Finite Group Databasebeta

← #30650586

Subgroups — #30650586

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550578C211yesmaximal, characteristic, complemented
2#22550517C211yesmaximal, characteristic, complemented
3#22550232C211yesmaximal, characteristic, complemented
4#22550209C211yesmaximal, characteristic, complemented
5#22550202C211yesmaximal, characteristic, complemented
6#22550175C211yesmaximal, characteristic, complemented
7#22550111C211yesmaximal, characteristic, complemented
8#22550093C211yesmaximal, characteristic, complemented
9#22550020C211yesmaximal, characteristic, complemented
10#22550009C211yesmaximal, characteristic, complemented
11#22549956C211yesmaximal, characteristic, complemented
12C8×C22⋊D127C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14C2×#16861131C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic, complemented
21#16861202—21no—
22#16861200C2211yescharacteristic, complemented
23#16861198C2211yescharacteristic
24#16861194—21no—
25#16861193—21no—
26#16861184—21no—
27#16861184C2211yescharacteristic
28#16861181—21no—
29#16861180C2211yescharacteristic, complemented
30#16861179—21no—
31D4×Dic127—21no—
32D4×Dic127C2211yescharacteristic, complemented
33#16861171C2211yescharacteristic
34C4×C22⋊D127C2211yescharacteristic
35#16861167C2211yescharacteristic, complemented
36C2×C4⋊Dic127C2211yescharacteristic
37C22⋊C4×D127—21no—
38#16861142C2211yescharacteristic, complemented
39#16861142C2211yescharacteristic, complemented
40Dic127⋊C8C2211yescharacteristic, complemented
41#16861133C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C2×SD16×C127C2211yescharacteristic, complemented
53C22×C8×C127C2211yescharacteristic
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#106×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523249—22no—
61#14523249—41no—
62C2×D508—22no—
63C2×D508—21no—
64C2×D508—21no—
65C2×D508—21no—
66C2×D508—41no—
67C2×D508C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127D411yescharacteristic
71C22×Dic127—21no—
72C22×Dic127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127C2311yescharacteristic
76#14523233—22no—
77#14523233—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C4⋊Dic127—21no—
82C4⋊Dic127—21no—
83C4⋊Dic127C2311yescharacteristic
84C4⋊Dic127C2311yescharacteristic
85Dic127⋊C4—21no—
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127D411yescharacteristic
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—
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127—21no—
97C4×Dic127C2311yescharacteristic
98C2×Q8×C127C2311yescharacteristic
99C2×D4×C127—21no—
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102SD16×C127—41no—
103C2×C8×C127—41no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127—21no—
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C22⋊C4×C127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117D508—21no—
118D508—21no—
119D508—21no—
120D508—42no—
121D508—42no—
122D508—81no—
123D508#4112yes—
124C22⋊D127—22no—
125C22⋊D127—81no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—81no—
129C2×Dic127—81no—
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—42no—
136C4×D127—81no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C23×C127—21no—
140C23×C127C2×D411yescharacteristic
141Q8×C127—21no—
142Q8×C127—21no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127—42no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127#13512yes—
161C22×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—22no—
164C2×D127—21no—
165C2×D127—21no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—84no—
172C4×C127—21no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127—41no—
176C4×C127C2×#4111yescharacteristic
177C4×C127H(5,𝔽2)11yescharacteristic
178Dic127—21no—
179Dic127—21no—
180Dic127—21no—
181Dic127—41no—
182Dic127—41no—
183C2×C127—22no—
184C2×C127—81no—
185C2×C127#54611yescharacteristic
186C2×C127C2×#13511yescharacteristic
187C2×C127#57611yescharacteristic
188D127—41no—
189D127—42no—
190D127—81no—
191#3297—1271nomaximal, Sylow
192C127#329711yescharacteristic, complemented, Sylow
193#553—1271no—
194#546—1271no—
195#488—1271no—
196#484—1271no—
197#482—1271no—
198#478—1271no—
199#466—1271no—
200#465D12711yescharacteristic, complemented
201#463—1271no—
202#449—1271no—
203#447—1271no—
204C4×D8—1271no—
205C8×D4—1271no—
206#416—1271no—
207C2×#102—1271no—
208C2×#41—1271no—
209C22×D4—1271no—
210C2×SD16C2×D12711yescharacteristic, complemented
211C2×D8—1271no—
212C2×D8—1271no—
213C22×C8—1271no—
214C22×C8C2×D12711yescharacteristic
215#126—2541no—
216#125—1271no—
217#124—1271no—
218#123—2541no—
219#123—2541no—
220C22⋊Q8C2×D12711yescharacteristic
221#121—1271no—
222#121—1271no—
223#121—2541no—
224#121—2541no—
225#121C2×D12711yescharacteristic
226#120—2541no—
227C4×D4—1271no—
228C4×D4—1271no—
229C4×D4—1271no—
230C4×D4—2541no—
231#117—1271no—
232C2×C4⋊C4—1271no—
233C2×C22⋊C4—2541no—
234#107—1271no—
235#107—1271no—
236#106C2×D12711yescharacteristic, complemented
237C4⋊C8—1271no—
238#103—1271no—
239#103—1271no—
240#103C2×D12711yescharacteristic, complemented
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102C2×D12711yescharacteristic, complemented
248C22⋊C8—1271no—
249C22⋊C8—1271no—
250C4×C8—1271no—
251C24—2541no—
252#41—5081no—
253#41—2542no—
254C2×Q8C22×D12711yescharacteristic
255C2×D4—21no—
256C2×D4—1271no—
257C2×D4—1271no—
258C2×D4—1271no—
259C2×D4—5081no—
260C2×D4—2542no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4C22×D12711yescharacteristic
266C22×C4—1271no—
267C22×C4—2541no—
268C22×C4—2541no—
269C22×C4—2541no—
270C22×C4—2541no—
271C22×C4C22×D12711yescharacteristic
272SD16—41no—
273D8—5081no—
274D8—2542no—
275C2×C8—41no—
276C2×C8—1271no—
277C2×C8—1271no—
278C2×C8—5081no—
279C2×C8C22×D12711yescharacteristic
280C2×C8C22×D12711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—1271no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4C22×D12711yescharacteristic
289C4⋊C4C22×D12711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1271no—
293C22⋊C4—1271no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C22⋊C4—2541no—
300C22⋊C4—2541no—
301C42—1271no—
302C42—2541no—
303C23—21no—
304C23—1271no—
305C23—10161no—
306C23—5082no—
307C23—5082no—
308C23—2541no—
309C23—2541no—
310C23D4×D12711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—1272no—
317D4—10161no—
318D4—10161no—
319D4—5082no—
320D4—5082no—
321D4—2542no—
322D4—2541no—
323D4—2541no—
324D4—2541no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—1271no—
331C2×C4—1271no—
332C2×C4—1271no—
333C2×C4—10161no—
334C2×C4—10161no—
335C2×C4—10161no—
336C2×C4—5082no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4D4×D12711yescharacteristic
342C2×C4C23×D12711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2541no—
346C8—2541no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—10164no—
353C22—5081no—
354C22—5081no—
355C22—5081no—
356C22—5081no—
357C22—5081no—
358C22—2541no—
359C22—2541no—
360C22—2542no—
361C22—2541no—
362C22#1686123012yes—
363C22C2×D4×D12711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—5081no—
369C4—5081no—
370C4—2541no—
371C4—2541no—
372C4—2541no—
373C4#41×D12711yescharacteristic
374C4#1686125411yescharacteristic
375C2—22no—
376C2—81no—
377C2—10161no—
378C2—5082no—
379C2—5081no—
380C2#2255051511yescharacteristic
381C2C2×#1686123011yescharacteristic
382C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
hypercenter#465D127
hypocenterC127#3297
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22550578C2semidirect
2. #22550517C2semidirect
3. #22550232C2semidirect
4. #22550209C2semidirect
5. #22550202C2semidirect
6. #22550175C2semidirect
7. #22550111C2semidirect
8. #22550093C2semidirect
9. #22550020C2semidirect
10. #22550009C2semidirect
11. #22549956C2semidirect
12. C8×C22⋊D127C2semidirect
13. #22549871C2semidirect
14. C2×#16861131C2semidirect
15. #465×C127C2semidirect
20. C2×C8×D127C22semidirect
22. #16861200C22semidirect
29. #16861180C22semidirect
32. D4×Dic127C22semidirect
35. #16861167C22semidirect
38. #16861142C22semidirect
39. #16861142C22semidirect
40. Dic127⋊C8C22semidirect
41. #16861133C22semidirect
42. #16861133C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. C8×Dic127C22semidirect
52. C2×SD16×C127C22semidirect
56. #106×C127C22semidirect
57. #103×C127C22semidirect
58. #102×C127C22semidirect
192. C127#3297semidirect
200. #465D127semidirect
210. C2×SD16C2×D127semidirect
236. #106C2×D127semidirect
240. #103C2×D127semidirect
247. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505782
#225505172
#225502322
#225502092
#225502022
#225501752
#225501112
#225500932
#225500202
#225500092
#225499562
C8×C22⋊D1272
#225498712
C2×#168611312
#465×C1272
#3297127