Finite Group Databasebeta

← #30648915

Subgroups — #30648915

Order 16256.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550499C211yesmaximal, characteristic, complemented
2#22550351C211yesmaximal, characteristic, complemented
3#22550214C211yesmaximal, characteristic, complemented
4#22550201C211yesmaximal, characteristic, complemented
5#22550164C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550093C211yesmaximal, characteristic, complemented
8#22550093C211yesmaximal, characteristic, complemented
9#22550010C211yesmaximal, characteristic
10#22549985C211yesmaximal, characteristic, complemented
11#22549977C211yesmaximal, characteristic, complemented
12SD16×Dic127C211yesmaximal, characteristic, complemented
13#22549956C211yesmaximal, characteristic, complemented
14#22549934C211yesmaximal, characteristic, complemented
15#450×C127C211yesmaximal, characteristic, complemented
16C2×#14523233C2211yescharacteristic
17C2×#14523233C2211yescharacteristic
18#16861208C2211yescharacteristic, complemented
19#16861204—21no—
20#16861198C2211yescharacteristic, complemented
21#16861195—21no—
22#16861191—21no—
23#16861189—21no—
24#16861183C2211yescharacteristic
25Q8×Dic127C2211yescharacteristic, complemented
26D4×Dic127—21no—
27D4×Dic127—21no—
28D4×Dic127C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32#16861171C2211yescharacteristic, complemented
33#16861168C2211yescharacteristic
34#16861168C2211yescharacteristic
35C2×C22⋊Dic127—21no—
36C2×C22⋊Dic127—21no—
37#16861143C2211yescharacteristic, complemented
38#16861142C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861133C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic
55C22⋊Q8×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×D508C2311yescharacteristic
61C2×C22⋊D127—21no—
62C22×Dic127—21no—
63C22×Dic127—21no—
64C22×Dic127—21no—
65C22×Dic127—21no—
66C2×C4×D127—21no—
67#14523233—22no—
68#14523233—22no—
69C2×C127⋊C8C2311yescharacteristic
70C2×C127⋊C8C2311yescharacteristic
71C4⋊Dic127—21no—
72C4⋊Dic127D411yescharacteristic
73C4⋊Dic127D411yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78C22⋊Dic127—21no—
79C22⋊Dic127—21no—
80C22⋊Dic127—21no—
81C22⋊Dic127—21no—
82C22⋊Dic127—21no—
83C22⋊Dic127D411yescharacteristic
84C22⋊Dic127D411yescharacteristic
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89C4×Dic127—21no—
90C4×Dic127—21no—
91C4×Dic127C2311yescharacteristic
92C4×Dic127C2311yescharacteristic
93C24×C127—21no—
94C2×Q8×C127C2311yescharacteristic
95C2×D4×C127—22no—
96C2×D4×C127—22no—
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101SD16×C127—22no—
102SD16×C127—22no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127—21no—
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22×D127—21no—
109Dic254—81no—
110D508—21no—
111D508—21no—
112C22⋊D127—81no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—81no—
117C2×Dic127—81no—
118C2×Dic127—81no—
119C2×Dic127—81no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C4×D127—81no—
125C127⋊C8—21no—
126C127⋊C8—21no—
127C23×C127—24no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127—81no—
131C23×C127—81no—
132C23×C127C2×D411yescharacteristic
133Q8×C127—21no—
134Q8×C127—21no—
135D4×C127—24no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127#4112yes—
139D4×C127#4112yes—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—21no—
149C22×C127—22no—
150C22×C127—22no—
151C22×C127—21no—
152C22×C127—21no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—84no—
158C22×C127C22×D411yescharacteristic
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C4×C127—21no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127C2×#4111yescharacteristic
166C4×C127C2×#4111yescharacteristic
167Dic127—21no—
168Dic127—21no—
169Dic127—21no—
170Dic127—21no—
171Dic127—41no—
172Dic127—41no—
173Dic127—41no—
174C2×C127—41no—
175C2×C127—42no—
176C2×C127—42no—
177C2×C127#52511yescharacteristic
178C2×C127#57611yescharacteristic
179C2×C127#57711yescharacteristic
180D127—81no—
181#3118—1271nomaximal, Sylow
182C127#311811yescharacteristic, complemented, Sylow
183#542—1271no—
184#518—1271no—
185#485—1271no—
186#482—1271no—
187#476—1271no—
188#474—1271no—
189#463—1271no—
190#463—1271no—
191#450D12711yescharacteristic, complemented
192#447—1271no—
193#442—1271no—
194#440—1271no—
195C4×SD16—1271no—
196C4×D8—1271no—
197#433—1271no—
198C22×D4C2×D12711yescharacteristic
199C2×SD16C2×D12711yescharacteristic
200C2×SD16C2×D12711yescharacteristic
201C2×D8—1271no—
202C2×D8—1271no—
203C4⋊Q8—1271no—
204#126—2541no—
205#124—1271no—
206#124—2541no—
207#123—2541no—
208C22⋊Q8—2541no—
209C22⋊Q8C2×D12711yescharacteristic
210#121—1271no—
211C4×Q8—1271no—
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—1271no—
217C4×D4—2541no—
218C4×D4—2541no—
219#117—1271no—
220#117—1271no—
221C2×C22⋊C4—2541no—
222C2×C22⋊C4—2541no—
223#107—1271no—
224#107—1271no—
225#106—1271no—
226#106—1271no—
227C4⋊C8—1271no—
228C4⋊C8—1271no—
229#103—1271no—
230#103—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102C2×D12711yescharacteristic
236#102C2×D12711yescharacteristic
237C22⋊C8—1271no—
238C22⋊C8C2×D12711yescharacteristic
239#97—1271no—
240C4×C8—1271no—
241C24—21no—
242C2×Q8—2541no—
243C2×Q8C22×D12711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—1271no—
248C2×D4—2541no—
249C2×D4C22×D12711yescharacteristic
250C2×D4C22×D12711yescharacteristic
251C22×C4—2541no—
252C22×C4—2541no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4C22×D12711yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—2542no—
260D8—2542no—
261C2×C8—1271no—
262C2×C8—1271no—
263C2×C8C22×D12711yescharacteristic
264C2×C8C22×D12711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—1271no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4C22×D12711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—1271no—
276C22⋊C4—1271no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C42—1271no—
287C42—1271no—
288C42—2541no—
289C42—2541no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—2541no—
296C23D4×D12711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—10161no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—10161no—
304D4—2541no—
305D4—2541no—
306D4#1452324712yes—
307D4#1452324712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—1271no—
313C2×C4—1271no—
314C2×C4—1271no—
315C2×C4—1271no—
316C2×C4—10161no—
317C2×C4—10161no—
318C2×C4—10161no—
319C2×C4—10161no—
320C2×C4—10161no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4D4×D12711yescharacteristic
325C2×C4C23×D12711yescharacteristic
326C8—21no—
327C8—21no—
328C8—2541no—
329C8—2541no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—5081no—
341C22—5081no—
342C22—5081no—
343C22C2×D4×D12711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—5081no—
348C4—5081no—
349C4—5081no—
350C4—2541no—
351C4—2541no—
352C4—2541no—
353C4—2541no—
354C4C2×#1452324711yescharacteristic
355C4C2×#1452324711yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—10161no—
360C2#2255038211yescharacteristic
361C2#2255068011yescharacteristic
362C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radical#22550010C2
hypercenter#450D127
hypocenterC127#3118
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22550499C2semidirect
2. #22550351C2semidirect
3. #22550214C2semidirect
4. #22550201C2semidirect
5. #22550164C2semidirect
6. #22550154C2semidirect
7. #22550093C2semidirect
8. #22550093C2semidirect
10. #22549985C2semidirect
11. #22549977C2semidirect
12. SD16×Dic127C2semidirect
13. #22549956C2semidirect
14. #22549934C2semidirect
15. #450×C127C2semidirect
18. #16861208C22semidirect
20. #16861198C22semidirect
25. Q8×Dic127C22semidirect
32. #16861171C22semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861133C22semidirect
44. #16861133C22semidirect
45. #16861131C22semidirect
46. #16861131C22semidirect
50. #16861119C22semidirect
51. C8×Dic127C22semidirect
182. C127#3118semidirect
191. #450D127semidirect

Maximal subgroups

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

subgroupindex
#225504992
#225503512
#225502142
#225502012
#225501642
#225501542
#225500932
#225500102
#225499852
#225499772
SD16×Dic1272
#225499562
#225499342
#450×C1272
#3118127