Finite Group Databasebeta

← #30648908

Subgroups — #30648908

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#22550350C211yesmaximal, characteristic, complemented
2#22550214C211yesmaximal, characteristic, complemented
3#22550197C211yesmaximal, characteristic, complemented
4#22550163C211yesmaximal, characteristic, complemented
5#22550153C211yesmaximal, characteristic, complemented
6#22550088C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550026C211yesmaximal, characteristic, complemented
9#22550009C211yesmaximal, characteristic
10#22549986C211yesmaximal, characteristic, complemented
11#22549974C211yesmaximal, characteristic, complemented
12C4×#14523235C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549937C211yesmaximal, characteristic, complemented
15#542×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×#14523235C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C4×D508C2211yescharacteristic
22C4×D508C2211yescharacteristic
23C4×D508C2211yescharacteristic
24C4×D508C2211yescharacteristic
25#16861171—21no—
26#16861171—21no—
27#16861168C2211yescharacteristic
28C22⋊C4×D127—21no—
29C22⋊C4×D127—21no—
30#16861142C2211yescharacteristic, complemented
31#16861142C2211yescharacteristic, complemented
32#16861140C2211yescharacteristic, complemented
33#16861140C2211yescharacteristic, complemented
34C4⋊(C127⋊C8)C2211yescharacteristic, complemented
35C4⋊(C127⋊C8)C2211yescharacteristic, complemented
36#16861133C2211yescharacteristic, complemented
37#16861133C2211yescharacteristic, complemented
38#16861130C2211yescharacteristic, complemented
39#16861130C2211yescharacteristic, complemented
40#16861129C2211yescharacteristic
41#16861129C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44C22⋊(C127⋊C8)C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46#16861120C2211yescharacteristic, complemented
47C4×C127⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C127C2211yescharacteristic, complemented
49#126×C127—21no—
50#124×C127—21no—
51#124×C127C2211yescharacteristic, complemented
52#123×C127—21no—
53C22⋊Q8×C127—21no—
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56C4×Q8×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic, complemented
58#117×C127C2211yescharacteristic
59C23×D127—21no—
60C2×D508—22no—
61C2×D508—22no—
62C2×D508—21no—
63C2×D508C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C4×D127—21no—
66C2×C4×D127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69#14523235—22no—
70#14523235—22no—
71#14523233—22no—
72#14523233—22no—
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C22⋊Dic127—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83(C2×D127)⋊C4—21no—
84C4×Dic127—21no—
85C2×Q8×C127—21no—
86C2×Q8×C127C2311yescharacteristic
87C2×D4×C127—21no—
88C2×D4×C127C2311yescharacteristic
89C22×C4×C127—21no—
90C22×C4×C127C2311yescharacteristic
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127D411yescharacteristic
95C4⋊C4×C127D411yescharacteristic
96C4⋊C4×C127C2311yescharacteristic
97C4⋊C4×C127C2311yescharacteristic
98C22⋊C4×C127—21no—
99C22⋊C4×C127—21no—
100C22⋊C4×C127—21no—
101C22⋊C4×C127—21no—
102C22⋊C4×C127—21no—
103C22⋊C4×C127D411yescharacteristic
104C22⋊C4×C127D411yescharacteristic
105C42×C127—21no—
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—24no—
111C22×D127—81no—
112C22×D127—81no—
113D508—24no—
114D508—21no—
115D508—21no—
116D508#4112yes—
117D508#4112yes—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C4×D127—81no—
121C4×D127—81no—
122C4×D127—81no—
123C4×D127—81no—
124C127⋊C8—21no—
125C127⋊C8—21no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C23×C127—21no—
129C23×C127C2×D411yescharacteristic
130Q8×C127—21no—
131Q8×C127—21no—
132Q8×C127—81no—
133D4×C127—21no—
134D4×C127—21no—
135D4×C127—81no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—81no—
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127C22×D411yescharacteristic
155C2×D127—22no—
156C2×D127—22no—
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×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127C2×#4111yescharacteristic
171C4×C127C2×#4111yescharacteristic
172Dic127—41no—
173Dic127—41no—
174C2×C127—41no—
175C2×C127—81no—
176C2×C127#52511yescharacteristic
177C2×C127#57611yescharacteristic
178C2×C127#57711yescharacteristic
179D127—42no—
180D127—42no—
181#3118—1271nomaximal, Sylow
182C127#311811yescharacteristic, complemented, Sylow
183#542D12711yescharacteristic, complemented
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#433—1271no—
198C22×D4—1271no—
199C2×SD16—1271no—
200C2×SD16—1271no—
201C2×D8—1271no—
202C2×D8—1271no—
203C4⋊Q8C2×D12711yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D12711yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D12711yescharacteristic
210#121C2×D12711yescharacteristic
211C4×Q8C2×D12711yescharacteristic, complemented
212C4×D4—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—2541no—
217C4×D4—2541no—
218C4×D4C2×D12711yescharacteristic, complemented
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⋊C8—1271no—
228C4⋊C8—1271no—
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⋊C8—1271no—
238C22⋊C8—1271no—
239#97—1271no—
240C4×C8—1271no—
241C24—2541no—
242C2×Q8—21no—
243C2×Q8C22×D12711yescharacteristic
244C2×D4—21no—
245C2×D4—1271no—
246C2×D4—1271no—
247C2×D4—2542no—
248C2×D4—2542no—
249C2×D4—2541no—
250C2×D4C22×D12711yescharacteristic
251C22×C4—21no—
252C22×C4—2541no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4C22×D12711yescharacteristic
257SD16—2542no—
258SD16—2542no—
259D8—2542no—
260D8—2542no—
261C2×C8—1271no—
262C2×C8—1271no—
263C2×C8—1271no—
264C2×C8—1271no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4C22⋊D12711yescharacteristic
271C4⋊C4C22⋊D12711yescharacteristic
272C4⋊C4C22×D12711yescharacteristic
273C4⋊C4C22×D12711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4C22⋊D12711yescharacteristic
285C22⋊C4C22⋊D12711yescharacteristic
286C42—21no—
287C42—2541no—
288C42C22×D12711yescharacteristic
289C42C22×D12711yescharacteristic
290C23—21no—
291C23—10161no—
292C23—10161no—
293C23—2544no—
294C23—2541no—
295C23—2541no—
296C23C2×C22⋊D12711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—1272no—
304D4—1272no—
305D4—2544no—
306D4—2541no—
307D4—2541no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—10161no—
315C2×C4—10161no—
316C2×C4—10161no—
317C2×C4—10161no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4C2×C22⋊D12711yescharacteristic
321C2×C4C2×C22⋊D12711yescharacteristic
322C2×C4C2×C22⋊D12711yescharacteristic
323C2×C4C2×C22⋊D12711yescharacteristic
324C2×C4C2×C22⋊D12711yescharacteristic
325C2×C4C23×D12711yescharacteristic
326C8—2541no—
327C8—2541no—
328C8—2541no—
329C8—2541no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—10164no—
337C22—5081no—
338C22—5081no—
339C22—5081no—
340C22—5081no—
341C22—2542no—
342C22—2542no—
343C22C22×C22⋊D12711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—5081no—
353C4—5081no—
354C4#41×D12711yescharacteristic
355C4#41×D12711yescharacteristic
356C2—41no—
357C2—81no—
358C2—5082no—
359C2—5082no—
360C2#2255038311yescharacteristic
361C2#2255067911yescharacteristic
362C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#542×C127C2
socleC22×C127C22×D4
prime radical#22550009C2
hypercenter#542D127
hypocenterC127#3118
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #22550350C2semidirect
2. #22550214C2semidirect
3. #22550197C2semidirect
4. #22550163C2semidirect
5. #22550153C2semidirect
6. #22550088C2semidirect
7. #22550088C2semidirect
8. #22550026C2semidirect
10. #22549986C2semidirect
11. #22549974C2semidirect
12. C4×#14523235C2semidirect
13. C4×#14523233C2semidirect
14. #22549937C2semidirect
15. #542×C127C2semidirect
30. #16861142C22semidirect
31. #16861142C22semidirect
32. #16861140C22semidirect
33. #16861140C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. C4⋊(C127⋊C8)C22semidirect
36. #16861133C22semidirect
37. #16861133C22semidirect
38. #16861130C22semidirect
39. #16861130C22semidirect
46. #16861120C22semidirect
47. C4×C127⋊C8C22semidirect
48. C4⋊Q8×C127C22semidirect
51. #124×C127C22semidirect
56. C4×Q8×C127C22semidirect
57. C4×D4×C127C22semidirect
182. C127#3118semidirect
183. #542D127semidirect
203. C4⋊Q8C2×D127semidirect
206. #124C2×D127semidirect
211. C4×Q8C2×D127semidirect
218. C4×D4C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503502
#225502142
#225501972
#225501632
#225501532
#225500882
#225500262
#225500092
#225499862
#225499742
C4×#145232352
C4×#145232332
#225499372
#542×C1272
#3118127