Finite Group Databasebeta

← #30642054

Subgroups — #30642054

Order 16256.

1116 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D127C211yesmaximal, characteristic
2C2×#16861192C211yesmaximal, characteristic, complemented
3C22×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
4#22549793C212yesmaximal, complemented
5#22549791C212yesmaximal, complemented
6#22549756C211yesmaximal, characteristic, complemented
7#22549753C211yesmaximal, characteristic, complemented
8#22549747C212yesmaximal, complemented
9#22549731C212yesmaximal, complemented
10#22549731C211yesmaximal, characteristic, complemented
11C2×#123×C127C211yesmaximal, characteristic, complemented
12C24×D127C2211yescharacteristic
13C24×D127C2211yescharacteristic
14C22×D508C2211yescharacteristic
15C22×C22⋊D127C2211yescharacteristic
16C22×C22⋊D127C2211yescharacteristic
17C2×D4×D127—24no—
18C2×D4×D127—24no—
19C2×D4×D127—24no—
20C23×Dic127C2211yescharacteristic, complemented
21C22×C4×D127C2211yescharacteristic
22#16861192—24no—
23C2×Dic127⋊C4C2212yescomplemented
24C2×C22⋊Dic127C2211yescharacteristic, complemented
25C2×C22⋊Dic127C2212yescomplemented
26C2×(C2×D127)⋊C4—24no—
27C2×(C2×D127)⋊C4C2212yes—
28C2×(C2×D127)⋊C4C2212yes—
29C2×(C2×D127)⋊C4C2212yescomplemented
30C2×(C2×D127)⋊C4C2212yescomplemented
31C2×(C2×D127)⋊C4C2212yescomplemented
32C2×(C2×D127)⋊C4C2212yescomplemented
33C127⋊#95C2212yescomplemented
34C127⋊#95C2212yescomplemented
35C22×D4×C127C2211yescharacteristic
36C23×C4×C127C2211yescharacteristic, complemented
37#123×C127—24no—
38C2×C4⋊C4×C127C2212yescomplemented
39C2×C22⋊C4×C127C2212yescomplemented
40C2×C22⋊C4×C127C2211yescharacteristic, complemented
41C23×D127—22no—
42C23×D127—22no—
43C23×D127—24no—
44C23×D127—24no—
45C23×D127—22no—
46C23×D127—22no—
47C23×D127—22no—
48C23×D127—42no—
49C23×D127—42no—
50C23×D127D414yes—
51C23×D127C2311yescharacteristic
52C23×D127C2311yescharacteristic
53C23×D127C2311yescharacteristic
54C2×D508—22no—
55C2×D508—22no—
56C2×D508—22no—
57C2×C22⋊D127—22no—
58C2×C22⋊D127—22no—
59C2×C22⋊D127—22no—
60C2×C22⋊D127—22no—
61C2×C22⋊D127—22no—
62C2×C22⋊D127D414yes—
63D4×D127—232no—
64C22×Dic127—22no—
65C22×Dic127—42no—
66C22×Dic127C2312yes—
67C22×Dic127C2312yes—
68C22×Dic127C2311yescharacteristic
69C2×C4×D127—22no—
70C2×C4×D127—22no—
71C2×C4×D127—22no—
72Dic127⋊C4—24no—
73C22⋊Dic127—22no—
74C22⋊Dic127—42no—
75(C2×D127)⋊C4—24no—
76(C2×D127)⋊C4—24no—
77(C2×D127)⋊C4—24no—
78(C2×D127)⋊C4—24no—
79(C2×D127)⋊C4—42no—
80(C2×D127)⋊C4—42no—
81C24×C127C2311yescharacteristic
82C24×C127C2311yescharacteristic
83C2×D4×C127—22no—
84C2×D4×C127—22no—
85C2×D4×C127D414yes—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127C2312yes—
90C22×C4×C127C2312yes—
91C22×C4×C127C2311yescharacteristic
92C4⋊C4×C127—24no—
93C22⋊C4×C127—22no—
94C22⋊C4×C127—24no—
95C22×D127—21no—
96C22×D127—21no—
97C22×D127—22no—
98C22×D127—28no—
99C22×D127—28no—
100C22×D127—21no—
101C22×D127—21no—
102C22×D127—21no—
103C22×D127—21no—
104C22×D127—22no—
105C22×D127—22no—
106C22×D127—22no—
107C22×D127—22no—
108C22×D127—28no—
109C22×D127—44no—
110C22×D127—48no—
111C22×D127—48no—
112C22×D127—48no—
113C22×D127—48no—
114C22×D127—44no—
115C22×D127—44no—
116C22×D127C2×D412yes—
117C22×D127C2×D412yes—
118C22×D127C2×D412yes—
119C22×D127#4114yes—
120D508—28no—
121C22⋊D127—28no—
122C22⋊D127—28no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—22no—
126C2×Dic127—22no—
127C2×Dic127—22no—
128C2×Dic127—22no—
129C2×Dic127—22no—
130C2×Dic127—22no—
131C2×Dic127—44no—
132C2×Dic127C2×D412yes—
133C4×D127—28no—
134C23×C127—22no—
135C23×C127—22no—
136C23×C127—21no—
137C23×C127—21no—
138C23×C127—22no—
139C23×C127—22no—
140C23×C127C2×D412yes—
141C23×C127C2×D412yes—
142C23×C127C2×D412yes—
143C23×C127C2×D412yes—
144C23×C127C2411yescharacteristic
145D4×C127—28no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—22no—
149C2×C4×C127—22no—
150C2×C4×C127—22no—
151C2×C4×C127—22no—
152C2×C4×C127—22no—
153C2×C4×C127—22no—
154C2×C4×C127—44no—
155C2×C4×C127C2×D412yes—
156C22×C127—22no—
157C22×C127—22no—
158C22×C127—24no—
159C22×C127—24no—
160C22×C127—22no—
161C22×C127—24no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—42no—
165C22×C127#12014yes—
166C22×C127C22×D411yescharacteristic
167C22×C127C22×D411yescharacteristic
168C22×C127C22×D411yescharacteristic
169C22×C127C2×#4112yes—
170C22×C127H(5,𝔽2)12yes—
171C2×D127—22no—
172C2×D127—22no—
173C2×D127—24no—
174C2×D127—24no—
175C2×D127—22no—
176C2×D127—22no—
177C2×D127—22no—
178C2×D127—22no—
179C2×D127—24no—
180C2×D127—416no—
181C2×D127—416no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—42no—
185C2×D127—42no—
186C2×D127—42no—
187C2×D127#12314yescomplemented
188C4×C127—22no—
189C4×C127—42no—
190C4×C127—42no—
191Dic127—22no—
192Dic127—42no—
193Dic127—42no—
194C2×C127—24no—
195C2×C127—42no—
196C2×C127C2×#12011yescharacteristic
197C2×C127C2×#12311yescharacteristic
198C2×C127#53411yescharacteristic
199C2×C127#54612yes—
200C2×C127#54612yes—
201D127—24no—
202D127—42no—
203D127—42no—
204#2405—1271nomaximal, Sylow
205C127#240511yescharacteristic, complemented, Sylow
206C23×D4—1271no—
207C2×#123—1271no—
208C2×#123D12711yescharacteristic, complemented
209C22×C22⋊C4—1271no—
210#394—1272no—
211#394—1272no—
212#386—1271no—
213#386—1271no—
214#385—1272no—
215#379—1272no—
216#379—1271no—
217C25—1271no—
218C25—1271no—
219C22×D4—1271no—
220C22×D4—1271no—
221C22×D4—1271no—
222C22×D4—2544no—
223C22×D4—2544no—
224C22×D4—2544no—
225C22×D4C2×D12711yescharacteristic
226C23×C4—1271no—
227C23×C4—1271no—
228C23×C4C2×D12711yescharacteristic, complemented
229#123—24no—
230#123—2544no—
231C2×C4⋊C4—1272no—
232C2×C4⋊C4C2×D12712yescomplemented
233C2×C22⋊C4—1272no—
234C2×C22⋊C4—1272no—
235C2×C22⋊C4—1272no—
236C2×C22⋊C4—1271no—
237C2×C22⋊C4—1272no—
238C2×C22⋊C4—1272no—
239C2×C22⋊C4—1272no—
240C2×C22⋊C4—1272no—
241C2×C22⋊C4—2544no—
242C2×C22⋊C4C2×D12711yescharacteristic, complemented
243C2×C22⋊C4C2×D12712yescomplemented
244#95—1272no—
245#95—1272no—
246C24—1271no—
247C24—1274no—
248C24—1271no—
249C24—1271no—
250C24—5082no—
251C24—5082no—
252C24—2544no—
253C24—2542no—
254C24—2542no—
255C24—2544no—
256C24—2542no—
257C24—2542no—
258C24—2542no—
259C24C22×D12711yescharacteristic
260C24C22×D12711yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—1274no—
264C2×D4—2542no—
265C2×D4—2542no—
266C2×D4—2542no—
267C2×D4—25432no—
268C2×D4—2542no—
269C2×D4—2542no—
270C2×D4—2542no—
271C2×D4—2542no—
272C2×D4—2542no—
273C2×D4C22⋊D12714yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—1271no—
278C22×C4—1272no—
279C22×C4—1272no—
280C22×C4—5082no—
281C22×C4—2542no—
282C22×C4—2542no—
283C22×C4—2542no—
284C22×C4—2542no—
285C22×C4C22×D12712yes—
286C22×C4C22×D12712yes—
287C22×C4C22×D12711yescharacteristic
288C4⋊C4—24no—
289C4⋊C4—2544no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—5082no—
293C22⋊C4—5082no—
294C22⋊C4—5082no—
295C22⋊C4—2544no—
296C22⋊C4—2544no—
297C22⋊C4—2544no—
298C22⋊C4—2544no—
299C22⋊C4—2542no—
300C23—22no—
301C23—22no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—1272no—
307C23—1274no—
308C23—1272no—
309C23—1272no—
310C23—5088no—
311C23—5088no—
312C23—5088no—
313C23—5084no—
314C23—5084no—
315C23—5084no—
316C23—5088no—
317C23—2548no—
318C23—2548no—
319C23—2541no—
320C23—2541no—
321C23—2541no—
322C23—2541no—
323C23—2542no—
324C23—2542no—
325C23—2542no—
326C23—2542no—
327C23—2548no—
328C23—2541no—
329C23—2541no—
330C23—2542no—
331C23D4×D12712yes—
332C23D4×D12712yes—
333C23C2×C22⋊D12712yes—
334C23C2×C22⋊D12712yes—
335C23C23×D12711yescharacteristic
336D4—28no—
337D4—2548no—
338D4—2548no—
339D4—2548no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—44no—
349C2×C4—1272no—
350C2×C4—5084no—
351C2×C4—2548no—
352C2×C4—2542no—
353C2×C4—2542no—
354C2×C4—2542no—
355C2×C4—2542no—
356C2×C4—2542no—
357C2×C4—2542no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4C2×C22⋊D12712yes—
361C22—22no—
362C22—22no—
363C22—24no—
364C22—24no—
365C22—22no—
366C22—24no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—1274no—
371C22—50816no—
372C22—50816no—
373C22—5081no—
374C22—5081no—
375C22—5082no—
376C22—5082no—
377C22—5082no—
378C22—2542no—
379C22—2542no—
380C22—2542no—
381C22—2542no—
382C22—2544no—
383C22—2542no—
384C22—2542no—
385C22—2544no—
386C22—2544no—
387C22#1686117714yes—
388C22C2×D4×D12711yescharacteristic
389C22C2×D4×D12711yescharacteristic
390C22C22×C22⋊D12711yescharacteristic
391C22#41×D12712yes—
392C22#1686125512yes—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—5082no—
397C4—5082no—
398C4—2542no—
399C2—24no—
400C2—42no—
401C2—5082no—
402C2—5082no—
403C2—2544no—
404C2C2×#1686117711yescharacteristic
405C2#123×D12711yescharacteristic
406C2#2255043711yescharacteristic
407C2#2255051412yes—
408C2#2255051812yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×#123×C127C2
prime radicalC22×D4×D127C2
hypercenterC2×#123D127
hypocenterC127#2405
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#16861192C2semidirect
3. C22×(C2×D127)⋊C4C2semidirect
4. #22549793C2semidirect
5. #22549791C2semidirect
6. #22549756C2semidirect
7. #22549753C2semidirect
8. #22549747C2semidirect
9. #22549731C2semidirect
10. #22549731C2semidirect
11. C2×#123×C127C2semidirect
20. C23×Dic127C22semidirect
23. C2×Dic127⋊C4C22semidirect
24. C2×C22⋊Dic127C22semidirect
25. C2×C22⋊Dic127C22semidirect
29. C2×(C2×D127)⋊C4C22semidirect
30. C2×(C2×D127)⋊C4C22semidirect
31. C2×(C2×D127)⋊C4C22semidirect
32. C2×(C2×D127)⋊C4C22semidirect
33. C127⋊#95C22semidirect
34. C127⋊#95C22semidirect
36. C23×C4×C127C22semidirect
38. C2×C4⋊C4×C127C22semidirect
39. C2×C22⋊C4×C127C22semidirect
40. C2×C22⋊C4×C127C22semidirect
187. C2×D127#123semidirect
205. C127#2405semidirect
208. C2×#123D127semidirect
228. C23×C4C2×D127semidirect
232. C2×C4⋊C4C2×D127semidirect
242. C2×C22⋊C4C2×D127semidirect
243. C2×C22⋊C4C2×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×#168611922
C22×(C2×D127)⋊C42
#225497932
#225497912
#225497562
#225497532
#225497472
#225497312
C2×#123×C1272
#2405127