Finite Group Databasebeta

← #30648912

Subgroups — #30648912

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#22550491C211yesmaximal, characteristic, complemented
2#22550348C211yesmaximal, characteristic, complemented
3#22550220C211yesmaximal, characteristic, complemented
4#22550198C211yesmaximal, characteristic, complemented
5#22550165C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550090C211yesmaximal, characteristic, complemented
8#22550089C211yesmaximal, characteristic, complemented
9#22550028C211yesmaximal, characteristic, complemented
10#22550008C211yesmaximal, characteristic
11#22549978C211yesmaximal, characteristic, complemented
12C4×#14523236C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549938C211yesmaximal, characteristic, complemented
15#442×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C2×#14523237C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20#16861208C2211yescharacteristic, complemented
21#16861202—21no—
22#16861196C2211yescharacteristic, complemented
23#16861195—21no—
24#16861192—21no—
25#16861189C2211yescharacteristic
26#16861188—21no—
27#16861180C2211yescharacteristic
28C4×Dic254C2211yescharacteristic, complemented
29C4×D508C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31#16861171C2211yescharacteristic
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167C2211yescharacteristic
35#16861166C2211yescharacteristic
36C2×(C2×D127)⋊C4—21no—
37C22⋊C4×D127—21no—
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
51C4×C127⋊C8C2211yescharacteristic, 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×Dic254—21no—
61C2×Dic254C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65D4×D127—22no—
66D4×D127—22no—
67C22×Dic127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127C2311yescharacteristic
71#14523237—22no—
72#14523236—22no—
73#14523233—22no—
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4D411yescharacteristic
82Dic127⋊C4D411yescharacteristic
83C22⋊Dic127—21no—
84C22⋊Dic127—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)⋊C4D411yescharacteristic
92(C2×D127)⋊C4D411yescharacteristic
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127C2311yescharacteristic
96C2×D4×C127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100D8×C127—22no—
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—24no—
110C22×D127—81no—
111C22×D127—81no—
112C22×D127C2×D411yescharacteristic
113Dic254—21no—
114Dic254—21no—
115Dic254—81no—
116D508—21no—
117D508#4112yes—
118C22⋊D127—24no—
119C22⋊D127—81no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—81no—
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C4×D127—21no—
128C4×D127—21no—
129C4×D127—81no—
130C4×D127—81no—
131C127⋊C8—21no—
132C127⋊C8—21no—
133C23×C127—21no—
134C23×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127#4112yes—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—81no—
142C2×C4×C127—81no—
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—22no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127C22×D411yescharacteristic
155C2×D127—21no—
156C2×D127—22no—
157C2×D127—21no—
158C2×D127—21no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—84no—
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127C2×#4111yescharacteristic
167C4×C127C2×#4111yescharacteristic
168Dic127—21no—
169Dic127—21no—
170Dic127—21no—
171Dic127—41no—
172Dic127—41no—
173Dic127—41no—
174C2×C127—42no—
175C2×C127—81no—
176C2×C127#52511yescharacteristic
177C2×C127#57611yescharacteristic
178C2×C127#57711yescharacteristic
179D127—41no—
180D127—42no—
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#442D12711yescharacteristic, complemented
194#440—1271no—
195C4×SD16—1271no—
196C4×D8—1271no—
197#433—1271no—
198C22×D4—1271no—
199C2×SD16—1271no—
200C2×SD16—1271no—
201C2×D8—1271no—
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#121—1271no—
211C4×Q8—1271no—
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215C4×D4—2541no—
216C4×D4—2541no—
217C4×D4C2×D12711yescharacteristic
218C4×D4C2×D12711yescharacteristic, complemented
219#117—1271no—
220#117—1271no—
221C2×C22⋊C4—2541no—
222C2×C22⋊C4—2541no—
223#107—1271no—
224#107—1271no—
225#106—1271no—
226#106C2×D12711yescharacteristic, complemented
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, complemented
237C22⋊C8—1271no—
238C22⋊C8—1271no—
239#97C2×D12711yescharacteristic, complemented
240C4×C8—1271no—
241C24—2541no—
242C2×Q8—1271no—
243C2×Q8—2541no—
244C2×D4—1271no—
245C2×D4—2541no—
246C2×D4—2542no—
247C2×D4—2541no—
248C2×D4—2542no—
249C2×D4C22×D12711yescharacteristic
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—1271no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257SD16—2542no—
258SD16—2542no—
259D8—22no—
260D8—2542no—
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—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4C22×D12711yescharacteristic
273C4⋊C4C22×D12711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—1271no—
277C22⋊C4—1271no—
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—2541no—
288C42—2541no—
289C42C22×D12711yescharacteristic
290C23—21no—
291C23—21no—
292C23—1271no—
293C23—10161no—
294C23—10161no—
295C23—2544no—
296C23—2541no—
297Q8—10161no—
298Q8—2541no—
299Q8—2541no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—1272no—
304D4—10161no—
305D4—2544no—
306D4—2541no—
307D4#1452324812yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—1271no—
313C2×C4—1271no—
314C2×C4—1271no—
315C2×C4—10161no—
316C2×C4—10161no—
317C2×C4—10161no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4D4×D12711yescharacteristic
324C2×C4D4×D12711yescharacteristic
325C2×C4C23×D12711yescharacteristic
326C8—21no—
327C8—21no—
328C8—2541no—
329C8—2541no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—10164no—
337C22—5081no—
338C22—5081no—
339C22—2541no—
340C22—2542no—
341C22—2541no—
342C22—2541no—
343C22C2×D4×D12711yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—5081no—
349C4—5081no—
350C4—5081no—
351C4—2541no—
352C4—2541no—
353C4—2541no—
354C4#41×D12711yescharacteristic
355C4C2×#1452324811yescharacteristic
356C2—42no—
357C2—81no—
358C2—5081no—
359C2—5082no—
360C2#2255038511yescharacteristic
361C2#2255068011yescharacteristic
362C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#442×C127C2
socleC22×C127C22×D4
prime radical#22550008C2
hypercenter#442D127
hypocenterC127#3118
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. #22550491C2semidirect
2. #22550348C2semidirect
3. #22550220C2semidirect
4. #22550198C2semidirect
5. #22550165C2semidirect
6. #22550154C2semidirect
7. #22550090C2semidirect
8. #22550089C2semidirect
9. #22550028C2semidirect
11. #22549978C2semidirect
12. C4×#14523236C2semidirect
13. C4×#14523233C2semidirect
14. #22549938C2semidirect
15. #442×C127C2semidirect
20. #16861208C22semidirect
22. #16861196C22semidirect
28. C4×Dic254C22semidirect
38. #16861143C22semidirect
39. #16861142C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861132C22semidirect
47. #16861130C22semidirect
51. C4×C127⋊C8C22semidirect
54. C4×D4×C127C22semidirect
55. #106×C127C22semidirect
57. #102×C127C22semidirect
58. #97×C127C22semidirect
182. C127#3118semidirect
193. #442D127semidirect
218. C4×D4C2×D127semidirect
226. #106C2×D127semidirect
236. #102C2×D127semidirect
239. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504912
#225503482
#225502202
#225501982
#225501652
#225501542
#225500902
#225500892
#225500282
#225500082
#225499782
C4×#145232362
C4×#145232332
#225499382
#442×C1272
#3118127