Finite Group Databasebeta

← #30648098

Subgroups — #30648098

Order 16256.

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

normal subgroups only

This group has 736 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
2AGL(1,ℤ/8)×D127C212yesmaximal, complemented
3#22550439C211yesmaximal, characteristic, complemented
4#22550050C212yesmaximal, complemented
5#22550047C212yesmaximal, complemented
6#22550046C211yesmaximal, characteristic, complemented
7#22550044C211yesmaximal, characteristic, complemented
8#22550043C211yesmaximal, characteristic, complemented
9#22549891C212yesmaximal, complemented
10#22549857C211yesmaximal, characteristic, complemented
11#453×C127C211yesmaximal, characteristic, complemented
12#16861256—22no—
13#16861256C2212yescomplemented
14#16861254—22no—
15#16861254—22no—
16#16861254C2211yescharacteristic, complemented
17C2×#14523247—22no—
18C2×#14523247—22no—
19C2×#14523247C2211yescharacteristic
20#41×D127—22no—
21#41×D127C2212yes—
22C2×D4×D127C2211yescharacteristic, complemented
23#16861237C2212yescomplemented
24#16861236—22no—
25#16861235C2212yescomplemented
26#16861234C2212yescomplemented
27#16861233—22no—
28SD16×D127—22no—
29D8×D127—22no—
30M16×D127C2212yescomplemented
31#16861205C2211yescharacteristic, complemented
32#16861197C2211yescharacteristic, complemented
33#16861196C2211yescharacteristic, complemented
34#16861181—22no—
35#16861177—22no—
36#16861166C2211yescharacteristic, complemented
37#16861137C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861127C2212yescomplemented
41#16861126C2211yescharacteristic, complemented
42H(5,𝔽2)×C127C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C127C2212yescomplemented
44#127×C127C2211yescharacteristic, complemented
45#104×C127C2212yescomplemented
46#100×C127C2211yescharacteristic, complemented
47C23×D127—21no—
48#14523249D412yescomplemented
49#14523248—24no—
50#14523248—22no—
51#14523248—22no—
52#14523248C2312yes—
53#14523247—24no—
54#14523247—24no—
55#14523247—24no—
56#14523247—22no—
57#14523247—22no—
58#14523247—24no—
59#14523247—22no—
60#14523247—22no—
61#14523247—22no—
62#14523247D412yescomplemented
63C2×Dic254—22no—
64C2×Dic254—22no—
65C2×Dic254C2311yescharacteristic
66Q8×D127—22no—
67Q8×D127D412yescomplemented
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—24no—
70C2×C22⋊D127—22no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—22no—
73C2×C22⋊D127—22no—
74C2×C22⋊D127D412yes—
75D4×D127—24no—
76D4×D127—22no—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127—22no—
80D4×D127D412yescomplemented
81C22×Dic127—22no—
82C22×Dic127D412yes—
83C2×C4×D127—22no—
84C2×C4×D127—22no—
85C2×C4×D127C2311yescharacteristic
86#14523237—22no—
87#14523236—22no—
88#14523235—22no—
89D1016—22no—
90#14523233—22no—
91#14523233—22no—
92#14523232C2312yes—
93#14523231—22no—
94C8×D127—22no—
95Dic127⋊C4—21no—
96C22⋊Dic127—22no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—22no—
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×D127—21no—
114C22×D127—42no—
115C22×D127—82no—
116C22×D127C2×D411yescharacteristic
117Dic254—24no—
118Dic254—22no—
119Dic254—22no—
120Dic254C2×D412yes—
121D508—22no—
122D508—22no—
123D508C2×D412yes—
124C22⋊D127—24no—
125C22⋊D127—24no—
126C22⋊D127—22no—
127C22⋊D127—22no—
128C22⋊D127—24no—
129C22⋊D127—22no—
130C22⋊D127—41no—
131C22⋊D127—41no—
132C22⋊D127—82no—
133C2×Dic127—22no—
134C2×Dic127—24no—
135C2×Dic127—22no—
136C2×Dic127—22no—
137C2×Dic127—22no—
138C2×Dic127—21no—
139C2×Dic127C2×D412yes—
140C2×Dic127C2×D411yescharacteristic
141C4×D127—24no—
142C4×D127—22no—
143C4×D127—22no—
144C4×D127—22no—
145C4×D127C2×D412yes—
146C4×D127C2×D412yes—
147C127⋊C8—22no—
148C23×C127—22no—
149C23×C127—21no—
150C23×C127C2×D412yes—
151Q8×C127C2×D412yes—
152D4×C127—24no—
153D4×C127—22no—
154D4×C127—22no—
155D4×C127—22no—
156D4×C127—82no—
157D4×C127C2×D412yes—
158C2×C4×C127—22no—
159C2×C4×C127—22no—
160C2×C4×C127—21no—
161C2×C4×C127C2411yescharacteristic
162C8×C127—22no—
163C22×C127—22no—
164C22×C127—22no—
165C22×C127—42no—
166C22×C127—42no—
167C22×C127—41no—
168C22×C127—82no—
169C22×C127C22×D411yescharacteristic
170C2×D127—22no—
171C2×D127—41no—
172C2×D127—42no—
173C2×D127—41no—
174C2×D127—82no—
175C2×D127—84no—
176C2×D127—82no—
177C2×D127#12012yes—
178C4×C127—22no—
179C4×C127—41no—
180C4×C127C22×D412yes—
181Dic127—22no—
182Dic127—22no—
183Dic127—41no—
184Dic127#12011yescharacteristic
185Dic127#12011yescharacteristic
186C2×C127—21no—
187C2×C127—42no—
188C2×C127—42no—
189C2×C127—81no—
190C2×C127C2×#12011yescharacteristic
191D127—22no—
192D127—42no—
193D127—81no—
194#3021—1271nomaximal, Sylow
195C127#302111yescharacteristic, complemented, Sylow
196#585—1271no—
197C2×AGL(1,ℤ/8)—1272no—
198#535—1271no—
199#454—1272no—
200#454—1272no—
201#453—1271no—
202#453—1271no—
203#453—1271no—
204#453D12711yescharacteristic, complemented
205#421—1272no—
206#413—1271no—
2072−1+4—1272no—
2082−1+4—2542no—
209H(5,𝔽2)—1271no—
210H(5,𝔽2)—2542no—
211H(5,𝔽2)—2542no—
212H(5,𝔽2)C2×D12711yescharacteristic, complemented
213C2×#41—1271no—
214C2×#41—1272no—
215C2×#41—2542no—
216C2×#41—2542no—
217C2×#41—2542no—
218C22×D4—1271no—
219AGL(1,ℤ/8)—1272no—
220AGL(1,ℤ/8)—1272no—
221AGL(1,ℤ/8)—1272no—
222AGL(1,ℤ/8)—2542no—
223AGL(1,ℤ/8)—2542no—
224AGL(1,ℤ/8)C2×D12712yescomplemented
225C2×SD16—2542no—
226C2×D8—2542no—
227C2×M16—1272no—
228#127—1271no—
229#127C2×D12711yescharacteristic, complemented
230#124—1271no—
231#124—1271no—
232#121—2542no—
233#120—2542no—
234#117—1271no—
235#104—1272no—
236#104—1272no—
237#104—1272no—
238#104C2×D12712yescomplemented
239#101—1272no—
240#100—1271no—
241#100C2×D12711yescharacteristic, complemented
242C24—2541no—
243#41—22no—
244#41—1272no—
245#41—1272no—
246#41—1272no—
247#41—2542no—
248#41—2542no—
249#41—2544no—
250#41—2542no—
251#41—2542no—
252#41—2542no—
253#41—2544no—
254#41—2542no—
255#41—2544no—
256#41—2544no—
257#41—2544no—
258#41—2542no—
259#41C22×D12712yes—
260C2×Q8—1271no—
261C2×Q8—1272no—
262C2×Q8—2542no—
263C2×Q8—2542no—
264C2×Q8—2542no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—1272no—
269C2×D4—1272no—
270C2×D4—1271no—
271C2×D4—2542no—
272C2×D4—2542no—
273C2×D4—2541no—
274C2×D4—2542no—
275C2×D4—2542no—
276C2×D4—2544no—
277C2×D4—2542no—
278C2×D4—2542no—
279C2×D4—2544no—
280C2×D4—2542no—
281C2×D4C22×D12711yescharacteristic
282C2×D4C22×D12711yescharacteristic
283C22×C4—1271no—
284C22×C4—1272no—
285C22×C4—2542no—
286C22×C4—2542no—
287C22×C4—2542no—
288SD16—22no—
289SD16—2542no—
290SD16—2542no—
291SD16—2542no—
292D8—22no—
293D8—2542no—
294D8—2542no—
295D8—2542no—
296M16—1272no—
297M16—2542no—
298M16C22×D12712yes—
299C2×C8—2542no—
300C4⋊C4—2541no—
301C22⋊C4—2542no—
302C22⋊C4—2541no—
303C22⋊C4—2542no—
304C42—1271no—
305C42C22×D12711yescharacteristic
306C23—22no—
307C23—21no—
308C23—1271no—
309C23—10162no—
310C23—5082no—
311C23—2542no—
312C23—2542no—
313C23—2541no—
314C23D4×D12712yes—
315Q8—1272no—
316Q8—2542no—
317Q8—2544no—
318Q8—2542no—
319Q8D4×D12712yes—
320D4—24no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—82no—
325D4—1272no—
326D4—10162no—
327D4—5081no—
328D4—5081no—
329D4—2542no—
330D4—2542no—
331D4—2544no—
332D4—2542no—
333D4—2544no—
334D4—2544no—
335D4—2542no—
336D4—2542no—
337D4D4×D12712yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—1271no—
342C2×C4—1272no—
343C2×C4—1272no—
344C2×C4—1272no—
345C2×C4—2542no—
346C2×C4—2542no—
347C2×C4—2544no—
348C2×C4—2542no—
349C2×C4—2542no—
350C2×C4—2542no—
351C2×C4—2541no—
352C2×C4—2544no—
353C2×C4—2542no—
354C2×C4—2542no—
355C2×C4C23×D12711yescharacteristic
356C8—22no—
357C8—2542no—
358C22—22no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—82no—
364C22—1272no—
365C22—10162no—
366C22—10162no—
367C22—10164no—
368C22—5082no—
369C22—5081no—
370C22—5081no—
371C22—2542no—
372C22C2×D4×D12711yescharacteristic
373C4—22no—
374C4—41no—
375C4—1271no—
376C4—1271no—
377C4—5081no—
378C4—2542no—
379C4—2542no—
380C4C2×D4×D12712yes—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—81no—
385C2—10161no—
386C2—5082no—
387C2—2542no—
388C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#453×C127C2
socleC2×C127C2×#120
hypercenter#453D127
hypocenterC127#3021
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. AGL(1,ℤ/8)×D127C2semidirect
3. #22550439C2semidirect
4. #22550050C2semidirect
5. #22550047C2semidirect
6. #22550046C2semidirect
7. #22550044C2semidirect
8. #22550043C2semidirect
9. #22549891C2semidirect
10. #22549857C2semidirect
11. #453×C127C2semidirect
13. #16861256C22semidirect
16. #16861254C22semidirect
22. C2×D4×D127C22semidirect
23. #16861237C22semidirect
25. #16861235C22semidirect
26. #16861234C22semidirect
30. M16×D127C22semidirect
31. #16861205C22semidirect
32. #16861197C22semidirect
33. #16861196C22semidirect
36. #16861166C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861127C22semidirect
41. #16861126C22semidirect
42. H(5,𝔽2)×C127C22semidirect
43. AGL(1,ℤ/8)×C127C22semidirect
44. #127×C127C22semidirect
45. #104×C127C22semidirect
46. #100×C127C22semidirect
48. #14523249D4semidirect
62. #14523247D4semidirect
67. Q8×D127D4semidirect
80. D4×D127D4semidirect
195. C127#3021semidirect
204. #453D127semidirect
212. H(5,𝔽2)C2×D127semidirect
224. AGL(1,ℤ/8)C2×D127semidirect
229. #127C2×D127semidirect
238. #104C2×D127semidirect
241. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507112
AGL(1,ℤ/8)×D1272
#225504392
#225500502
#225500472
#225500462
#225500442
#225500432
#225498912
#225498572
#453×C1272
#3021127