Finite Group Databasebeta

← #30650689

Subgroups — #30650689

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#22550527C211yesmaximal, characteristic, complemented
2#22550517C211yesmaximal, characteristic, complemented
3#22550237C211yesmaximal, characteristic, complemented
4#22550196C211yesmaximal, characteristic, complemented
5#22550165C211yesmaximal, characteristic, complemented
6#22550130C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550094C211yesmaximal, characteristic, complemented
9#22550024C211yesmaximal, characteristic, complemented
10#22550012C211yesmaximal, characteristic, complemented
11#22549987C211yesmaximal, characteristic, complemented
12#22549944C211yesmaximal, characteristic, complemented
13#22549876C211yesmaximal, characteristic, complemented
14C2×#16861129C211yesmaximal, characteristic, complemented
15#483×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248—21no—
18C22×D508C2211yescharacteristic
19C2×#14523235C2211yescharacteristic, complemented
20C2×D1016C2211yescharacteristic
21C2×D1016C2211yescharacteristic
22C2×#14523231C2211yescharacteristic, complemented
23C22×C127⋊C8C2211yescharacteristic
24#16861208C2211yescharacteristic, complemented
25#16861198C2211yescharacteristic
26#16861194—21no—
27#16861194—21no—
28#16861192—21no—
29#16861187—21no—
30#16861184—21no—
31#16861184C2211yescharacteristic
32#16861184C2211yescharacteristic, complemented
33#16861183C2211yescharacteristic
34#16861179—21no—
35#16861171C2211yescharacteristic
36#16861171C2211yescharacteristic, complemented
37C4×C22⋊D127C2211yescharacteristic
38C4⋊C4×D127C2211yescharacteristic, complemented
39C22⋊C4×D127—21no—
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51#16861119C2211yescharacteristic, complemented
52C22⋊Q8×C127C2211yescharacteristic
53C2×C4⋊C4×C127C2211yescharacteristic
54#106×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic
59C23×D127—21no—
60#14523249—22no—
61#14523249—41no—
62#14523248—44no—
63C2×Dic254—21no—
64C2×D508—22no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508—21no—
68C2×D508—21no—
69C2×D508—41no—
70C2×D508C2311yescharacteristic
71C2×D508C2311yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127D411yescharacteristic
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79C2×C4×D127C2311yescharacteristic
80#14523235—41no—
81D1016—22no—
82D1016—41no—
83#14523231—41no—
84C2×C127⋊C8—41no—
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C4⋊Dic127C2311yescharacteristic
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4D411yescharacteristic
91C22⋊Dic127D411yescharacteristic
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
100C4×Dic127C2311yescharacteristic
101C2×Q8×C127C2311yescharacteristic
102C22×C4×C127—21no—
103C22×C4×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127—21no—
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C4⋊C4×C127C2311yescharacteristic
111C22⋊C4×C127—21no—
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—21no—
115C22×D127—42no—
116C22×D127—42no—
117C22×D127—81no—
118C22×D127C2×D411yescharacteristic
119Dic254—42no—
120D508—21no—
121D508—21no—
122D508—21no—
123D508—21no—
124D508—21no—
125D508—42no—
126D508—42no—
127D508—42no—
128D508—81no—
129D508#4112yescomplemented
130C22⋊D127—22no—
131C22⋊D127—42no—
132C2×Dic127—21no—
133C2×Dic127C2×D411yescharacteristic
134C2×Dic127C2×D411yescharacteristic
135C2×Dic127C2×D411yescharacteristic
136C4×D127—21no—
137C4×D127—21no—
138C4×D127—42no—
139C4×D127—42no—
140C4×D127—42no—
141C4×D127—81no—
142C127⋊C8—21no—
143C127⋊C8—21no—
144C23×C127C2×D411yescharacteristic
145Q8×C127—21no—
146Q8×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—42no—
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127AGL(1,ℤ/8)12yes—
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—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—84no—
175C4×C127—21no—
176C4×C127—41no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127—41no—
180C4×C127C2×#4111yescharacteristic
181C4×C127H(5,𝔽2)11yescharacteristic
182Dic127—21no—
183Dic127—21no—
184Dic127—21no—
185Dic127—41no—
186C2×C127—22no—
187C2×C127#54611yescharacteristic
188C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C127#57711yescharacteristic
190D127—41no—
191D127—42no—
192D127—81no—
193D127—81no—
194#3306—1271nomaximal, Sylow
195C127#330611yescharacteristic, complemented, Sylow
196#547—1271no—
197#546—1271no—
198#489—1271no—
199#483D12711yescharacteristic, complemented
200#481—1271no—
201#476—1271no—
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⋊Q8—1271no—
220#124—1271no—
221#123—2541no—
222#123—2541no—
223#123—2541no—
224C22⋊Q8—2541no—
225C22⋊Q8C2×D12711yescharacteristic
226#121—1271no—
227#121—1271no—
228#121—1271no—
229#121—2541no—
230#120—2541no—
231C4×D4—1271no—
232C4×D4—1271no—
233C4×D4—1271no—
234C2×C4⋊C4—1271no—
235C2×C4⋊C4C2×D12711yescharacteristic
236C2×C22⋊C4—2541no—
237#106—1271no—
238#106C2×D12711yescharacteristic, complemented
239#106C2×D12711yescharacteristic, complemented
240C4⋊C8—1271no—
241#103—1271no—
242#103C2×D12711yescharacteristic, complemented
243#103C2×D12711yescharacteristic, complemented
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102—1271no—
251C22⋊C8—1271no—
252C22⋊C8C2×D12711yescharacteristic
253#97—1271no—
254C24—2541no—
255#41—5084no—
256#41—5081no—
257#41—2542no—
258C2×Q8—2541no—
259C2×Q8C22×D12711yescharacteristic
260C2×D4—1271no—
261C2×D4—1271no—
262C2×D4—1271no—
263C2×D4—1271no—
264C2×D4—5081no—
265C2×D4—2542no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4—2541no—
271C22×C4—21no—
272C22×C4—1271no—
273C22×C4—2541no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4—2541no—
277C22×C4C22×D12711yescharacteristic
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—21no—
289C4⋊C4—1271no—
290C4⋊C4—1271no—
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—
306C42—1271no—
307C23—1271no—
308C23—10161no—
309C23—5082no—
310C23—5082no—
311C23—2541no—
312C23—2541no—
313C23—2541no—
314C23D4×D12711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—5082no—
318D4—1272no—
319D4—10161no—
320D4—5082no—
321D4—5082no—
322D4—5082no—
323D4—5082no—
324D4—2542no—
325D4—2541no—
326D4—2541no—
327D4—2541no—
328D4—2541no—
329D4—2541no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—1271no—
338C2×C4—1271no—
339C2×C4—1271no—
340C2×C4—10161no—
341C2×C4—5082no—
342C2×C4—5082no—
343C2×C4—5082no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4D4×D12711yescharacteristic
348C2×C4C23×D12711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2541no—
352C8—2541no—
353C22—21no—
354C22—21no—
355C22—10164no—
356C22—5081no—
357C22—5081no—
358C22—5081no—
359C22—5081no—
360C22—5081no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—2541no—
365C22—2541no—
366C22—2542no—
367C22—2541no—
368C22#1686123612yes—
369C22C2×D4×D12711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—5081no—
376C4—2541no—
377C4—2541no—
378C4—2541no—
379C4#41×D12711yescharacteristic
380C4#1686125511yescharacteristic
381C2—22no—
382C2—10161no—
383C2—10161no—
384C2—5082no—
385C2—5081no—
386C2#2255051811yescharacteristic
387C2C2×#1686123611yescharacteristic
388C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#483×C127C2
socleC22×C127C22×D4
hypercenter#483D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #22550527C2semidirect
2. #22550517C2semidirect
3. #22550237C2semidirect
4. #22550196C2semidirect
5. #22550165C2semidirect
6. #22550130C2semidirect
7. #22550107C2semidirect
8. #22550094C2semidirect
9. #22550024C2semidirect
10. #22550012C2semidirect
11. #22549987C2semidirect
12. #22549944C2semidirect
13. #22549876C2semidirect
14. C2×#16861129C2semidirect
15. #483×C127C2semidirect
19. C2×#14523235C22semidirect
22. C2×#14523231C22semidirect
24. #16861208C22semidirect
32. #16861184C22semidirect
36. #16861171C22semidirect
38. C4⋊C4×D127C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861133C22semidirect
49. #16861129C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. #16861119C22semidirect
54. #106×C127C22semidirect
55. #106×C127C22semidirect
56. #103×C127C22semidirect
57. #103×C127C22semidirect
129. D508#41semidirect
195. C127#3306semidirect
199. #483D127semidirect
238. #106C2×D127semidirect
239. #106C2×D127semidirect
242. #103C2×D127semidirect
243. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505272
#225505172
#225502372
#225501962
#225501652
#225501302
#225501072
#225500942
#225500242
#225500122
#225499872
#225499442
#225498762
C2×#168611292
#483×C1272
#3306127