Finite Group Databasebeta

← #30648075

Subgroups — #30648075

Order 16256.

724 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550711C211yesmaximal, characteristic, complemented
2#22550672C212yesmaximal, complemented
3#22550424C211yesmaximal, characteristic, complemented
4#22550060C212yesmaximal, complemented
5#22550056C211yesmaximal, characteristic, complemented
6#22550054C211yesmaximal, characteristic, complemented
7#22550047C212yesmaximal, complemented
8#22550043C211yesmaximal, characteristic, complemented
9#104×D127C212yesmaximal, complemented
10#22549857C211yesmaximal, characteristic, complemented
11#453×C127C211yesmaximal, characteristic, complemented
12#16861256—22no—
13#16861256C2212yes—
14#16861254—22no—
15#16861254—22no—
16#16861254C2211yescharacteristic
17C2×#14523247—22no—
18C2×#14523247—22no—
19C2×#14523247C2211yescharacteristic
20C2×#14523247C2211yescharacteristic, complemented
21#41×D127—22no—
22#41×D127C2212yes—
23#16861242C2212yescomplemented
24#16861239C2212yescomplemented
25#16861238—22no—
26#16861235C2212yescomplemented
27#16861233—22no—
28#16861230—22no—
29#16861228—22no—
30M16×D127C2212yescomplemented
31#16861209C2211yescharacteristic, complemented
32#16861196C2211yescharacteristic, complemented
33#16861196C2211yescharacteristic, complemented
34#16861180—22no—
35D4×Dic127—22no—
36C42×D127C2211yescharacteristic, complemented
37#16861137C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861127C2212yescomplemented
41#16861126C2211yescharacteristic, complemented
42H(5,𝔽2)×C127C2211yescharacteristic
43AGL(1,ℤ/8)×C127C2212yescomplemented
44#127×C127C2211yescharacteristic, complemented
45#104×C127C2212yescomplemented
46#100×C127C2211yescharacteristic, complemented
47#14523249D412yescomplemented
48#14523248—24no—
49#14523248—22no—
50#14523248—22no—
51#14523248C2312yes—
52#14523247—24no—
53#14523247—24no—
54#14523247—24no—
55#14523247—22no—
56#14523247—22no—
57#14523247—24no—
58#14523247—22no—
59#14523247—22no—
60#14523247—22no—
61#14523247—22no—
62#14523247—22no—
63#14523247D412yescomplemented
64C2×Dic254—22no—
65C2×Dic254—22no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127D412yescomplemented
70C2×C22⋊D127—24no—
71C2×C22⋊D127—22no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—22no—
74C2×C22⋊D127D412yes—
75D4×D127—24no—
76D4×D127—22no—
77D4×D127—22no—
78D4×D127D412yescomplemented
79C22×Dic127—22no—
80C22×Dic127—21no—
81C22×Dic127D412yes—
82C2×C4×D127—22no—
83C2×C4×D127—22no—
84C2×C4×D127—21no—
85C2×C4×D127C2311yescharacteristic
86Dic508—22no—
87#14523238—22no—
88#14523237—22no—
89#14523236—22no—
90#14523236—22no—
91#14523233—22no—
92#14523232C2312yes—
93#14523231—22no—
94C8×D127—22no—
95C4⋊Dic127—22no—
96C22⋊Dic127—22no—
97C22⋊Dic127—22no—
98C4×Dic127—21no—
99C4×Dic127C2311yescharacteristic
100#41×C127—22no—
101#41×C127C2312yes—
102C2×D4×C127—22no—
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127C2311yescharacteristic
106C2×D4×C127C2311yescharacteristic
107SD16×C127—22no—
108D8×C127—22no—
109M16×C127C2312yes—
110C42×C127C2311yescharacteristic
111C22×D127—22no—
112C22×D127—22no—
113C22×D127C2×D411yescharacteristic
114Dic254—24no—
115Dic254—22no—
116Dic254—22no—
117Dic254—22no—
118Dic254C2×D412yes—
119D508—22no—
120D508C2×D412yes—
121C22⋊D127—24no—
122C22⋊D127—24no—
123C22⋊D127—22no—
124C22⋊D127—22no—
125C22⋊D127—24no—
126C22⋊D127—22no—
127C22⋊D127—42no—
128C2×Dic127—22no—
129C2×Dic127—24no—
130C2×Dic127—22no—
131C2×Dic127—22no—
132C2×Dic127—22no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—41no—
136C2×Dic127—41no—
137C2×Dic127C2×D412yes—
138C2×Dic127C2×D411yescharacteristic
139C4×D127—24no—
140C4×D127—22no—
141C4×D127—22no—
142C4×D127—22no—
143C4×D127—42no—
144C4×D127C2×D412yes—
145C4×D127C2×D412yes—
146C127⋊C8—22no—
147C23×C127—22no—
148C23×C127—21no—
149C23×C127C2×D412yes—
150Q8×C127C2×D412yes—
151D4×C127—24no—
152D4×C127—22no—
153D4×C127—22no—
154D4×C127—22no—
155D4×C127—44no—
156D4×C127C2×D412yes—
157C2×C4×C127—22no—
158C2×C4×C127—21no—
159C2×C4×C127—22no—
160C2×C4×C127C2411yescharacteristic
161C8×C127—22no—
162C22×C127—22no—
163C22×C127—22no—
164C22×C127—42no—
165C22×C127—42no—
166C22×C127—41no—
167C22×C127—82no—
168C22×C127C22×D411yescharacteristic
169C2×D127—22no—
170C2×D127—41no—
171C2×D127—82no—
172C2×D127—82no—
173C2×D127#12012yes—
174C4×C127—22no—
175C4×C127—22no—
176C4×C127C22×D412yes—
177Dic127—22no—
178Dic127—22no—
179Dic127—41no—
180Dic127—41no—
181Dic127#12011yescharacteristic
182Dic127#12011yescharacteristic
183C2×C127—21no—
184C2×C127—42no—
185C2×C127—42no—
186C2×C127—81no—
187C2×C127C2×#12011yescharacteristic
188D127—22no—
189D127—42no—
190#3018—1271nomaximal, Sylow
191C127#301811yescharacteristic, complemented, Sylow
192#585—1271no—
193#575—1272no—
194#532—1271no—
195#456—1272no—
196#455—1271no—
197#455—1271no—
198#454—1272no—
199#453—1271no—
200#453D12711yescharacteristic, complemented
201C2×#104—1272no—
202#413—1271no—
2032−1+4—1272no—
2042−1+4—2542no—
205H(5,𝔽2)—1271no—
206H(5,𝔽2)—2542no—
207H(5,𝔽2)—2542no—
208H(5,𝔽2)C2×D12711yescharacteristic
209C2×#41—1271no—
210C2×#41—1272no—
211C2×#41—1271no—
212C2×#41—2542no—
213C2×#41—2542no—
214C2×#41—2542no—
215#137—1272no—
216#137—1272no—
217#137—2542no—
218AGL(1,ℤ/8)—1272no—
219AGL(1,ℤ/8)—2542no—
220AGL(1,ℤ/8)C2×D12712yescomplemented
221#135—2542no—
222#135—2542no—
223C2×M16—1272no—
224C4⋊Q8—1271no—
225#127C2×D12711yescharacteristic, complemented
226#124—1271no—
227#124—1271no—
228#121—2542no—
229C4×D4—2542no—
230C2×C42—1271no—
231#104—1272no—
232#104—1272no—
233#104—1272no—
234#104C2×D12712yescomplemented
235#101—1272no—
236#100—1271no—
237#100C2×D12711yescharacteristic, complemented
238#41—22no—
239#41—1272no—
240#41—1272no—
241#41—1272no—
242#41—2542no—
243#41—2542no—
244#41—2544no—
245#41—2542no—
246#41—2542no—
247#41—2542no—
248#41—2542no—
249#41—2542no—
250#41—2544no—
251#41—2542no—
252#41—2544no—
253#41—2544no—
254#41—2544no—
255#41—2542no—
256#41C22×D12712yes—
257C2×Q8—1271no—
258C2×Q8—1272no—
259C2×Q8—1271no—
260C2×Q8—2542no—
261C2×Q8—2542no—
262C2×Q8—2542no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—1272no—
267C2×D4—1272no—
268C2×D4—2542no—
269C2×D4—2542no—
270C2×D4—2544no—
271C2×D4—2541no—
272C2×D4—2542no—
273C2×D4—2544no—
274C2×D4—2542no—
275C2×D4C22×D12711yescharacteristic
276C2×D4C22×D12711yescharacteristic
277C22×C4—1271no—
278C22×C4—1272no—
279C22×C4—2542no—
280C22×C4—2541no—
281C22×C4—2542no—
282C22×C4—2541no—
283C22×C4—2542no—
284Q16—2542no—
285Q16—2542no—
286SD16—22no—
287SD16—2542no—
288SD16—2542no—
289SD16—2542no—
290D8—22no—
291D8—2542no—
292M16—1272no—
293M16—2542no—
294M16C22×D12712yes—
295C2×C8—2542no—
296C4⋊C4—2542no—
297C22⋊C4—2542no—
298C22⋊C4—2542no—
299C42—1271no—
300C42—2541no—
301C42C22×D12711yescharacteristic
302C23—22no—
303C23—21no—
304C23—1271no—
305C23—2542no—
306C23—2542no—
307C23D4×D12712yes—
308Q8—1272no—
309Q8—2542no—
310Q8—2544no—
311Q8—2542no—
312Q8—2542no—
313Q8D4×D12712yes—
314D4—24no—
315D4—22no—
316D4—22no—
317D4—22no—
318D4—44no—
319D4—1272no—
320D4—5082no—
321D4—2542no—
322D4—2542no—
323D4—2544no—
324D4—2542no—
325D4—2544no—
326D4—2544no—
327D4—2542no—
328D4D4×D12712yes—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—1271no—
333C2×C4—1272no—
334C2×C4—1272no—
335C2×C4—1272no—
336C2×C4—5082no—
337C2×C4—5081no—
338C2×C4—5081no—
339C2×C4—2542no—
340C2×C4—2542no—
341C2×C4—2544no—
342C2×C4—2542no—
343C2×C4—2542no—
344C2×C4—2542no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2544no—
348C2×C4—2542no—
349C2×C4—2542no—
350C2×C4C23×D12711yescharacteristic
351C8—22no—
352C8—2542no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—41no—
358C22—82no—
359C22—1272no—
360C22—10162no—
361C22—10162no—
362C22—5081no—
363C22—2542no—
364C22C2×D4×D12711yescharacteristic
365C4—22no—
366C4—22no—
367C4—1271no—
368C4—1271no—
369C4—5081no—
370C4—5081no—
371C4—2542no—
372C4—2542no—
373C4C2×D4×D12712yes—
374C2—21no—
375C2—42no—
376C2—42no—
377C2—81no—
378C2—5082no—
379C2—2542no—
380C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#453×C127C2
socleC2×C127C2×#120
hypercenter#453D127
hypocenterC127#3018
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22550711C2semidirect
2. #22550672C2semidirect
3. #22550424C2semidirect
4. #22550060C2semidirect
5. #22550056C2semidirect
6. #22550054C2semidirect
7. #22550047C2semidirect
8. #22550043C2semidirect
9. #104×D127C2semidirect
10. #22549857C2semidirect
11. #453×C127C2semidirect
20. C2×#14523247C22semidirect
23. #16861242C22semidirect
24. #16861239C22semidirect
26. #16861235C22semidirect
30. M16×D127C22semidirect
31. #16861209C22semidirect
32. #16861196C22semidirect
33. #16861196C22semidirect
36. C42×D127C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861127C22semidirect
41. #16861126C22semidirect
43. AGL(1,ℤ/8)×C127C22semidirect
44. #127×C127C22semidirect
45. #104×C127C22semidirect
46. #100×C127C22semidirect
47. #14523249D4semidirect
63. #14523247D4semidirect
69. Q8×D127D4semidirect
78. D4×D127D4semidirect
191. C127#3018semidirect
200. #453D127semidirect
220. AGL(1,ℤ/8)C2×D127semidirect
225. #127C2×D127semidirect
234. #104C2×D127semidirect
237. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507112
#225506722
#225504242
#225500602
#225500562
#225500542
#225500472
#225500432
#104×D1272
#225498572
#453×C1272
#3018127