Finite Group Databasebeta

← #30650552

Subgroups — #30650552

Order 16256.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550581C211yesmaximal, characteristic, complemented
2#22550510C211yesmaximal, characteristic
3#22550221C211yesmaximal, characteristic, complemented
4#22550196C211yesmaximal, characteristic, complemented
5#22550165C211yesmaximal, characteristic, complemented
6#22550106C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550075C211yesmaximal, characteristic, complemented
9#22550029C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22549963C211yesmaximal, characteristic, complemented
12(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
13C2×#16861129C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#481×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×#14523237C2211yescharacteristic
20C2×#14523237C2211yescharacteristic, complemented
21C2×#14523236C2211yescharacteristic, complemented
22C22×C127⋊C8C2211yescharacteristic, complemented
23C2×C8×D127C2211yescharacteristic, complemented
24#16861208C2211yescharacteristic, complemented
25#16861205C2211yescharacteristic
26#16861199—21no—
27#16861189C2211yescharacteristic, complemented
28#16861188—21no—
29#16861188C2211yescharacteristic, complemented
30#16861187—21no—
31#16861184C2211yescharacteristic
32#16861182—21no—
33#16861179—21no—
34#16861177—21no—
35#16861174C2211yescharacteristic, complemented
36#16861171C2211yescharacteristic
37C4×C22⋊D127—21no—
38C4×C22⋊D127C2211yescharacteristic
39C4⋊C4×D127C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861132C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic, complemented
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52#121×C127C2211yescharacteristic
53C2×C4⋊C4×C127C2211yescharacteristic, complemented
54#106×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C23×D127—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63C2×D508—22no—
64C2×D508—21no—
65C2×D508—21no—
66C2×D508—41no—
67C2×D508C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127D411yescharacteristic
73D4×D127—22no—
74D4×D127—41no—
75D4×D127—44no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127C2311yescharacteristic
79#14523237—22no—
80#14523237—41no—
81#14523236—41no—
82C2×C127⋊C8—41no—
83C2×C127⋊C8C2311yescharacteristic
84C2×C127⋊C8C2311yescharacteristic
85C8×D127—41no—
86C4⋊Dic127—21no—
87C4⋊Dic127C2311yescharacteristic
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92Dic127⋊C4D411yescharacteristic
93C22⋊Dic127—21no—
94C22⋊Dic127D411yescharacteristic
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4D411yescharacteristic
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic
101C2×D4×C127—21no—
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104C22×C4×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
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—42no—
115C22×D127—42no—
116C22×D127—42no—
117C22×D127—42no—
118C22×D127—81no—
119C22×D127—81no—
120C22×D127C2×D411yescharacteristic
121Dic254—21no—
122Dic254—21no—
123Dic254—81no—
124D508—21no—
125D508—21no—
126D508—21no—
127D508—42no—
128D508—42no—
129D508#4112yes—
130C22⋊D127—42no—
131C22⋊D127—42no—
132C22⋊D127—81no—
133C22⋊D127SD1614yescomplemented
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127C2×D411yescharacteristic
137C2×Dic127C2×D411yescharacteristic
138C2×Dic127C2×D411yescharacteristic
139C4×D127—21no—
140C4×D127—21no—
141C4×D127—42no—
142C4×D127—81no—
143C127⋊C8—21no—
144C127⋊C8—21no—
145C23×C127—21no—
146C23×C127C2×D411yescharacteristic
147D4×C127—21no—
148D4×C127—21no—
149D4×C127—42no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—81no—
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2411yescharacteristic
158C8×C127—21no—
159C8×C127—21no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127C2×SD1612yes—
166C22×C127C22×D411yescharacteristic
167C2×D127—21no—
168C2×D127—21no—
169C2×D127—22no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—84no—
176C2×D127—84no—
177C2×D127C2×SD1612yes—
178C4×C127—21no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127C2×#4111yescharacteristic
183C4×C127H(5,𝔽2)11yescharacteristic
184Dic127—21no—
185Dic127—21no—
186Dic127—41no—
187Dic127—41no—
188Dic127C2×SD1612yes—
189C2×C127—22no—
190C2×C127—81no—
191C2×C127#54611yescharacteristic
192C2×C127C22×SD1611yescharacteristic
193C2×C127#57611yescharacteristic
194D127—22no—
195D127—42no—
196D127—81no—
197#3295—1271nomaximal, Sylow
198C127#329511yescharacteristic, complemented, Sylow
199#554—1271no—
200D42—1271no—
201#486—1271no—
202#481—1271no—
203#481D12711yescharacteristic, complemented
204#476—1271no—
205#465—1271no—
206#465—1271no—
207#460—1271no—
208#450—1271no—
209#450—1271no—
210C4×SD16—1271no—
211C8×D4—1271no—
212C2×#102—1271no—
213C2×#102—1271no—
214C22×D4—1271no—
215C22×D4—1271no—
216C22×D4—2541no—
217C2×SD16—1271no—
218C2×SD16—1271no—
219C2×SD16—1271no—
220C22×C8—1271no—
221C22×C8—1271no—
222C4⋊Q8—1271no—
223#127—1271no—
224#125—2541no—
225C22⋊Q8—1271no—
226C22⋊Q8—1271no—
227C22⋊Q8—2541no—
228C22⋊Q8—2541no—
229#121—1271no—
230#121—2541no—
231#121C2×D12711yescharacteristic
232#120—2541no—
233#120—2541no—
234C4×Q8—1271no—
235C4×D4—1271no—
236C4×D4—1271no—
237C4×D4—2541no—
238C2×C4⋊C4—1271no—
239C2×C4⋊C4C2×D12711yescharacteristic, complemented
240#106—1271no—
241#106C2×D12711yescharacteristic, complemented
242#106C2×D12711yescharacteristic, complemented
243C4⋊C8—1271no—
244#103—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102—1271no—
251#102—1271no—
252#102C2×D12711yescharacteristic, complemented
253#102C2×D12711yescharacteristic, complemented
254C22⋊C8—1271no—
255C22⋊C8C2×D12711yescharacteristic, complemented
256C4×C8—1271no—
257C24—2541no—
258C24—2541no—
259C2×Q8—1271no—
260C2×Q8—2541no—
261C2×D4—21no—
262C2×D4—1271no—
263C2×D4—1271no—
264C2×D4—1271no—
265C2×D4—5081no—
266C2×D4—5084no—
267C2×D4—5081no—
268C2×D4—2542no—
269C2×D4—2541no—
270C2×D4—2541no—
271C2×D4—2541no—
272C2×D4—2541no—
273C2×D4—2542no—
274C2×D4—2541no—
275C2×D4C22×D12711yescharacteristic
276C22×C4—21no—
277C22×C4—1271no—
278C22×C4—2541no—
279C22×C4—2541no—
280C22×C4C22×D12711yescharacteristic
281SD16—5081no—
282SD16—5081no—
283SD16—2542no—
284C2×C8—1271no—
285C2×C8—1271no—
286C2×C8—5081no—
287C2×C8—5081no—
288C2×C8C22×D12711yescharacteristic
289C2×C8C22×D12711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—1271no—
292C4⋊C4—1271no—
293C4⋊C4—2541no—
294C4⋊C4—2541no—
295C4⋊C4—2541no—
296C4⋊C4—2541no—
297C4⋊C4—2541no—
298C4⋊C4C22×D12711yescharacteristic
299C4⋊C4C22×D12711yescharacteristic
300C4⋊C4C22×D12711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—1271no—
303C22⋊C4—1271no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C22⋊C4—2541no—
307C22⋊C4—2541no—
308C42—1271no—
309C42—2541no—
310C23—21no—
311C23—1271no—
312C23—10161no—
313C23—10161no—
314C23—5082no—
315C23—5082no—
316C23—5082no—
317C23—5082no—
318C23—2541no—
319C23—2541no—
320C23D4×D12711yescharacteristic
321Q8—10161no—
322Q8—2541no—
323Q8—2541no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—1274no—
328D4—1272no—
329D4—10161no—
330D4—5082no—
331D4—5082no—
332D4—5082no—
333D4—5082no—
334D4—2541no—
335D4—2541no—
336D4—2541no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—81no—
343C2×C4—1271no—
344C2×C4—1271no—
345C2×C4—1271no—
346C2×C4—10161no—
347C2×C4—5082no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4D4×D12711yescharacteristic
353C2×C4C23×D12711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2541no—
357C8—2541no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—1272no—
364C22—10164no—
365C22—10164no—
366C22—5081no—
367C22—5081no—
368C22—5081no—
369C22—5081no—
370C22—5081no—
371C22—2542no—
372C22—2541no—
373C22—2541no—
374C22SD16×D12712yes—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—1272no—
381C4—5081no—
382C4—5081no—
383C4—2541no—
384C4—2541no—
385C4#41×D12711yescharacteristic
386C4#1686125511yescharacteristic
387C2—22no—
388C2—81no—
389C2—10161no—
390C2—5082no—
391C2—2542no—
392C2#2255051811yescharacteristic
393C2C2×SD16×D12711yescharacteristic
394C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#481×C127C2
socleC22×C127C22×D4
prime radical#22550510C2
hypercenter#481D127
hypocenterC127#3295
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #22550581C2semidirect
3. #22550221C2semidirect
4. #22550196C2semidirect
5. #22550165C2semidirect
6. #22550106C2semidirect
7. #22550103C2semidirect
8. #22550075C2semidirect
9. #22550029C2semidirect
10. #22550028C2semidirect
11. #22549963C2semidirect
12. (C22⋊D127)⋊C8C2semidirect
13. C2×#16861129C2semidirect
14. #102×D127C2semidirect
15. #481×C127C2semidirect
20. C2×#14523237C22semidirect
21. C2×#14523236C22semidirect
22. C22×C127⋊C8C22semidirect
23. C2×C8×D127C22semidirect
24. #16861208C22semidirect
27. #16861189C22semidirect
29. #16861188C22semidirect
35. #16861174C22semidirect
39. C4⋊C4×D127C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861132C22semidirect
44. #16861131C22semidirect
45. #16861130C22semidirect
48. #16861129C22semidirect
49. #16861129C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. C8×Dic127C22semidirect
53. C2×C4⋊C4×C127C22semidirect
54. #106×C127C22semidirect
55. #106×C127C22semidirect
56. #102×C127C22semidirect
57. #102×C127C22semidirect
58. C22⋊C8×C127C22semidirect
133. C22⋊D127SD16semidirect
198. C127#3295semidirect
203. #481D127semidirect
239. C2×C4⋊C4C2×D127semidirect
241. #106C2×D127semidirect
242. #106C2×D127semidirect
252. #102C2×D127semidirect
253. #102C2×D127semidirect
255. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505812
#225505102
#225502212
#225501962
#225501652
#225501062
#225501032
#225500752
#225500292
#225500282
#225499632
(C22⋊D127)⋊C82
C2×#168611292
#102×D1272
#481×C1272
#3295127