Finite Group Databasebeta

← #30647986

Subgroups — #30647986

Order 16256.

754 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D127C211yesmaximal, characteristic
2C2×#16861240C211yesmaximal, characteristic, complemented
3C2×#16861185C211yesmaximal, characteristic, complemented
4#22550037C212yesmaximal, complemented
5#22550035C212yesmaximal, complemented
6#22550030C212yesmaximal, complemented
7#22550025C212yesmaximal, complemented
8#22549878C211yesmaximal, characteristic, complemented
9#22549877C211yesmaximal, characteristic, complemented
10#22549839C211yesmaximal, characteristic, complemented
11C2×#137×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2211yescharacteristic
13C2×#14523248—21no—
14C2×#14523248C2211yescharacteristic
15C2×#14523247—21no—
16C2×#14523247C2211yescharacteristic
17#41×D127—24no—
18#41×D127—24no—
19C2×Q8×D127C2211yescharacteristic
20C2×D4×D127—21no—
21C2×D4×D127C2211yescharacteristic
22C22×C4×D127—21no—
23C22×C4×D127C2211yescharacteristic
24#16861240—24no—
25C2×#14523238C2212yescomplemented
26C2×#14523235C2212yescomplemented
27C2×#14523232C2211yescharacteristic, complemented
28#16861185—21no—
29#16861185—21no—
30#16861185C2212yes—
31#16861185C2212yescomplemented
32C2×C4⋊Dic127C2211yescharacteristic, complemented
33C2×Dic127⋊C4—21no—
34C2×(C2×D127)⋊C4—21no—
35#16861134C2212yescomplemented
36#16861133C2212yescomplemented
37#16861131C2212yescomplemented
38#16861130C2212yescomplemented
39(C2×D127)⋊C8C2212yes—
40(C2×D127)⋊C8C2212yescomplemented
41C2×#41×C127C2211yescharacteristic
42C22×Q8×C127C2211yescharacteristic, complemented
43#137×C127—24no—
44C2×Q16×C127C2212yescomplemented
45C2×SD16×C127C2212yescomplemented
46C2×M16×C127C2211yescharacteristic, complemented
47C23×D127—21no—
48C23×D127D411yescharacteristic
49#14523249—22no—
50#14523249—22no—
51#14523248—28no—
52#14523248—22no—
53#14523248D414yes—
54#14523247—28no—
55#14523247—22no—
56#14523247—22no—
57C2×Dic254—21no—
58C2×Dic254C2311yescharacteristic
59Q8×D127—22no—
60Q8×D127—22no—
61C2×D508—21no—
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66D4×D127—28no—
67D4×D127—22no—
68D4×D127—22no—
69C22×Dic127—21no—
70C22×Dic127—21no—
71C22×Dic127D411yescharacteristic
72C2×C4×D127—28no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—22no—
77C2×C4×D127—22no—
78C2×C4×D127D411yescharacteristic
79C2×C4×D127D411yescharacteristic
80C2×C4×D127C2311yescharacteristic
81C2×C4×D127C2311yescharacteristic
82#14523238—24no—
83#14523235—24no—
84#14523232—22no—
85C2×C127⋊C8C2312yes—
86C4⋊Dic127—21no—
87C4⋊Dic127C2312yes—
88Dic127⋊C4—22no—
89Dic127⋊C4—22no—
90(C2×D127)⋊C4—22no—
91(C2×D127)⋊C4—22no—
92#41×C127—22no—
93#41×C127D414yes—
94C2×Q8×C127—21no—
95C2×Q8×C127—22no—
96C2×Q8×C127—22no—
97C2×Q8×C127C2312yes—
98C2×Q8×C127C2311yescharacteristic
99C2×D4×C127—21no—
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127—21no—
102C22×C4×C127—21no—
103C22×C4×C127C2311yescharacteristic
104Q16×C127—24no—
105SD16×C127—24no—
106M16×C127—22no—
107C2×C8×C127C2312yes—
108C22×D127—24no—
109C22×D127—22no—
110C22×D127—21no—
111C22×D127—22no—
112C22×D127—41no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127C2×D411yescharacteristic
116C22×D127C2×D411yescharacteristic
117Dic254—24no—
118Dic254—21no—
119Dic254C2×D412yes—
120D508—24no—
121D508—21no—
122D508C2×D412yes—
123C22⋊D127—24no—
124C22⋊D127—24no—
125C22⋊D127—24no—
126C2×Dic127—24no—
127C2×Dic127—22no—
128C2×Dic127—21no—
129C2×Dic127—22no—
130C2×Dic127—42no—
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C4×D127—24no—
134C4×D127—24no—
135C4×D127—22no—
136C4×D127—24no—
137C4×D127—22no—
138C4×D127—21no—
139C4×D127—21no—
140C4×D127—21no—
141C4×D127—21no—
142C127⋊C8—22no—
143C23×C127—21no—
144C23×C127C2×D411yescharacteristic
145Q8×C127—21no—
146Q8×C127—22no—
147Q8×C127—22no—
148Q8×C127—42no—
149Q8×C127C2×D412yes—
150D4×C127—24no—
151D4×C127—21no—
152D4×C127C2×D412yes—
153C2×C4×C127—21no—
154C2×C4×C127—24no—
155C2×C4×C127—22no—
156C2×C4×C127—42no—
157C2×C4×C127C2×D411yescharacteristic
158C2×C4×C127C2×D412yes—
159C2×C4×C127C2×D412yes—
160C2×C4×C127C2411yescharacteristic
161C8×C127—22no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—22no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127#12012yes—
168C22×C127C22×D411yescharacteristic
169C2×D127—21no—
170C2×D127—21no—
171C2×D127—21no—
172C2×D127—22no—
173C2×D127—21no—
174C2×D127—21no—
175C2×D127—44no—
176C2×D127—48no—
177C2×D127—41no—
178C2×D127—41no—
179C2×D127#13712yescomplemented
180C4×C127—22no—
181C4×C127—42no—
182C4×C127#12012yes—
183C4×C127C22×D411yescharacteristic
184C4×C127C22×D411yescharacteristic
185Dic127—21no—
186Dic127—22no—
187Dic127—21no—
188Dic127—42no—
189C2×C127—22no—
190C2×C127—42no—
191C2×C127C2×#12011yescharacteristic
192C2×C127C2×#13711yescharacteristic
193C2×C127#57511yescharacteristic
194D127—22no—
195D127—41no—
196D127—42no—
197#3007—1271nomaximal, Sylow
198C127#300711yescharacteristic, complemented, Sylow
199C22×#41—1271no—
200C2×#137—1271no—
201C2×#137D12711yescharacteristic, complemented
202C2×C22⋊Q8—1271no—
203#452—1272no—
204#452—1272no—
205#451—1272no—
206#450—1272no—
207#417—1271no—
208#417—1271no—
209#407—1271no—
210C2×#41—1271no—
211C2×#41—1271no—
212C2×#41—1271no—
213C2×#41—2541no—
214C2×#41—2541no—
215C2×#41—2544no—
216C2×#41—2544no—
217C2×#41C2×D12711yescharacteristic
218C22×Q8—1271no—
219C22×Q8C2×D12711yescharacteristic, complemented
220C22×D4—1271no—
221C22×D4—2541no—
222C23×C4—1271no—
223C23×C4—2541no—
224#137—24no—
225#137—2544no—
226C2×Q16—1272no—
227C2×Q16C2×D12712yescomplemented
228C2×SD16—1272no—
229C2×SD16C2×D12712yescomplemented
230C2×M16—1271no—
231C2×M16C2×D12711yescharacteristic, complemented
232C22⋊Q8—1272no—
233C22⋊Q8—1272no—
234C22⋊Q8—2541no—
235C22⋊Q8—2541no—
236C2×C4⋊C4—1271no—
237C2×C4⋊C4—2541no—
238C2×C22⋊C4—2541no—
239#103—1272no—
240#103—1272no—
241#102—1272no—
242#102—1272no—
243C22⋊C8—1272no—
244C22⋊C8—1272no—
245C24—1271no—
246C24—2541no—
247#41—22no—
248#41—1274no—
249#41—2548no—
250#41—2542no—
251#41—2548no—
252#41—2542no—
253#41—2542no—
254#41—2542no—
255#41—2542no—
256#41C22⋊D12714yes—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—1271no—
261C2×Q8—2541no—
262C2×Q8—2542no—
263C2×Q8—2542no—
264C2×Q8C22×D12712yes—
265C2×Q8C22×D12711yescharacteristic
266C2×D4—21no—
267C2×D4—1271no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4—2542no—
271C2×D4—2548no—
272C2×D4—2541no—
273C2×D4—2541no—
274C2×D4—2542no—
275C2×D4C22×D12711yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—1271no—
279C22×C4—1271no—
280C22×C4—1271no—
281C22×C4—1271no—
282C22×C4—1271no—
283C22×C4—2541no—
284C22×C4—2541no—
285C22×C4—2548no—
286C22×C4—2541no—
287C22×C4—2541no—
288C22×C4—2542no—
289C22×C4—2541no—
290C22×C4—2542no—
291C22×C4C22×D12711yescharacteristic
292Q16—24no—
293Q16—2544no—
294SD16—24no—
295SD16—2544no—
296M16—22no—
297M16—2542no—
298C2×C8—1272no—
299C2×C8C22×D12712yes—
300C4⋊C4—1272no—
301C4⋊C4—2542no—
302C4⋊C4—2542no—
303C4⋊C4—2541no—
304C22⋊C4—2542no—
305C22⋊C4—2542no—
306C23—21no—
307C23—1271no—
308C23—1271no—
309C23—5081no—
310C23—5082no—
311C23—5082no—
312C23—2542no—
313C23—2544no—
314C23—2541no—
315C23—2542no—
316C23D4×D12711yescharacteristic
317Q8—21no—
318Q8—22no—
319Q8—22no—
320Q8—42no—
321Q8—1272no—
322Q8—2544no—
323Q8—2541no—
324Q8C2×C22⋊D12712yes—
325D4—24no—
326D4—21no—
327D4—1272no—
328D4—2544no—
329D4—2544no—
330D4—2544no—
331D4—2544no—
332D4—2541no—
333D4C2×C22⋊D12712yes—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—1271no—
339C2×C4—1271no—
340C2×C4—5082no—
341C2×C4—2544no—
342C2×C4—2544no—
343C2×C4—2542no—
344C2×C4—2544no—
345C2×C4—2541no—
346C2×C4—2542no—
347C2×C4—2541no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2542no—
352C2×C4—2542no—
353C2×C4—2544no—
354C2×C4D4×D12712yes—
355C2×C4D4×D12711yescharacteristic
356C2×C4C2×C22⋊D12712yes—
357C2×C4C23×D12711yescharacteristic
358C8—22no—
359C8—2542no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22—1272no—
366C22—5084no—
367C22—5088no—
368C22—5081no—
369C22—5081no—
370C22—2542no—
371C22—2541no—
372C22—2541no—
373C22—2541no—
374C22—2541no—
375C22—2541no—
376C22#1686117712yes—
377C22C2×D4×D12711yescharacteristic
378C4—22no—
379C4—42no—
380C4—5082no—
381C4—2541no—
382C4—2541no—
383C4—2542no—
384C4#1686117712yes—
385C4C2×D4×D12711yescharacteristic
386C4C22×C22⋊D12711yescharacteristic
387C2—22no—
388C2—42no—
389C2—5082no—
390C2—5081no—
391C2—2542no—
392C2C2×#1686117711yescharacteristic
393C2#137×D12711yescharacteristic
394C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×D127C2
hypercenterC2×#137D127
hypocenterC127#3007
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. C2×#16861185C2semidirect
4. #22550037C2semidirect
5. #22550035C2semidirect
6. #22550030C2semidirect
7. #22550025C2semidirect
8. #22549878C2semidirect
9. #22549877C2semidirect
10. #22549839C2semidirect
11. C2×#137×C127C2semidirect
25. C2×#14523238C22semidirect
26. C2×#14523235C22semidirect
27. C2×#14523232C22semidirect
31. #16861185C22semidirect
32. C2×C4⋊Dic127C22semidirect
35. #16861134C22semidirect
36. #16861133C22semidirect
37. #16861131C22semidirect
38. #16861130C22semidirect
40. (C2×D127)⋊C8C22semidirect
42. C22×Q8×C127C22semidirect
44. C2×Q16×C127C22semidirect
45. C2×SD16×C127C22semidirect
46. C2×M16×C127C22semidirect
179. C2×D127#137semidirect
198. C127#3007semidirect
201. C2×#137D127semidirect
219. C22×Q8C2×D127semidirect
227. C2×Q16C2×D127semidirect
229. C2×SD16C2×D127semidirect
231. C2×M16C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1272
C2×#168612402
C2×#168611852
#225500372
#225500352
#225500302
#225500252
#225498782
#225498772
#225498392
C2×#137×C1272
#3007127