Finite Group Databasebeta

← #30648106

Subgroups — #30648106

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×#16861240C211yesmaximal, characteristic, complemented
3#22550450C211yesmaximal, characteristic, complemented
4#22550057C214yesmaximal, complemented
5#22550052C214yesmaximal, complemented
6#22549893C211yesmaximal, characteristic, complemented
7#22549892C211yesmaximal, characteristic, complemented
8#22549858C211yesmaximal, characteristic, complemented
9C2×#137×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—
22#16861240—22no—
23#16861240C2214yescomplemented
24C2×#14523238—21no—
25C2×#14523235—21no—
26C2×#14523232C2211yescharacteristic, complemented
27#16861207C2212yescomplemented
28#16861198C2212yescomplemented
29#16861192—22no—
30#16861185—22no—
31#16861168C2211yescharacteristic, complemented
32#16861136C2214yescomplemented
33#16861135C2214yescomplemented
34#16861127C2212yescomplemented
35#16861125C2212yescomplemented
36C2×#41×C127C2211yescharacteristic
37C22×Q8×C127C2211yescharacteristic, complemented
38#137×C127—22no—
39#137×C127C2214yescomplemented
40C2×Q16×C127—21no—
41C2×SD16×C127—21no—
42C2×M16×C127C2211yescharacteristic, complemented
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×Dic254D411yescharacteristic
63C2×Dic254C2311yescharacteristic
64Q8×D127—22no—
65Q8×D127—22no—
66C2×D508—22no—
67C2×D508D411yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—24no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127D412yes—
72D4×D127—24no—
73D4×D127—22no—
74D4×D127—24no—
75D4×D127—22no—
76C2×C4×D127—24no—
77C2×C4×D127—21no—
78C2×C4×D127D412yes—
79#14523238—24no—
80#14523235—24no—
81#14523232—21no—
82#14523232C2312yes—
83C2×C127⋊C8—21no—
84C4⋊Dic127—21no—
85Dic127⋊C4—24no—
86C22⋊Dic127—21no—
87(C2×D127)⋊C4—24no—
88C4×Dic127C2312yes—
89#41×C127—21no—
90#41×C127—21no—
91#41×C127D412yes—
92#41×C127C2312yes—
93C2×Q8×C127—21no—
94C2×Q8×C127—21no—
95C2×Q8×C127—21no—
96C2×Q8×C127—21no—
97C2×Q8×C127—21no—
98C2×Q8×C127D411yescharacteristic
99C2×Q8×C127C2312yes—
100C2×D4×C127—21no—
101C2×D4×C127D411yescharacteristic
102C22×C4×C127—21no—
103C22×C4×C127—21no—
104C22×C4×C127C2311yescharacteristic
105Q16×C127—24no—
106SD16×C127—24no—
107M16×C127—21no—
108M16×C127C2312yes—
109C2×C8×C127—21no—
110C22×D127—24no—
111C22×D127—21no—
112C22×D127C2×D412yes—
113Dic254—24no—
114Dic254—21no—
115Dic254—21no—
116Dic254—21no—
117Dic254C2×D412yes—
118D508—24no—
119D508—21no—
120D508—21no—
121D508—21no—
122D508C2×D412yes—
123C22⋊D127—24no—
124C22⋊D127—24no—
125C22⋊D127—22no—
126C22⋊D127—22no—
127C22⋊D127—22no—
128C22⋊D127—22no—
129C2×Dic127—24no—
130C2×Dic127—21no—
131C2×Dic127—22no—
132C2×Dic127C2×D412yes—
133C4×D127—24no—
134C4×D127—24no—
135C4×D127—22no—
136C4×D127—22no—
137C4×D127—22no—
138C4×D127—22no—
139C127⋊C8—22no—
140C23×C127—21no—
141C23×C127C2×D411yescharacteristic
142Q8×C127—21no—
143Q8×C127—22no—
144Q8×C127—22no—
145Q8×C127—41no—
146Q8×C127—41no—
147Q8×C127C2×D412yes—
148D4×C127—22no—
149D4×C127—22no—
150D4×C127—21no—
151D4×C127C2×D412yes—
152C2×C4×C127—21no—
153C2×C4×C127—22no—
154C2×C4×C127—22no—
155C2×C4×C127—22no—
156C2×C4×C127—41no—
157C2×C4×C127—41no—
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×#137D12711yescharacteristic, complemented
197#536—1271no—
198#456—1274no—
199#455—1274no—
200#421—1271no—
201#421—1271no—
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×Q8C2×D12711yescharacteristic, complemented
217#137—22no—
218#137—1274no—
219#137—2542no—
220#137C2×D12714yescomplemented
221C2×Q16—21no—
222C2×Q16—2541no—
223C2×SD16—21no—
224C2×SD16—2541no—
225C2×M16—1271no—
226C2×M16C2×D12711yescharacteristic, complemented
227C4⋊Q8—1272no—
228#124—1272no—
229#123—2542no—
230C22⋊Q8—2542no—
231#117—1271no—
232#104—1274no—
233#104—1274no—
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#41C22⋊D12712yes—
257#41C22×D12712yes—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—1271no—
264C2×Q8—1271no—
265C2×Q8—2542no—
266C2×Q8—2542no—
267C2×Q8—2542no—
268C2×Q8C22⋊D12711yescharacteristic
269C2×Q8C22×D12712yes—
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×D4C22⋊D12711yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—1272no—
285C22×C4—2544no—
286C22×C4—2541no—
287C22×C4C22×D12711yescharacteristic
288Q16—24no—
289Q16—2544no—
290SD16—24no—
291SD16—2544no—
292M16—21no—
293M16—1272no—
294M16—2541no—
295M16C22×D12712yes—
296C2×C8—21no—
297C2×C8—2541no—
298C4⋊C4—2544no—
299C4⋊C4—2541no—
300C22⋊C4—2544no—
301C22⋊C4—2541no—
302C42—1272no—
303C23—21no—
304C23—1272no—
305C23—2544no—
306C23—2541no—
307C23D4×D12711yescharacteristic
308Q8—21no—
309Q8—22no—
310Q8—22no—
311Q8—41no—
312Q8—41no—
313Q8—1272no—
314Q8—2544no—
315Q8—2541no—
316Q8—2541no—
317Q8—2541no—
318Q8C2×C22⋊D12712yes—
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×C22⋊D12712yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—1272no—
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×C22⋊D12711yescharacteristic
354C2×C4C2×C22⋊D12711yescharacteristic
355C2×C4C23×D12711yescharacteristic
356C8—22no—
357C8—2542no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—10161no—
362C22—10164no—
363C22—5082no—
364C22—2542no—
365C22—2542no—
366C22#1686117711yescharacteristic
367C22#1686117711yescharacteristic
368C22C2×D4×D12711yescharacteristic
369C4—22no—
370C4—42no—
371C4—5082no—
372C4—2542no—
373C4—2542no—
374C4#1686117711yescharacteristic
375C4#1686117711yescharacteristic
376C4C2×D4×D12711yescharacteristic
377C4C22×C22⋊D12711yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—5082no—
383C2—5082no—
384C2C2×#1686117711yescharacteristic
subgroupG/H
centerC2C2×#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC2×C127C2×#120
prime radical#22550712C2
hypercenterC2×#137D127
hypocenterC127#3022
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22550450C2semidirect
4. #22550057C2semidirect
5. #22550052C2semidirect
6. #22549893C2semidirect
7. #22549892C2semidirect
8. #22549858C2semidirect
9. C2×#137×C127C2semidirect
23. #16861240C22semidirect
26. C2×#14523232C22semidirect
27. #16861207C22semidirect
28. #16861198C22semidirect
31. #16861168C22semidirect
32. #16861136C22semidirect
33. #16861135C22semidirect
34. #16861127C22semidirect
35. #16861125C22semidirect
37. C22×Q8×C127C22semidirect
39. #137×C127C22semidirect
42. C2×M16×C127C22semidirect
193. C127#3022semidirect
196. C2×#137D127semidirect
216. C22×Q8C2×D127semidirect
220. #137C2×D127semidirect
226. C2×M16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
C2×#168612402
#225504502
#225500572
#225500522
#225498932
#225498922
#225498582
C2×#137×C1272
#3022127