Finite Group Databasebeta

← #30648907

Subgroups — #30648907

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#22550493C211yesmaximal, characteristic, complemented
2#22550351C211yesmaximal, characteristic, complemented
3#22550218C211yesmaximal, characteristic, complemented
4#22550197C211yesmaximal, characteristic, complemented
5#22550164C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550089C211yesmaximal, characteristic, complemented
8#22550089C211yesmaximal, characteristic, complemented
9#22550027C211yesmaximal, characteristic, complemented
10#22549988C211yesmaximal, characteristic, complemented
11#22549973C211yesmaximal, characteristic, complemented
12#22549967C211yesmaximal, characteristic, complemented
13D8×Dic127C211yesmaximal, characteristic, complemented
14#22549934C211yesmaximal, characteristic, complemented
15#447×C127C211yesmaximal, characteristic
16C2×#14523236C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18#16861208C2211yescharacteristic, complemented
19#16861202—21no—
20#16861196—21no—
21#16861196C2211yescharacteristic, complemented
22#16861190—21no—
23#16861188C2211yescharacteristic
24#16861186—21no—
25#16861174C2211yescharacteristic, complemented
26D4×Dic127—21no—
27D4×Dic127—21no—
28D4×Dic127C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32D4×Dic127C2211yescharacteristic, 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#16861134C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic, complemented
48#16861130C2211yescharacteristic, complemented
49C22⋊(C127⋊C8)C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54C2×D8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61C22×Dic127—21no—
62C22×Dic127—21no—
63C22×Dic127—21no—
64C22×Dic127—21no—
65C22×Dic127—21no—
66#14523236—22no—
67#14523236—22no—
68C2×C127⋊C8C2311yescharacteristic
69C2×C127⋊C8C2311yescharacteristic
70C4⋊Dic127D411yescharacteristic
71C4⋊Dic127D411yescharacteristic
72C4⋊Dic127C2311yescharacteristic
73C4⋊Dic127C2311yescharacteristic
74C4⋊Dic127C2311yescharacteristic
75Dic127⋊C4—21no—
76Dic127⋊C4—21no—
77Dic127⋊C4—21no—
78C22⋊Dic127—21no—
79C22⋊Dic127—21no—
80C22⋊Dic127—21no—
81C22⋊Dic127—21no—
82C22⋊Dic127—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127D411yescharacteristic
88C22⋊Dic127D411yescharacteristic
89C4×Dic127—21no—
90C4×Dic127—21no—
91C4×Dic127C2311yescharacteristic
92C4×Dic127C2311yescharacteristic
93C24×C127—21no—
94C2×D4×C127—22no—
95C2×D4×C127—22no—
96C2×D4×C127—21no—
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102D8×C127—22no—
103D8×C127—22no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108Dic254—21no—
109Dic254—21no—
110Dic254—81no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—81no—
116C2×Dic127—81no—
117C2×Dic127—81no—
118C2×Dic127—81no—
119C2×Dic127—81no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C127⋊C8—21no—
125C127⋊C8—21no—
126C23×C127—24no—
127C23×C127—21no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127—81no—
131C23×C127—81no—
132C23×C127C2×D411yescharacteristic
133D4×C127—24no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—81no—
139D4×C127#4112yes—
140D4×C127#4112yes—
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—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—84no—
161C22×C127C22×D411yescharacteristic
162C4×C127—21no—
163C4×C127—41no—
164C4×C127C2×#4111yescharacteristic
165C4×C127C2×#4111yescharacteristic
166Dic127—21no—
167Dic127—21no—
168Dic127—21no—
169Dic127—21no—
170Dic127—41no—
171Dic127—41no—
172Dic127—41no—
173Dic127—41no—
174C2×C127—41no—
175C2×C127—42no—
176C2×C127—42no—
177C2×C127—81no—
178C2×C127#52511yescharacteristic
179C2×C127#57611yescharacteristic
180C2×C127#57711yescharacteristic
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#450—1271no—
192#447D12711yescharacteristic
193#442—1271no—
194#440—1271no—
195C4×SD16—1271no—
196C4×D8—1271no—
197#433—1271no—
198C22×D4C2×D12711yescharacteristic
199C2×SD16—1271no—
200C2×SD16—1271no—
201C2×D8C2×D12711yescharacteristic
202C2×D8C2×D12711yescharacteristic
203C4⋊Q8—1271no—
204#126—2541no—
205#124—1271no—
206#124—2541no—
207#123—2541no—
208C22⋊Q8—1271no—
209C22⋊Q8—2541no—
210#121C2×D12711yescharacteristic
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—1271no—
243C2×Q8—2541no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22×D12711yescharacteristic
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—2542no—
258SD16—2542no—
259D8—22no—
260D8—22no—
261C2×C8—1271no—
262C2×C8—1271no—
263C2×C8C22×D12711yescharacteristic
264C2×C8C22×D12711yescharacteristic
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—2541no—
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—21no—
294C23—81no—
295C23—81no—
296C23D4×D12711yescharacteristic
297Q8—10161no—
298Q8—2541no—
299Q8—2541no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—81no—
306D4#1452324712yes—
307D4#1452324712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—1271no—
314C2×C4—1271no—
315C2×C4—10161no—
316C2×C4—10161no—
317C2×C4—10161no—
318C2×C4—10161no—
319C2×C4—10161no—
320C2×C4—2541no—
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—41no—
340C22—41no—
341C22—41no—
342C22—84no—
343C22C2×D4×D12711yescharacteristic
344C4—21no—
345C4—41no—
346C4—5081no—
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—81no—
360C2#2255038211yescharacteristic
361C2#2255067711yescharacteristic
362C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#447×C127C2
socleC22×C127C22×D4
hypercenter#447D127
hypocenterC127#3118
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22550493C2semidirect
2. #22550351C2semidirect
3. #22550218C2semidirect
4. #22550197C2semidirect
5. #22550164C2semidirect
6. #22550154C2semidirect
7. #22550089C2semidirect
8. #22550089C2semidirect
9. #22550027C2semidirect
10. #22549988C2semidirect
11. #22549973C2semidirect
12. #22549967C2semidirect
13. D8×Dic127C2semidirect
14. #22549934C2semidirect
18. #16861208C22semidirect
21. #16861196C22semidirect
25. #16861174C22semidirect
32. D4×Dic127C22semidirect
37. #16861143C22semidirect
38. #16861142C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861134C22semidirect
47. #16861130C22semidirect
48. #16861130C22semidirect
50. #16861119C22semidirect
51. C8×Dic127C22semidirect
182. C127#3118semidirect

Maximal subgroups

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

subgroupindex
#225504932
#225503512
#225502182
#225501972
#225501642
#225501542
#225500892
#225500272
#225499882
#225499732
#225499672
D8×Dic1272
#225499342
#447×C1272
#3118127