Finite Group Databasebeta

← #30650688

Subgroups — #30650688

Order 16256.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550529C211yesmaximal, characteristic, complemented
2#22550521C211yesmaximal, characteristic, complemented
3#22550238C211yesmaximal, characteristic, complemented
4#22550208C211yesmaximal, characteristic, complemented
5#22550196C211yesmaximal, characteristic, complemented
6#22550130C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550022C211yesmaximal, characteristic, complemented
10#22550011C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549946C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#476×C127C211yesmaximal, characteristic, complemented
16C2×#14523249—21no—
17C2×#14523247C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19C2×#14523237C2211yescharacteristic, complemented
20C2×D1016C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22C2×#14523231C2211yescharacteristic, complemented
23C2×C8×D127C2211yescharacteristic
24#16861197C2211yescharacteristic
25#16861194—21no—
26#16861194—21no—
27#16861193—21no—
28#16861189C2211yescharacteristic
29#16861185—21no—
30#16861184C2211yescharacteristic
31#16861184C2211yescharacteristic
32#16861184C2211yescharacteristic, complemented
33#16861183—21no—
34#16861179—21no—
35C4×D508C2211yescharacteristic
36C4×D508C2211yescharacteristic, complemented
37C4⋊C4×D127C2211yescharacteristic
38C4⋊C4×D127C2211yescharacteristic, complemented
39C2×(C2×D127)⋊C4—21no—
40#16861140C2211yescharacteristic, complemented
41#16861134C2211yescharacteristic, complemented
42#16861132C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic, complemented
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51#16861120C2211yescharacteristic, complemented
52C4⋊Q8×C127C2211yescharacteristic, complemented
53C4×D4×C127C2211yescharacteristic
54#106×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56C4⋊C8×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C23×D127—21no—
60#14523249—44no—
61#14523247—22no—
62#14523247—41no—
63C2×Dic254C2311yescharacteristic
64C2×D508—21no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508D411yescharacteristic
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72D4×D127—22no—
73D4×D127—41no—
74C22×Dic127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127C2311yescharacteristic
79C2×C4×D127C2311yescharacteristic
80#14523237—41no—
81D1016—41no—
82#14523233—22no—
83#14523231—41no—
84C2×C127⋊C8C2311yescharacteristic
85C2×C127⋊C8C2311yescharacteristic
86C8×D127—41no—
87C4⋊Dic127—21no—
88C4⋊Dic127D411yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4D411yescharacteristic
100(C2×D127)⋊C4D411yescharacteristic
101C2×Q8×C127—21no—
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127—21no—
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C42×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—42no—
115C22×D127—42no—
116C22×D127—81no—
117C22×D127C2×D411yescharacteristic
118C22×D127C2×D411yescharacteristic
119Dic254—21no—
120Dic254—21no—
121D508—22no—
122D508—21no—
123D508—21no—
124D508—21no—
125D508—21no—
126D508—42no—
127D508—42no—
128D508—81no—
129C22⋊D127—42no—
130C22⋊D127—42no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—42no—
134C2×Dic127C2×D411yescharacteristic
135C2×Dic127C2×D411yescharacteristic
136C4×D127—21no—
137C4×D127—21no—
138C4×D127—21no—
139C4×D127—21no—
140C4×D127—42no—
141C4×D127—42no—
142C4×D127—42no—
143C127⋊C8—21no—
144C127⋊C8—21no—
145C23×C127—21no—
146Q8×C127—42no—
147D4×C127—21no—
148D4×C127#4112yescomplemented
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—81no—
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—22no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—21no—
165C2×D127—21no—
166C2×D127—21no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—84no—
174C2×D127AGL(1,ℤ/8)12yes—
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127—41no—
180C4×C127C2×#4111yescharacteristic
181C4×C127H(5,𝔽2)11yescharacteristic
182Dic127—21no—
183Dic127—21no—
184Dic127—41no—
185Dic127—41no—
186C2×C127—42no—
187C2×C127#54611yescharacteristic
188C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C127#57711yescharacteristic
190D127—22no—
191D127—41no—
192D127—81no—
193D127—81no—
194#3306—1271nomaximal, Sylow
195C127#330611yescharacteristic, complemented, Sylow
196#547—1271no—
197#546—1271no—
198#489—1271no—
199#483—1271no—
200#481—1271no—
201#476D12711yescharacteristic, complemented
202#469—1271no—
203#465—1271no—
204#463—1271no—
205#449—1271no—
206#447—1271no—
207#442—1271no—
208#435—1271no—
209#417—1271no—
210C2×#102—1271no—
211C2×#41—1271no—
212C2×#41—2541no—
213C22×D4—1271no—
214C2×SD16—1271no—
215C2×D8—1271no—
216C2×D8—1271no—
217C2×M16—1271no—
218C22×C8—1271no—
219C4⋊Q8C2×D12711yescharacteristic, complemented
220#124—1271no—
221#123—2541no—
222#123—2541no—
223#123—2541no—
224C22⋊Q8—1271no—
225C22⋊Q8—2541no—
226#121—1271no—
227#121—1271no—
228#121—1271no—
229#121—2541no—
230#120—2541no—
231C4×D4—1271no—
232C4×D4—1271no—
233C4×D4C2×D12711yescharacteristic
234C2×C4⋊C4—1271no—
235C2×C4⋊C4—1271no—
236C2×C22⋊C4—2541no—
237#106—1271no—
238#106C2×D12711yescharacteristic, complemented
239#106C2×D12711yescharacteristic, complemented
240C4⋊C8C2×D12711yescharacteristic, complemented
241#103—1271no—
242#103—1271no—
243#103—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102C2×D12711yescharacteristic
250#102C2×D12711yescharacteristic
251C22⋊C8—1271no—
252C22⋊C8—1271no—
253#97—1271no—
254C24—2541no—
255#41—5084no—
256#41—5081no—
257#41—2542no—
258C2×Q8—21no—
259C2×Q8—1271no—
260C2×D4—1271no—
261C2×D4—1271no—
262C2×D4—1271no—
263C2×D4—5081no—
264C2×D4—2541no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4—2542no—
270C2×D4C22×D12711yescharacteristic
271C22×C4—21no—
272C22×C4—1271no—
273C22×C4—1271no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4—2541no—
277C22×C4—2541no—
278SD16—5081no—
279D8—5081no—
280D8—2542no—
281M16—5081no—
282C2×C8—1271no—
283C2×C8—1271no—
284C2×C8—5081no—
285C2×C8C22×D12711yescharacteristic
286C2×C8C22×D12711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—1271no—
289C4⋊C4—1271no—
290C4⋊C4—2541no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4C22×D12711yescharacteristic
294C4⋊C4C22×D12711yescharacteristic
295C4⋊C4C22×D12711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—1271no—
298C22⋊C4—1271no—
299C22⋊C4—2541no—
300C22⋊C4—2541no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C42C22×D12711yescharacteristic
307C23—21no—
308C23—1271no—
309C23—1271no—
310C23—10161no—
311C23—5082no—
312C23—5082no—
313C23—2541no—
314C23—2541no—
315Q8—42no—
316Q8—2541no—
317Q8—2541no—
318D4—21no—
319D4—10161no—
320D4—5082no—
321D4—5082no—
322D4—5082no—
323D4—5082no—
324D4—2542no—
325D4—2541no—
326D4—2541no—
327D4—2541no—
328D4—2541no—
329D4#1452324912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—81no—
334C2×C4—1271no—
335C2×C4—1271no—
336C2×C4—5082no—
337C2×C4—5082no—
338C2×C4—5082no—
339C2×C4—5082no—
340C2×C4—2541no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4D4×D12711yescharacteristic
347C2×C4D4×D12711yescharacteristic
348C2×C4C23×D12711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2541no—
352C8—2541no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—1272no—
357C22—10164no—
358C22—5081no—
359C22—5081no—
360C22—5081no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—2541no—
365C22—2541no—
366C22—2541no—
367C22—2541no—
368C22—2541no—
369C22C2×D4×D12711yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—5081no—
376C4—5081no—
377C4—2541no—
378C4—2541no—
379C4C2×#1452324911yescharacteristic
380C4#1686125511yescharacteristic
381C2—42no—
382C2—10161no—
383C2—10161no—
384C2—5081no—
385C2—2542no—
386C2#2255051711yescharacteristic
387C2AGL(1,ℤ/8)×D12711yescharacteristic
388C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#476×C127C2
socleC22×C127C22×D4
hypercenter#476D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #22550529C2semidirect
2. #22550521C2semidirect
3. #22550238C2semidirect
4. #22550208C2semidirect
5. #22550196C2semidirect
6. #22550130C2semidirect
7. #22550106C2semidirect
8. #22550092C2semidirect
9. #22550022C2semidirect
10. #22550011C2semidirect
11. #22549986C2semidirect
12. #22549946C2semidirect
13. #22549874C2semidirect
14. #102×D127C2semidirect
15. #476×C127C2semidirect
19. C2×#14523237C22semidirect
22. C2×#14523231C22semidirect
32. #16861184C22semidirect
36. C4×D508C22semidirect
38. C4⋊C4×D127C22semidirect
40. #16861140C22semidirect
41. #16861134C22semidirect
42. #16861132C22semidirect
43. #16861132C22semidirect
48. #16861129C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. #16861120C22semidirect
52. C4⋊Q8×C127C22semidirect
54. #106×C127C22semidirect
55. #106×C127C22semidirect
56. C4⋊C8×C127C22semidirect
148. D4×C127#41semidirect
195. C127#3306semidirect
201. #476D127semidirect
219. C4⋊Q8C2×D127semidirect
238. #106C2×D127semidirect
239. #106C2×D127semidirect
240. C4⋊C8C2×D127semidirect
329. D4#14523249semidirect

Maximal subgroups

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

subgroupindex
#225505292
#225505212
#225502382
#225502082
#225501962
#225501302
#225501062
#225500922
#225500222
#225500112
#225499862
#225499462
#225498742
#102×D1272
#476×C1272
#3306127