Finite Group Databasebeta

← #30648916

Subgroups — #30648916

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#22550497C211yesmaximal, characteristic, complemented
2#22550350C211yesmaximal, characteristic, complemented
3#22550218C211yesmaximal, characteristic, complemented
4#22550201C211yesmaximal, characteristic, complemented
5#22550167C211yesmaximal, characteristic, complemented
6#22550157C211yesmaximal, characteristic, complemented
7#22550092C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550029C211yesmaximal, characteristic, complemented
10#22550012C211yesmaximal, characteristic
11#22549989C211yesmaximal, characteristic, complemented
12#22549978C211yesmaximal, characteristic, complemented
13C4×#14523237C211yesmaximal, characteristic, complemented
14C4×D1016C211yesmaximal, characteristic, complemented
15#433×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C2×#14523237C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×D1016C2211yescharacteristic
20C2×D1016C2211yescharacteristic
21#16861209C2211yescharacteristic, complemented
22#16861204—21no—
23#16861197C2211yescharacteristic, complemented
24#16861195—21no—
25#16861191—21no—
26#16861188C2211yescharacteristic
27#16861187—21no—
28#16861183C2211yescharacteristic
29C4×Dic254C2211yescharacteristic, complemented
30C4×D508C2211yescharacteristic
31C4×D508C2211yescharacteristic
32C4×D508C2211yescharacteristic
33C4×D508C2211yescharacteristic
34C4×D508C2211yescharacteristic, complemented
35#16861171—21no—
36#16861171—21no—
37#16861168C2211yescharacteristic
38C22⋊C4×D127—21no—
39C22⋊C4×D127—21no—
40#16861143C2211yescharacteristic, complemented
41#16861143C2211yescharacteristic, complemented
42#16861141C2211yescharacteristic, complemented
43#16861141C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861134C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic, complemented
51#16861131C2211yescharacteristic, complemented
52#117×C127C2211yescharacteristic
53C4⋊C8×C127C2211yescharacteristic, complemented
54C4⋊C8×C127C2211yescharacteristic, complemented
55C22⋊C8×C127C2211yescharacteristic
56C22⋊C8×C127C2211yescharacteristic
57#97×C127C2211yescharacteristic, complemented
58C4×C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C2×Dic254—21no—
61C2×Dic254C2311yescharacteristic
62C2×D508—22no—
63C2×D508—22no—
64C2×D508—21no—
65C2×D508C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74#14523237—22no—
75#14523237—22no—
76D1016—22no—
77D1016—22no—
78C4⋊Dic127—21no—
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C22×C4×C127C2311yescharacteristic
98C2×C8×C127C2311yescharacteristic
99C2×C8×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127D411yescharacteristic
103C4⋊C4×C127D411yescharacteristic
104C22⋊C4×C127D411yescharacteristic
105C22⋊C4×C127D411yescharacteristic
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—24no—
111C22×D127—21no—
112C22×D127—81no—
113C22×D127—81no—
114Dic254—21no—
115Dic254—21no—
116Dic254—81no—
117D508—24no—
118D508—21no—
119D508—21no—
120D508—21no—
121D508—21no—
122D508#4112yes—
123D508#4112yes—
124C22⋊D127—81no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C4×D127—81no—
131C4×D127—81no—
132C4×D127—81no—
133C4×D127—81no—
134C4×D127—81no—
135C23×C127C2×D411yescharacteristic
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2411yescharacteristic
144C8×C127—21no—
145C8×C127—21no—
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127C22×D411yescharacteristic
152C2×D127—22no—
153C2×D127—22no—
154C2×D127—41no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—84no—
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127C2×#4111yescharacteristic
168C4×C127C2×#4111yescharacteristic
169Dic127—41no—
170Dic127—41no—
171Dic127—41no—
172Dic127—41no—
173Dic127—41no—
174C2×C127—41no—
175C2×C127#52511yescharacteristic
176C2×C127#57611yescharacteristic
177C2×C127#57711yescharacteristic
178D127—42no—
179D127—42no—
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#450—1271no—
192#447—1271no—
193#442—1271no—
194#440—1271no—
195C4×SD16—1271no—
196C4×D8—1271no—
197#433D12711yescharacteristic, complemented
198C22×D4—1271no—
199C2×SD16—1271no—
200C2×SD16—1271no—
201C2×D8—1271no—
202C2×D8—1271no—
203C4⋊Q8—1271no—
204#126—2541no—
205#124—1271no—
206#124—2541no—
207#123—2541no—
208C22⋊Q8—1271no—
209C22⋊Q8—2541no—
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#117C2×D12711yescharacteristic
221C2×C22⋊C4—2541no—
222C2×C22⋊C4—2541no—
223#107—1271no—
224#107—1271no—
225#106—1271no—
226#106—1271no—
227C4⋊C8C2×D12711yescharacteristic, complemented
228C4⋊C8C2×D12711yescharacteristic, complemented
229#103—1271no—
230#103—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237C22⋊C8C2×D12711yescharacteristic
238C22⋊C8C2×D12711yescharacteristic
239#97C2×D12711yescharacteristic, complemented
240C4×C8C2×D12711yescharacteristic, complemented
241C24—2541no—
242C2×Q8—1271no—
243C2×Q8—2541no—
244C2×D4—1271no—
245C2×D4—1271no—
246C2×D4—1271no—
247C2×D4—2542no—
248C2×D4—2542no—
249C2×D4—2541no—
250C2×D4—2541no—
251C22×C4—2541no—
252C22×C4—2541no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4C22×D12711yescharacteristic
257SD16—2542no—
258SD16—2542no—
259D8—2542no—
260D8—2542no—
261C2×C8C22×D12711yescharacteristic
262C2×C8C22×D12711yescharacteristic
263C2×C8C22×D12711yescharacteristic
264C2×C8C22×D12711yescharacteristic
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4D50811yescharacteristic
273C4⋊C4D50811yescharacteristic
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4D50811yescharacteristic
285C22⋊C4D50811yescharacteristic
286C42—2541no—
287C42—2541no—
288C42C22×D12711yescharacteristic
289C42C22×D12711yescharacteristic
290C23—10161no—
291C23—10161no—
292C23—2544no—
293C23—2541no—
294C23—2541no—
295C23—2541no—
296C23C2×D50811yescharacteristic
297Q8—10161no—
298Q8—2541no—
299Q8—2541no—
300D4—1272no—
301D4—1272no—
302D4—10161no—
303D4—2544no—
304D4—2541no—
305D4—2541no—
306D4—2541no—
307D4—2541no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—10161no—
311C2×C4—10161no—
312C2×C4—10161no—
313C2×C4—10161no—
314C2×C4—10161no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4C2×D50811yescharacteristic
321C2×C4C2×D50811yescharacteristic
322C2×C4C2×D50811yescharacteristic
323C2×C4C2×D50811yescharacteristic
324C2×C4C2×D50811yescharacteristic
325C2×C4C23×D12711yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—10164no—
334C22—5081no—
335C22—5081no—
336C22—5081no—
337C22—5081no—
338C22—5081no—
339C22—5081no—
340C22—5081no—
341C22—2542no—
342C22—2542no—
343C22C22×D50811yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—5081no—
350C4—5081no—
351C4—5081no—
352C4—5081no—
353C4—5081no—
354C4#41×D12711yescharacteristic
355C4#41×D12711yescharacteristic
356C2—41no—
357C2—10161no—
358C2—5082no—
359C2—5082no—
360C2#2255038811yescharacteristic
361C2#2255068111yescharacteristic
362C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#433×C127C2
socleC22×C127C22×D4
prime radical#22550012C2
hypercenter#433D127
hypocenterC127#3118
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #22550497C2semidirect
2. #22550350C2semidirect
3. #22550218C2semidirect
4. #22550201C2semidirect
5. #22550167C2semidirect
6. #22550157C2semidirect
7. #22550092C2semidirect
8. #22550092C2semidirect
9. #22550029C2semidirect
11. #22549989C2semidirect
12. #22549978C2semidirect
13. C4×#14523237C2semidirect
14. C4×D1016C2semidirect
15. #433×C127C2semidirect
21. #16861209C22semidirect
23. #16861197C22semidirect
29. C4×Dic254C22semidirect
34. C4×D508C22semidirect
40. #16861143C22semidirect
41. #16861143C22semidirect
42. #16861141C22semidirect
43. #16861141C22semidirect
44. #16861134C22semidirect
45. #16861134C22semidirect
50. #16861131C22semidirect
51. #16861131C22semidirect
53. C4⋊C8×C127C22semidirect
54. C4⋊C8×C127C22semidirect
57. #97×C127C22semidirect
58. C4×C8×C127C22semidirect
182. C127#3118semidirect
197. #433D127semidirect
227. C4⋊C8C2×D127semidirect
228. C4⋊C8C2×D127semidirect
239. #97C2×D127semidirect
240. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504972
#225503502
#225502182
#225502012
#225501672
#225501572
#225500922
#225500292
#225500122
#225499892
#225499782
C4×#145232372
C4×D10162
#433×C1272
#3118127