Finite Group Databasebeta

← #30648110

Subgroups — #30648110

Order 16256.

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

normal subgroups only

This group has 704 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
1#22550712C211yesmaximal, characteristic
2C2×#16861242C211yesmaximal, characteristic, complemented
3C2×#16861239C211yesmaximal, characteristic, complemented
4#22550452C211yesmaximal, characteristic, complemented
5#22550060C214yesmaximal, complemented
6#22550055C214yesmaximal, complemented
7#22549894C211yesmaximal, characteristic, complemented
8#22549858C211yesmaximal, characteristic, complemented
9#421×C127C211yesmaximal, characteristic, complemented
10#16861257—21no—
11#16861256—21no—
12#16861256C2212yes—
13#16861255—21no—
14#16861255C2212yes—
15#16861254—22no—
16#16861254—21no—
17C2×#14523248—22no—
18C2×#14523248C2211yescharacteristic
19C2×#14523248C2211yescharacteristic
20#41×D127—22no—
21#41×D127—22no—
22C22×Dic254C2211yescharacteristic, complemented
23#16861242—22no—
24#16861242C2214yescomplemented
25#16861239—22no—
26#16861239C2214yescomplemented
27C2×Dic508—21no—
28C2×#14523238—21no—
29C2×#14523237—21no—
30C2×#14523236—21no—
31C2×#14523232C2211yescharacteristic, complemented
32#16861209C2212yescomplemented
33#16861197C2212yescomplemented
34#16861193—22no—
35#16861189—22no—
36#16861137C2214yescomplemented
37#16861127C2212yescomplemented
38#16861125C2212yescomplemented
39C2×#41×C127C2211yescharacteristic
40C2×M16×C127C2211yescharacteristic, complemented
41#117×C127C2211yescharacteristic, complemented
42#104×C127C2214yescomplemented
43#14523249—22no—
44#14523249—22no—
45#14523248—24no—
46#14523248—24no—
47#14523248—22no—
48#14523248—22no—
49#14523248—21no—
50#14523248—21no—
51#14523248—21no—
52#14523248—21no—
53#14523248—21no—
54#14523248—21no—
55#14523248D412yes—
56#14523248C2312yes—
57#14523247—24no—
58#14523247—22no—
59#14523247—24no—
60#14523247—22no—
61C2×Dic254—22no—
62C2×Dic254—21no—
63C2×Dic254—21no—
64C2×Dic254—21no—
65C2×Dic254—21no—
66C2×Dic254—21no—
67C2×Dic254D411yescharacteristic
68C2×Dic254C2312yes—
69C2×Dic254C2311yescharacteristic
70Q8×D127—22no—
71Q8×D127—22no—
72C2×D508—22no—
73C2×D508D411yescharacteristic
74C2×D508C2311yescharacteristic
75C2×C22⋊D127—24no—
76C2×C22⋊D127—21no—
77C2×C22⋊D127D412yes—
78D4×D127—24no—
79D4×D127—22no—
80D4×D127—24no—
81D4×D127—22no—
82C22×Dic127—21no—
83C2×C4×D127—24no—
84C2×C4×D127—21no—
85C2×C4×D127D412yes—
86Dic508—24no—
87#14523238—24no—
88#14523237—24no—
89#14523236—24no—
90#14523232—21no—
91#14523232C2312yes—
92C2×C127⋊C8—21no—
93C4⋊Dic127—24no—
94(C2×D127)⋊C4—24no—
95#41×C127—21no—
96#41×C127—21no—
97#41×C127D412yes—
98#41×C127C2312yes—
99C2×Q8×C127D411yescharacteristic
100C2×D4×C127—21no—
101C2×D4×C127D411yescharacteristic
102C22×C4×C127—21no—
103C22×C4×C127C2311yescharacteristic
104M16×C127—21no—
105M16×C127C2312yes—
106C2×C8×C127—21no—
107C4⋊C4×C127—21no—
108C22⋊C4×C127—21no—
109C42×C127C2312yes—
110C22×D127—24no—
111C22×D127—21no—
112C22×D127C2×D412yes—
113Dic254—24no—
114Dic254—21no—
115Dic254—21no—
116Dic254—21no—
117Dic254—22no—
118Dic254—22no—
119Dic254—41no—
120Dic254—41no—
121Dic254C2×D412yes—
122D508—24no—
123D508—21no—
124D508—21no—
125D508—21no—
126D508C2×D412yes—
127C22⋊D127—24no—
128C22⋊D127—24no—
129C22⋊D127—22no—
130C22⋊D127—22no—
131C22⋊D127—22no—
132C22⋊D127—22no—
133C2×Dic127—24no—
134C2×Dic127—21no—
135C2×Dic127—22no—
136C2×Dic127—41no—
137C2×Dic127—41no—
138C2×Dic127C2×D412yes—
139C4×D127—24no—
140C4×D127—24no—
141C4×D127—22no—
142C4×D127—22no—
143C4×D127—22no—
144C4×D127—22no—
145C127⋊C8—22no—
146C23×C127—21no—
147C23×C127C2×D411yescharacteristic
148Q8×C127—21no—
149Q8×C127C2×D412yes—
150D4×C127—22no—
151D4×C127—22no—
152D4×C127—21no—
153D4×C127C2×D412yes—
154C2×C4×C127—21no—
155C2×C4×C127—22no—
156C2×C4×C127—22no—
157C2×C4×C127—22no—
158C2×C4×C127C2×D411yescharacteristic
159C2×C4×C127C2×D411yescharacteristic
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2×D411yescharacteristic
162C2×C4×C127C2×D411yescharacteristic
163C2×C4×C127C2411yescharacteristic
164C8×C127—22no—
165C22×C127—22no—
166C22×C127—41no—
167C22×C127—81no—
168C22×C127#12011yescharacteristic
169C22×C127#12011yescharacteristic
170C22×C127C22×D411yescharacteristic
171C2×D127—22no—
172C2×D127—22no—
173C2×D127—42no—
174C2×D127—81no—
175C2×D127—84no—
176C4×C127—22no—
177C4×C127—42no—
178C4×C127#12011yescharacteristic
179C4×C127#12011yescharacteristic
180C4×C127C22×D411yescharacteristic
181C4×C127C22×D411yescharacteristic
182Dic127—22no—
183Dic127—22no—
184Dic127—42no—
185C2×C127—21no—
186C2×C127—21no—
187C2×C127—21no—
188C2×C127—42no—
189C2×C127C2×#12011yescharacteristic
190D127—42no—
191D127—42no—
192#3022—1271nomaximal, Sylow
193C127#302211yescharacteristic, complemented, Sylow
194#585—1271no—
195C2×#137—1271no—
196C2×#137—1271no—
197#536—1271no—
198#456—1274no—
199#455—1274no—
200#421—1271no—
201#421D12711yescharacteristic, complemented
202#413—1271no—
2032−1+4—1272no—
2042−1+4—2541no—
2052−1+4—2541no—
206H(5,𝔽2)—1272no—
207H(5,𝔽2)—2542no—
208H(5,𝔽2)—2541no—
209H(5,𝔽2)—2541no—
210C2×#41—1271no—
211C2×#41—1271no—
212C2×#41—2542no—
213C2×#41—2542no—
214C2×#41—2542no—
215C2×#41C2×D12711yescharacteristic
216C22×Q8—1271no—
217#137—1274no—
218#137—1274no—
219#137—2542no—
220#137—2542no—
221C2×Q16—2541no—
222C2×Q16—2541no—
223C2×SD16—2541no—
224C2×SD16—2541no—
225C2×M16—1271no—
226C2×M16C2×D12711yescharacteristic, complemented
227C4⋊Q8—1272no—
228#124—1272no—
229#123—2542no—
230C22⋊Q8—2542no—
231#117C2×D12711yescharacteristic, complemented
232#104—1274no—
233#104C2×D12714yescomplemented
234#101—1272no—
235#100—1272no—
236#41—21no—
237#41—21no—
238#41—1272no—
239#41—1272no—
240#41—2542no—
241#41—2542no—
242#41—2544no—
243#41—2544no—
244#41—2544no—
245#41—2544no—
246#41—2542no—
247#41—2542no—
248#41—2542no—
249#41—2542no—
250#41—2541no—
251#41—2541no—
252#41—2541no—
253#41—2541no—
254#41—2541no—
255#41—2541no—
256#41D50812yes—
257#41C22×D12712yes—
258C2×Q8—1271no—
259C2×Q8—1271no—
260C2×Q8—1272no—
261C2×Q8—2542no—
262C2×Q8—2541no—
263C2×Q8—2541no—
264C2×Q8—2541no—
265C2×Q8—2541no—
266C2×Q8—2542no—
267C2×Q8—2542no—
268C2×Q8—2541no—
269C2×Q8D50811yescharacteristic
270C2×D4—21no—
271C2×D4—1272no—
272C2×D4—1271no—
273C2×D4—1271no—
274C2×D4—2542no—
275C2×D4—2544no—
276C2×D4—2544no—
277C2×D4—2542no—
278C2×D4—2544no—
279C2×D4—2541no—
280C2×D4—2542no—
281C2×D4D50811yescharacteristic
282C22×C4—21no—
283C22×C4—1272no—
284C22×C4—2544no—
285C22×C4—2541no—
286C22×C4—2541no—
287C22×C4C22×D12711yescharacteristic
288Q16—2544no—
289Q16—2544no—
290SD16—2544no—
291SD16—2544no—
292M16—21no—
293M16—1272no—
294M16—2541no—
295M16C22×D12712yes—
296C2×C8—21no—
297C2×C8—2541no—
298C4⋊C4—21no—
299C4⋊C4—2544no—
300C22⋊C4—21no—
301C22⋊C4—2544no—
302C42C22×D12712yes—
303C23—21no—
304C23—1272no—
305C23—2544no—
306C23—2541no—
307C23D4×D12711yescharacteristic
308Q8—21no—
309Q8—1272no—
310Q8—5081no—
311Q8—5081no—
312Q8—2544no—
313Q8—2542no—
314Q8—2542no—
315Q8—2541no—
316Q8—2541no—
317Q8—2541no—
318Q8C2×D50812yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—1272no—
323D4—2544no—
324D4—2544no—
325D4—2544no—
326D4—2542no—
327D4—2542no—
328D4—2542no—
329D4—2542no—
330D4—2541no—
331D4—2541no—
332D4—2541no—
333D4C2×D50812yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—1272no—
339C2×C4—5081no—
340C2×C4—5081no—
341C2×C4—2544no—
342C2×C4—2544no—
343C2×C4—2542no—
344C2×C4—2544no—
345C2×C4—2541no—
346C2×C4—2542no—
347C2×C4—2542no—
348C2×C4—2542no—
349C2×C4—2542no—
350C2×C4D4×D12711yescharacteristic
351C2×C4D4×D12711yescharacteristic
352C2×C4D4×D12711yescharacteristic
353C2×C4C2×D50811yescharacteristic
354C2×C4C2×D50811yescharacteristic
355C2×C4C23×D12711yescharacteristic
356C8—22no—
357C8—2542no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—10161no—
362C22—10164no—
363C22—5082no—
364C22—2542no—
365C22—2542no—
366C22#1686117911yescharacteristic
367C22#1686117911yescharacteristic
368C22C2×D4×D12711yescharacteristic
369C4—22no—
370C4—42no—
371C4—5082no—
372C4—2542no—
373C4—2542no—
374C4#1686117911yescharacteristic
375C4#1686117911yescharacteristic
376C4C2×D4×D12711yescharacteristic
377C4C22×D50811yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—5082no—
383C2—5082no—
384C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#421×C127C2
socleC2×C127C2×#120
prime radical#22550712C2
hypercenter#421D127
hypocenterC127#3022
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×#16861242C2semidirect
3. C2×#16861239C2semidirect
4. #22550452C2semidirect
5. #22550060C2semidirect
6. #22550055C2semidirect
7. #22549894C2semidirect
8. #22549858C2semidirect
9. #421×C127C2semidirect
22. C22×Dic254C22semidirect
24. #16861242C22semidirect
26. #16861239C22semidirect
31. C2×#14523232C22semidirect
32. #16861209C22semidirect
33. #16861197C22semidirect
36. #16861137C22semidirect
37. #16861127C22semidirect
38. #16861125C22semidirect
40. C2×M16×C127C22semidirect
41. #117×C127C22semidirect
42. #104×C127C22semidirect
193. C127#3022semidirect
201. #421D127semidirect
226. C2×M16C2×D127semidirect
231. #117C2×D127semidirect
233. #104C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
C2×#168612422
C2×#168612392
#225504522
#225500602
#225500552
#225498942
#225498582
#421×C1272
#3022127