Finite Group Databasebeta

← #30650745

Subgroups — #30650745

Order 16256.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D508C211yesmaximal, characteristic
2#22550242C211yesmaximal, characteristic, complemented
3#22550200C211yesmaximal, characteristic, complemented
4#22550189C211yesmaximal, characteristic, complemented
5#22550173C211yesmaximal, characteristic, complemented
6#22550124C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550076C211yesmaximal, characteristic, complemented
9#22550026C211yesmaximal, characteristic, complemented
10#22550026C211yesmaximal, characteristic, complemented
11#22549974C211yesmaximal, characteristic, complemented
12#22549948C211yesmaximal, characteristic, complemented
13#22549880C211yesmaximal, characteristic, complemented
14C2×#16861129C211yesmaximal, characteristic, complemented
15#552×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C22×D508C2211yescharacteristic
18C2×D4×D127—21no—
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic, complemented
21C2×#14523235C2211yescharacteristic, complemented
22C2×#14523232C2211yescharacteristic, complemented
23C22×C127⋊C8C2211yescharacteristic, complemented
24#16861206C2211yescharacteristic
25#16861184—21no—
26#16861183C2211yescharacteristic
27#16861183C2211yescharacteristic
28#16861179—21no—
29#16861179—21no—
30C4×D508C2211yescharacteristic
31#16861142C2211yescharacteristic, complemented
32#16861142C2211yescharacteristic, complemented
33#16861140C2211yescharacteristic, complemented
34C4⋊(C127⋊C8)C2211yescharacteristic, complemented
35#16861133C2211yescharacteristic, complemented
36#16861129C2211yescharacteristic
37#16861129C2211yescharacteristic
38#16861129C2211yescharacteristic
39#16861129C2211yescharacteristic, complemented
40#16861129C2211yescharacteristic, complemented
41#16861129C2211yescharacteristic, complemented
42#16861129C2211yescharacteristic, complemented
43#16861129C2211yescharacteristic, complemented
44#16861129C2211yescharacteristic, complemented
45C22⋊(C127⋊C8)C2211yescharacteristic, complemented
46C22⋊(C127⋊C8)C2211yescharacteristic, complemented
47#16861120C2211yescharacteristic, complemented
48#126×C127—21no—
49#125×C127—21no—
50#125×C127C2211yescharacteristic, complemented
51#123×C127—21no—
52C22⋊Q8×C127C2211yescharacteristic, complemented
53C22⋊Q8×C127C2211yescharacteristic, complemented
54C4×Q8×C127C2211yescharacteristic, complemented
55C4×D4×C127C2211yescharacteristic
56#117×C127—21no—
57#117×C127C2211yescharacteristic, complemented
58C2×C4⋊C4×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C23×D127—21no—
61C2×D508—22no—
62C2×D508—22no—
63C2×D508—21no—
64C2×D508—21no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508—41no—
68C2×D508—41no—
69C2×D508C2311yescharacteristic
70C2×D508C2311yescharacteristic
71C2×D508C2311yescharacteristic
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74D4×D127—44no—
75C2×C4×D127—21no—
76#14523235—22no—
77#14523235—41no—
78#14523235—41no—
79#14523232—41no—
80C2×C127⋊C8—41no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C2×C127⋊C8C2311yescharacteristic
84C2×C127⋊C8C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89C2×Q8×C127C2311yescharacteristic
90C2×D4×C127D411yescharacteristic
91C22×C4×C127—21no—
92C22×C4×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127—21no—
100C4⋊C4×C127D411yescharacteristic
101C4⋊C4×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C22⋊C4×C127D411yescharacteristic
108C22⋊C4×C127D411yescharacteristic
109C42×C127—21no—
110C42×C127—21no—
111C42×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—21no—
115C22×D127—42no—
116C22×D127—42no—
117C22×D127—42no—
118C22×D127—42no—
119C22×D127—81no—
120C22×D127—81no—
121D508—21no—
122D508—21no—
123D508—21no—
124D508—21no—
125D508—21no—
126D508—42no—
127D508—42no—
128D508—42no—
129D508—81no—
130D508#4112yes—
131C22⋊D127—42no—
132C22⋊D127—42no—
133C2×Dic127—21no—
134C4×D127—42no—
135C127⋊C8—21no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C23×C127C2×D411yescharacteristic
140C23×C127C2×D411yescharacteristic
141Q8×C127—21no—
142Q8×C127—21no—
143D4×C127—22no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×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—42no—
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2×D411yescharacteristic
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2411yescharacteristic
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—21no—
163C22×C127AGL(1,ℤ/8)12yes—
164C22×C127C22×D411yescharacteristic
165C2×D127—22no—
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—
175C2×D127—84no—
176C4×C127—21no—
177C4×C127—21no—
178C4×C127—21no—
179C4×C127—21no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127—41no—
184C4×C127C2×#4111yescharacteristic
185C4×C127H(5,𝔽2)11yescharacteristic
186Dic127—41no—
187C2×C127—22no—
188C2×C127—41no—
189C2×C127#54611yescharacteristic
190C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C127#57711yescharacteristic
192D127—42no—
193D127—81no—
194D127—81no—
195#3310—1271nomaximal, Sylow
196C127#331011yescharacteristic, complemented, Sylow
197#552D12711yescharacteristic, complemented
198D42—1271no—
199#490—1271no—
200#482—1271no—
201#480—1271no—
202#478—1271no—
203#468—1271no—
204#465—1271no—
205#460—1271no—
206#450—1271no—
207#450—1271no—
208#440—1271no—
209#435—1271no—
210#418—1271no—
211C2×#102—1271no—
212C22×D4—1271no—
213C22×D4—1271no—
214C22×D4—2541no—
215C2×SD16—1271no—
216C2×SD16—1271no—
217C2×SD16—1271no—
218C2×M16—1271no—
219C22×C8—1271no—
220#127—1271no—
221#126—21no—
222#125—21no—
223#125C2×D12711yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D12711yescharacteristic, complemented
226C22⋊Q8C2×D12711yescharacteristic, complemented
227#121—1271no—
228#121—1271no—
229#121—2541no—
230#120—2541no—
231#120—2541no—
232C4×Q8C2×D12711yescharacteristic, complemented
233C4×D4—1271no—
234C4×D4C2×D12711yescharacteristic
235#117—21no—
236#117C2×D12711yescharacteristic, complemented
237C2×C4⋊C4C2×D12711yescharacteristic, complemented
238#107—1271no—
239#107—1271no—
240#106—1271no—
241C4⋊C8—1271no—
242#103—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102—1271no—
251#102—1271no—
252C22⋊C8—1271no—
253C22⋊C8—1271no—
254#97—1271no—
255C24—2541no—
256C24—2541no—
257C2×Q8C22×D12711yescharacteristic
258C2×D4—1271no—
259C2×D4—1271no—
260C2×D4—1271no—
261C2×D4—5081no—
262C2×D4—5081no—
263C2×D4—5084no—
264C2×D4—2542no—
265C2×D4—2542no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4—2541no—
271C2×D4—2541no—
272C2×D4C22⋊D12711yescharacteristic
273C22×C4—21no—
274C22×C4—2541no—
275C22×C4C22×D12711yescharacteristic
276C22×C4C22×D12711yescharacteristic
277SD16—5081no—
278SD16—5081no—
279SD16—2542no—
280M16—5081no—
281C2×C8—1271no—
282C2×C8—1271no—
283C2×C8—1271no—
284C2×C8—1271no—
285C2×C8—5081no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1271no—
293C4⋊C4C22⋊D12711yescharacteristic
294C4⋊C4C22×D12711yescharacteristic
295C4⋊C4C22×D12711yescharacteristic
296C4⋊C4C22×D12711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—2541no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4C22⋊D12711yescharacteristic
304C22⋊C4C22⋊D12711yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D12711yescharacteristic
308C23—10161no—
309C23—10161no—
310C23—5082no—
311C23—5082no—
312C23—5082no—
313C23—5082no—
314C23—2541no—
315C23—2541no—
316C23—2541no—
317C23C2×C22⋊D12711yescharacteristic
318C23C2×C22⋊D12711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—1272no—
323D4—10161no—
324D4—5082no—
325D4—5082no—
326D4—5082no—
327D4—5082no—
328D4—5082no—
329D4—2541no—
330D4—2541no—
331D4—2541no—
332D4—2541no—
333D4—2541no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—5082no—
344C2×C4—2541no—
345C2×C4C2×C22⋊D12711yescharacteristic
346C2×C4C2×C22⋊D12711yescharacteristic
347C2×C4C2×C22⋊D12711yescharacteristic
348C2×C4C2×C22⋊D12711yescharacteristic
349C2×C4C23×D12711yescharacteristic
350C8—2541no—
351C8—2541no—
352C8—2541no—
353C8—2541no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—10164no—
360C22—10164no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—5081no—
365C22—5081no—
366C22—5081no—
367C22—5081no—
368C22—5081no—
369C22—2542no—
370C22#1686123412yes—
371C22C22×C22⋊D12711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—5081no—
381C4#41×D12711yescharacteristic
382C4#1686125511yescharacteristic
383C2—22no—
384C2—41no—
385C2—10161no—
386C2—10161no—
387C2—5082no—
388C2#2255051411yescharacteristic
389C2C2×#1686123411yescharacteristic
390C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#552×C127C2
socleC22×C127C22×D4
prime radicalD4×D508C2
hypercenter#552D127
hypocenterC127#3310
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #22550242C2semidirect
3. #22550200C2semidirect
4. #22550189C2semidirect
5. #22550173C2semidirect
6. #22550124C2semidirect
7. #22550107C2semidirect
8. #22550076C2semidirect
9. #22550026C2semidirect
10. #22550026C2semidirect
11. #22549974C2semidirect
12. #22549948C2semidirect
13. #22549880C2semidirect
14. C2×#16861129C2semidirect
15. #552×C127C2semidirect
20. C2×#14523235C22semidirect
21. C2×#14523235C22semidirect
22. C2×#14523232C22semidirect
23. C22×C127⋊C8C22semidirect
31. #16861142C22semidirect
32. #16861142C22semidirect
33. #16861140C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. #16861133C22semidirect
39. #16861129C22semidirect
40. #16861129C22semidirect
41. #16861129C22semidirect
42. #16861129C22semidirect
43. #16861129C22semidirect
44. #16861129C22semidirect
45. C22⋊(C127⋊C8)C22semidirect
46. C22⋊(C127⋊C8)C22semidirect
47. #16861120C22semidirect
50. #125×C127C22semidirect
52. C22⋊Q8×C127C22semidirect
53. C22⋊Q8×C127C22semidirect
54. C4×Q8×C127C22semidirect
57. #117×C127C22semidirect
58. C2×C4⋊C4×C127C22semidirect
196. C127#3310semidirect
197. #552D127semidirect
223. #125C2×D127semidirect
225. C22⋊Q8C2×D127semidirect
226. C22⋊Q8C2×D127semidirect
232. C4×Q8C2×D127semidirect
236. #117C2×D127semidirect
237. C2×C4⋊C4C2×D127semidirect

Maximal subgroups

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

subgroupindex
D4×D5082
#225502422
#225502002
#225501892
#225501732
#225501242
#225501072
#225500762
#225500262
#225499742
#225499482
#225498802
C2×#168611292
#552×C1272
#3310127