect
90. #41×C127D4semidirect
92. C2×Q8×C127D4semidirect
93. C2×Q8×C127C23semidirect
96. C2×D4×C127D4semidirect
97. C2×D4×C127C23semidirect
106. M16×C127C23semidirect
194. C127#3019semidirect
197. #575D127semidirect
212. C2×#41C2×D127semidirect
220. #137C2×D127semidirect
224. AGL(1,ℤ/8)C2×D127semidirect
228. C2×M16C2×D127semidirect
253. #41C22⋊D127semidirect
256. C2×Q8C22⋊D127semidirect
257. C2×Q8C22×D127semidirect
277. C2×D4C22⋊D127semidirect
278. C2×D4C22×D127semidirect
295. M16C22×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
#225506742
#225504402
#225500522
#225500422
#225498932
#225498922
C2×#168611252
#575×C1272
#3019127