Finite Group Databasebeta

← #30642039

Subgroups — #30642039

Order 16256.

1564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×D4×D127C212yesmaximal, complemented
2C2×#16861179C212yesmaximal, complemented
3C2×#16861177C212yesmaximal, complemented
4#22549785C211yesmaximal, characteristic, complemented
5#22549784C212yesmaximal, complemented
6#22549754C212yesmaximal, complemented
7#22549752C212yesmaximal, complemented
8#22549731C211yesmaximal, characteristic, complemented
9#392×C127C211yesmaximal, characteristic, complemented
10C24×D127C2211yescharacteristic
11C24×D127C2212yescomplemented
12C22×D508C2212yes—
13C22×D508C2211yescharacteristic, complemented
14C22×C22⋊D127C2212yes—
15C22×C22⋊D127C2212yescomplemented
16C22×C22⋊D127C2212yescomplemented
17C2×D4×D127—28no—
18C2×D4×D127—28no—
19C2×D4×D127—28no—
20C22×C4×D127C2212yescomplemented
21#16861179—28no—
22#16861177—28no—
23C2×C22⋊Dic127C2212yescomplemented
24C2×C22⋊Dic127C2211yescharacteristic, complemented
25C2×(C2×D127)⋊C4C2212yescomplemented
26C2×(C2×D127)⋊C4C2212yescomplemented
27C2×(C2×D127)⋊C4C2212yescomplemented
28C2×(C2×D127)⋊C4C2212yescomplemented
29C127⋊#95C2212yescomplemented
30C127⋊#95C2211yescharacteristic, complemented
31C22×D4×C127C2212yescomplemented
32C22×D4×C127C2211yescharacteristic, complemented
33C2×C22⋊C4×C127C2212yescomplemented
34C2×C22⋊C4×C127C2211yescharacteristic, complemented
35#95×C127C2211yescharacteristic, complemented
36C23×D127—24no—
37C23×D127—24no—
38C23×D127—24no—
39C23×D127—24no—
40C23×D127—24no—
41C23×D127—24no—
42C23×D127—24no—
43C23×D127—41no—
44C23×D127—41no—
45C23×D127—42no—
46C23×D127—42no—
47C23×D127D414yescomplemented
48C23×D127C2312yes—
49C23×D127C2312yes—
50C2×D508—24no—
51C2×D508—24no—
52C2×D508—24no—
53C2×D508—41no—
54C2×D508D418yescomplemented
55C2×C22⋊D127—24no—
56C2×C22⋊D127—24no—
57C2×C22⋊D127—24no—
58C2×C22⋊D127—24no—
59C2×C22⋊D127—24no—
60C2×C22⋊D127—24no—
61C2×C22⋊D127—24no—
62C2×C22⋊D127—42no—
63C2×C22⋊D127D418yescomplemented
64D4×D127—264no—
65C22×Dic127C2312yes—
66C22×Dic127C2312yes—
67C2×C4×D127—24no—
68C2×C4×D127—24no—
69C2×C4×D127—24no—
70C22⋊Dic127—24no—
71C22⋊Dic127—41no—
72(C2×D127)⋊C4—24no—
73(C2×D127)⋊C4—24no—
74(C2×D127)⋊C4—24no—
75(C2×D127)⋊C4—24no—
76C24×C127C2311yescharacteristic
77C24×C127C2312yes—
78C2×D4×C127—24no—
79C2×D4×C127—24no—
80C2×D4×C127—24no—
81C2×D4×C127—24no—
82C2×D4×C127—41no—
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2312yes—
86C22⋊C4×C127—24no—
87C22⋊C4×C127—41no—
88C22×D127—22no—
89C22×D127—22no—
90C22×D127—22no—
91C22×D127—22no—
92C22×D127—216no—
93C22×D127—22no—
94C22×D127—22no—
95C22×D127—22no—
96C22×D127—22no—
97C22×D127—22no—
98C22×D127—22no—
99C22×D127—216no—
100C22×D127—44no—
101C22×D127—44no—
102C22×D127—48no—
103C22×D127—44no—
104C22×D127—48no—
105C22×D127—48no—
106C22×D127—48no—
107C22×D127—48no—
108C22×D127—48no—
109C22×D127—48no—
110C22×D127C2×D414yes—
111C22×D127C2×D414yes—
112C22×D127C2×D414yes—
113C22×D127C2×D414yescomplemented
114D508—216no—
115D508—44no—
116C22⋊D127—216no—
117C22⋊D127—216no—
118C22⋊D127—48no—
119C2×Dic127—22no—
120C2×Dic127—22no—
121C2×Dic127—22no—
122C2×Dic127—22no—
123C2×Dic127—22no—
124C2×Dic127C2×D414yescomplemented
125C4×D127—216no—
126C23×C127—22no—
127C23×C127—22no—
128C23×C127—22no—
129C23×C127—22no—
130C23×C127—22no—
131C23×C127—22no—
132C23×C127—21no—
133C23×C127—22no—
134C23×C127—21no—
135C23×C127—22no—
136C23×C127C2×D412yes—
137C23×C127C2×D414yescomplemented
138C23×C127C2411yescharacteristic
139D4×C127—216no—
140D4×C127—44no—
141C2×C4×C127—22no—
142C2×C4×C127—22no—
143C2×C4×C127—22no—
144C2×C4×C127—22no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127C2×D414yescomplemented
148C22×C127—24no—
149C22×C127—24no—
150C22×C127—22no—
151C22×C127—24no—
152C22×C127—24no—
153C22×C127—42no—
154C22×C127—42no—
155C22×C127—41no—
156C22×C127—42no—
157C22×C127—42no—
158C22×C127—42no—
159C22×C127—41no—
160C22×C127C22×D411yescharacteristic
161C22×C127C22×D412yes—
162C22×C127C22×D412yes—
163C22×C127H(5,𝔽2)11yescharacteristic
164C22×C127H(5,𝔽2)11yescharacteristic
165C2×D127—24no—
166C2×D127—24no—
167C2×D127—24no—
168C2×D127—24no—
169C2×D127—24no—
170C2×D127—24no—
171C2×D127—24no—
172C2×D127—24no—
173C2×D127—24no—
174C2×D127—416no—
175C2×D127—432no—
176C2×D127—42no—
177C2×D127—42no—
178C2×D127—42no—
179C2×D127—42no—
180C2×D127#12018yescomplemented
181C4×C127—24no—
182C4×C127—41no—
183C4×C127—41no—
184Dic127—24no—
185Dic127—42no—
186C2×C127—42no—
187C2×C127—44no—
188C2×C127C2×#12012yes—
189C2×C127#53411yescharacteristic
190C2×C127D4211yescharacteristic
191C2×C127D4212yes—
192C2×C127D4211yescharacteristic
193D127—28no—
194D127—44no—
195#2403—1271nomaximal, Sylow
196C127#240311yescharacteristic, complemented, Sylow
197C23×D4—1272no—
198C2×#120—1272no—
199C2×#120—1272no—
200#392—1272no—
201#392—1271no—
202#392D12711yescharacteristic, complemented
203#386—1272no—
204#386—1272no—
205#379—1271no—
206C25—1271no—
207C25—1272no—
208C22×D4—1272no—
209C22×D4—1272no—
210C22×D4—1272no—
211C22×D4—1272no—
212C22×D4—1271no—
213C22×D4—2548no—
214C22×D4—2548no—
215C22×D4—2548no—
216C22×D4C2×D12712yescomplemented
217C22×D4C2×D12711yescharacteristic, complemented
218C23×C4—1272no—
219#120—2548no—
220#120—2548no—
221C2×C22⋊C4—1271no—
222C2×C22⋊C4—1272no—
223C2×C22⋊C4—1272no—
224C2×C22⋊C4—1272no—
225C2×C22⋊C4—1272no—
226C2×C22⋊C4—1272no—
227C2×C22⋊C4C2×D12712yescomplemented
228C2×C22⋊C4C2×D12711yescharacteristic, complemented
229#95—1271no—
230#95—1272no—
231#95C2×D12711yescharacteristic, complemented
232C24—1272no—
233C24—1274no—
234C24—1272no—
235C24—5081no—
236C24—5082no—
237C24—5081no—
238C24—5082no—
239C24—2544no—
240C24—2544no—
241C24—2544no—
242C24—2544no—
243C24—2544no—
244C24—2544no—
245C24—2544no—
246C24C22×D12711yescharacteristic
247C24C22×D12712yes—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—41no—
253C2×D4—1278no—
254C2×D4—1278no—
255C2×D4—5081no—
256C2×D4—5082no—
257C2×D4—2544no—
258C2×D4—2544no—
259C2×D4—2544no—
260C2×D4—2544no—
261C2×D4—25464no—
262C2×D4—2544no—
263C2×D4—2544no—
264C2×D4—2544no—
265C2×D4—2544no—
266C2×D4—2544no—
267C2×D4—2544no—
268C22×C4—1272no—
269C22×C4—1272no—
270C22×C4—2544no—
271C22×C4—2544no—
272C22×C4—2544no—
273C22×C4C22×D12711yescharacteristic
274C22×C4C22×D12711yescharacteristic
275C22×C4C22×D12712yes—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C22⋊C4—5081no—
279C22⋊C4—2544no—
280C22⋊C4—2544no—
281C22⋊C4—2544no—
282C22⋊C4—2544no—
283C22⋊C4—2544no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—1274no—
295C23—1274no—
296C23—1274no—
297C23—1274no—
298C23—5088no—
299C23—5088no—
300C23—5084no—
301C23—5088no—
302C23—5088no—
303C23—5084no—
304C23—5088no—
305C23—5088no—
306C23—5088no—
307C23—5084no—
308C23—25416no—
309C23—25416no—
310C23—2542no—
311C23—2542no—
312C23—2542no—
313C23—2542no—
314C23—2542no—
315C23—2542no—
316C23—2542no—
317C23—2542no—
318C23—2542no—
319C23—2542no—
320C23D4×D12712yes—
321C23D4×D12714yescomplemented
322C23C23×D12711yescharacteristic
323D4—216no—
324D4—44no—
325D4—5084no—
326D4—5088no—
327D4—25416no—
328D4—25416no—
329D4—25416no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—1274no—
337C2×C4—25416no—
338C2×C4—2542no—
339C2×C4—2542no—
340C2×C4—2542no—
341C2×C4—2542no—
342C2×C4—2542no—
343C2×C4D4×D12714yescomplemented
344C22—24no—
345C22—24no—
346C22—22no—
347C22—24no—
348C22—24no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—41no—
356C22—1278no—
357C22—50832no—
358C22—50816no—
359C22—5082no—
360C22—5082no—
361C22—5082no—
362C22—5082no—
363C22—2544no—
364C22—2544no—
365C22—2544no—
366C22—2544no—
367C22—2544no—
368C22—2544no—
369C22—2544no—
370C22—2544no—
371C22—2544no—
372C22C2×D4×D12711yescharacteristic
373C22C2×D4×D12712yes—
374C22C2×D4×D12712yes—
375C22#1686125411yescharacteristic
376C22#1686125511yescharacteristic
377C4—24no—
378C4—41no—
379C4—41no—
380C4—5082no—
381C4—2544no—
382C2—42no—
383C2—44no—
384C2—5084no—
385C2—2548no—
386C2#120×D12712yes—
387C2#2255043411yescharacteristic
388C2#2255050811yescharacteristic
389C2#2255051012yes—
390C2#2255051111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#392×C127C2
hypercenter#392D127
hypocenterC127#2403
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D127C2semidirect
2. C2×#16861179C2semidirect
3. C2×#16861177C2semidirect
4. #22549785C2semidirect
5. #22549784C2semidirect
6. #22549754C2semidirect
7. #22549752C2semidirect
8. #22549731C2semidirect
9. #392×C127C2semidirect
11. C24×D127C22semidirect
13. C22×D508C22semidirect
15. C22×C22⋊D127C22semidirect
16. C22×C22⋊D127C22semidirect
20. C22×C4×D127C22semidirect
23. C2×C22⋊Dic127C22semidirect
24. C2×C22⋊Dic127C22semidirect
25. C2×(C2×D127)⋊C4C22semidirect
26. C2×(C2×D127)⋊C4C22semidirect
27. C2×(C2×D127)⋊C4C22semidirect
28. C2×(C2×D127)⋊C4C22semidirect
29. C127⋊#95C22semidirect
30. C127⋊#95C22semidirect
31. C22×D4×C127C22semidirect
32. C22×D4×C127C22semidirect
33. C2×C22⋊C4×C127C22semidirect
34. C2×C22⋊C4×C127C22semidirect
35. #95×C127C22semidirect
47. C23×D127D4semidirect
54. C2×D508D4semidirect
63. C2×C22⋊D127D4semidirect
113. C22×D127C2×D4semidirect
124. C2×Dic127C2×D4semidirect
137. C23×C127C2×D4semidirect
147. C2×C4×C127C2×D4semidirect
180. C2×D127#120semidirect
196. C127#2403semidirect
202. #392D127semidirect
216. C22×D4C2×D127semidirect
217. C22×D4C2×D127semidirect
227. C2×C22⋊C4C2×D127semidirect
228. C2×C22⋊C4C2×D127semidirect
231. #95C2×D127semidirect
321. C23D4×D127semidirect
343. C2×C4D4×D127semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×#168611792
C2×#168611772
#225497852
#225497842
#225497542
#225497522
#225497312
#392×C1272
#2403127