Finite Group Databasebeta

← #30648866

Subgroups — #30648866

Order 16256.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550408C211yesmaximal, characteristic, complemented
2C4×D4×D127C211yesmaximal, characteristic, complemented
3#22550208C211yesmaximal, characteristic, complemented
4#22550188C211yesmaximal, characteristic, complemented
5#22550175C211yesmaximal, characteristic, complemented
6#22550174C211yesmaximal, characteristic, complemented
7#22550090C211yesmaximal, characteristic, complemented
8#22550089C211yesmaximal, characteristic, complemented
9#22550028C211yesmaximal, characteristic, complemented
10#22550008C211yesmaximal, characteristic
11#22549986C211yesmaximal, characteristic, complemented
12#22549978C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549931C211yesmaximal, characteristic, complemented
15#442×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C22×C4×D127—21no—
18C2×#14523237C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21#16861202—21no—
22#16861200C2211yescharacteristic, complemented
23#16861196C2211yescharacteristic, complemented
24#16861191—21no—
25#16861189C2211yescharacteristic
26#16861180C2211yescharacteristic
27C4×Dic254C2211yescharacteristic, complemented
28C4×D508C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30#16861171C2211yescharacteristic
31C4×C22⋊D127—21no—
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861168—21no—
35C4⋊C4×D127C2211yescharacteristic
36C22⋊C4×D127—21no—
37C42×D127C2211yescharacteristic
38#16861143C2211yescharacteristic, complemented
39#16861142C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic, complemented
52C2×D8×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic, complemented
58#97×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64D4×D127—22no—
65D4×D127—22no—
66C22×Dic127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—44no—
71C2×C4×D127D411yescharacteristic
72C2×C4×D127D411yescharacteristic
73C2×C4×D127C2311yescharacteristic
74#14523237—22no—
75#14523236—22no—
76#14523233—22no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C4⋊Dic127—21no—
80C4⋊Dic127—21no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93C4×Dic127—21no—
94C4×Dic127D411yescharacteristic
95C4×Dic127D411yescharacteristic
96C4×Dic127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100C22×C4×C127—21no—
101D8×C127—22no—
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C22⋊C4×C127—21no—
108C42×C127C2311yescharacteristic
109C22×D127—24no—
110C22×D127—21no—
111C22×D127—42no—
112C22×D127—42no—
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116D508—21no—
117D508#4112yes—
118C22⋊D127—24no—
119C22⋊D127—42no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—42no—
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C4×D127—22no—
128C4×D127—21no—
129C4×D127—21no—
130C4×D127—41no—
131C4×D127—42no—
132C4×D127—42no—
133C4×D127—42no—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127—21no—
137C23×C127—21no—
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—21no—
141D4×C127#4112yes—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—42no—
145C2×C4×C127—42no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
149C8×C127—21no—
150C8×C127—21no—
151C22×C127—22no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127C22×D411yescharacteristic
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—22no—
161C2×D127—48no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127AGL(1,ℤ/8)12yes—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127C2×#4111yescharacteristic
170C4×C127C2×#4111yescharacteristic
171Dic127—21no—
172Dic127—21no—
173Dic127—21no—
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—42no—
178C2×C127—81no—
179C2×C127#52511yescharacteristic
180C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C127#57511yescharacteristic
182D127—22no—
183D127—42no—
184#3114—1271nomaximal, Sylow
185C127#311411yescharacteristic, complemented, Sylow
186#529—1271no—
187C2×C4×D4—1271no—
188#483—1271no—
189#480—1271no—
190#478—1271no—
191#478—1271no—
192#463—1271no—
193#463—1271no—
194#450—1271no—
195#447—1271no—
196#442—1271no—
197#442D12711yescharacteristic, complemented
198#440—1271no—
199#440—1271no—
200#432—1271no—
201C22×D4—1271no—
202C23×C4—2541no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C2×D8C2×D12711yescharacteristic
207#126—2541no—
208#125—1271no—
209#124—1271no—
210#123—2541no—
211C22⋊Q8—1271no—
212#121—1271no—
213C4×Q8—1271no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—1271no—
217C4×D4—2541no—
218C4×D4—2541no—
219C4×D4—2541no—
220C4×D4C2×D12711yescharacteristic
221C4×D4C2×D12711yescharacteristic, complemented
222#117—2541no—
223C2×C4⋊C4—1271no—
224C2×C22⋊C4—2541no—
225C2×C42—1271no—
226#107—1271no—
227#107—1271no—
228#106—1271no—
229#106C2×D12711yescharacteristic, complemented
230C4⋊C8—1271no—
231C4⋊C8—1271no—
232#103—1271no—
233#103—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102—1271no—
238#102C2×D12711yescharacteristic
239#102C2×D12711yescharacteristic, complemented
240C22⋊C8—1271no—
241C22⋊C8—1271no—
242#97—1271no—
243#97C2×D12711yescharacteristic, complemented
244C24—2541no—
245C2×Q8—1271no—
246C2×D4—1271no—
247C2×D4—2541no—
248C2×D4—2542no—
249C2×D4—2541no—
250C2×D4—2542no—
251C2×D4C22×D12711yescharacteristic
252C2×D4C22×D12711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1271no—
256C22×C4—1271no—
257C22×C4—1271no—
258C22×C4—5084no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263SD16—2542no—
264SD16—2542no—
265D8—22no—
266D8—2542no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8C22×D12711yescharacteristic
270C2×C8C22×D12711yescharacteristic
271C4⋊C4—1271no—
272C4⋊C4—1271no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4C22×D12711yescharacteristic
279C4⋊C4C22×D12711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C42—1271no—
290C42—1271no—
291C42—1271no—
292C42—2541no—
293C42C22×D12711yescharacteristic
294C23—21no—
295C23—21no—
296C23—1271no—
297C23—5082no—
298C23—5082no—
299C23—2544no—
300C23—2541no—
301Q8—2541no—
302Q8—2541no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—1272no—
307D4—5082no—
308D4—2544no—
309D4—2541no—
310D4#1452324812yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—1271no—
316C2×C4—1271no—
317C2×C4—1271no—
318C2×C4—5082no—
319C2×C4—5081no—
320C2×C4—5082no—
321C2×C4—5082no—
322C2×C4—5082no—
323C2×C4—2542no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4D4×D12711yescharacteristic
330C2×C4D4×D12711yescharacteristic
331C2×C4C23×D12711yescharacteristic
332C8—21no—
333C8—21no—
334C8—2541no—
335C8—2541no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—1272no—
343C22—5088no—
344C22—5081no—
345C22—5081no—
346C22—2542no—
347C22—2541no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—5081no—
355C4—5081no—
356C4—5081no—
357C4—2541no—
358C4—2541no—
359C4—2541no—
360C4#41×D12711yescharacteristic
361C4C2×#1452324811yescharacteristic
362C2—42no—
363C2—81no—
364C2—5082no—
365C2—2542no—
366C2#2255038511yescharacteristic
367C2AGL(1,ℤ/8)×D12711yescharacteristic
368C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#442×C127C2
socleC22×C127C22×D4
prime radical#22550008C2
hypercenter#442D127
hypocenterC127#3114
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #22550408C2semidirect
2. C4×D4×D127C2semidirect
3. #22550208C2semidirect
4. #22550188C2semidirect
5. #22550175C2semidirect
6. #22550174C2semidirect
7. #22550090C2semidirect
8. #22550089C2semidirect
9. #22550028C2semidirect
11. #22549986C2semidirect
12. #22549978C2semidirect
13. #22549976C2semidirect
14. #22549931C2semidirect
15. #442×C127C2semidirect
22. #16861200C22semidirect
23. #16861196C22semidirect
27. C4×Dic254C22semidirect
38. #16861143C22semidirect
39. #16861142C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861132C22semidirect
47. #16861130C22semidirect
51. #16861120C22semidirect
54. C4×D4×C127C22semidirect
55. #106×C127C22semidirect
57. #102×C127C22semidirect
58. #97×C127C22semidirect
185. C127#3114semidirect
197. #442D127semidirect
221. C4×D4C2×D127semidirect
229. #106C2×D127semidirect
239. #102C2×D127semidirect
243. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504082
C4×D4×D1272
#225502082
#225501882
#225501752
#225501742
#225500902
#225500892
#225500282
#225500082
#225499862
#225499782
#225499762
#225499312
#442×C1272
#3114127