Finite Group Databasebeta

← #30650547

Subgroups — #30650547

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#22550584C211yesmaximal, characteristic, complemented
2D4×C22⋊D127C211yesmaximal, characteristic
3#22550221C211yesmaximal, characteristic, complemented
4#22550195C211yesmaximal, characteristic, complemented
5#22550193C211yesmaximal, characteristic, complemented
6#22550164C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550102C211yesmaximal, characteristic, complemented
9#22550074C211yesmaximal, characteristic, complemented
10#22550025C211yesmaximal, characteristic, complemented
11SD16×Dic127C211yesmaximal, characteristic, complemented
12(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
13C2×#16861130C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#450×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127—21no—
17C2×D4×D127C2211yescharacteristic
18C2×#14523235C2211yescharacteristic, complemented
19C22×C127⋊C8C2211yescharacteristic, complemented
20C2×C8×D127C2211yescharacteristic, complemented
21#16861208C2211yescharacteristic, complemented
22#16861205C2211yescharacteristic
23#16861199—21no—
24#16861189—21no—
25#16861186—21no—
26#16861185C2211yescharacteristic, complemented
27#16861183C2211yescharacteristic
28#16861181—21no—
29#16861180C2211yescharacteristic
30#16861178—21no—
31#16861177—21no—
32Q8×Dic127C2211yescharacteristic, complemented
33D4×Dic127C2211yescharacteristic
34C4×C22⋊D127C2211yescharacteristic
35(C22⋊D127)⋊C4—21no—
36C2×C4⋊Dic127C2211yescharacteristic, complemented
37C4⋊C4×D127C2211yescharacteristic, complemented
38#16861141C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic, complemented
48#16861130C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic, complemented
55C22⋊Q8×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C2×Dic254—21no—
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—44no—
68C2×C22⋊D127D411yescharacteristic
69D4×D127—22no—
70D4×D127—41no—
71C22×Dic127—21no—
72C22×Dic127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2311yescharacteristic
75#14523235—41no—
76C2×C127⋊C8—41no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C8×D127—41no—
80C4⋊Dic127—21no—
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84C4⋊Dic127C2311yescharacteristic
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4D411yescharacteristic
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127D411yescharacteristic
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4D411yescharacteristic
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97C24×C127—21no—
98C2×Q8×C127C2311yescharacteristic
99C2×D4×C127—22no—
100C2×D4×C127—21no—
101C2×D4×C127—41no—
102C2×D4×C127C2311yescharacteristic
103C2×D4×C127C2311yescharacteristic
104C22×C4×C127C2311yescharacteristic
105SD16×C127—22no—
106SD16×C127—41no—
107C2×C8×C127C2311yescharacteristic
108C2×C8×C127C2311yescharacteristic
109C4⋊C4×C127—21no—
110C4⋊C4×C127C2311yescharacteristic
111C22⋊C4×C127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117Dic254—81no—
118D508—21no—
119D508—21no—
120C22⋊D127—42no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C22⋊D127—42no—
124C22⋊D127—81no—
125C22⋊D127SD1614yescomplemented
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—42no—
130C2×Dic127—81no—
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C2×Dic127C2×D411yescharacteristic
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—81no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127—42no—
142C23×C127—42no—
143C23×C127—81no—
144C23×C127C2×D411yescharacteristic
145Q8×C127—21no—
146Q8×C127—21no—
147D4×C127—21no—
148D4×C127—21no—
149D4×C127—21no—
150D4×C127—42no—
151D4×C127#4112yes—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2411yescharacteristic
158C8×C127—21no—
159C8×C127—21no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—22no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—84no—
169C22×C127C2×SD1612yes—
170C22×C127C22×D411yescharacteristic
171C2×D127—21no—
172C2×D127—21no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—84no—
177C2×D127C2×SD1612yes—
178C4×C127—21no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127C2×#4111yescharacteristic
182C4×C127H(5,𝔽2)11yescharacteristic
183Dic127—21no—
184Dic127—21no—
185Dic127—41no—
186Dic127—41no—
187Dic127—41no—
188Dic127C2×SD1612yes—
189C2×C127—22no—
190C2×C127—42no—
191C2×C127—81no—
192C2×C127#54611yescharacteristic
193C2×C127C22×SD1611yescharacteristic
194C2×C127#57611yescharacteristic
195D127—22no—
196D127—81no—
197#3295—1271nomaximal, Sylow
198C127#329511yescharacteristic, complemented, Sylow
199#554—1271no—
200D42—1271no—
201#486—1271no—
202#481—1271no—
203#481—1271no—
204#476—1271no—
205#465—1271no—
206#465—1271no—
207#460—1271no—
208#450—1271no—
209#450D12711yescharacteristic, complemented
210C4×SD16—1271no—
211C8×D4—1271no—
212C2×#102—1271no—
213C2×#102—1271no—
214C22×D4—1271no—
215C22×D4—2541no—
216C22×D4C2×D12711yescharacteristic
217C2×SD16—1271no—
218C2×SD16C2×D12711yescharacteristic
219C2×SD16C2×D12711yescharacteristic, complemented
220C22×C8—1271no—
221C22×C8—1271no—
222C4⋊Q8—1271no—
223#127—1271no—
224#125—2541no—
225C22⋊Q8—1271no—
226C22⋊Q8—2541no—
227C22⋊Q8—2541no—
228C22⋊Q8C2×D12711yescharacteristic, complemented
229#121—1271no—
230#121—1271no—
231#121—2541no—
232#120—2541no—
233#120—2541no—
234C4×Q8—1271no—
235C4×D4—1271no—
236C4×D4—1271no—
237C4×D4—2541no—
238C2×C4⋊C4—1271no—
239C2×C4⋊C4—1271no—
240#106—1271no—
241#106—1271no—
242#106—1271no—
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
253#102C2×D12711yescharacteristic, complemented
254C22⋊C8—1271no—
255C22⋊C8C2×D12711yescharacteristic, complemented
256C4×C8—1271no—
257C24—21no—
258C24—2541no—
259C2×Q8—2541no—
260C2×Q8C22×D12711yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—1271no—
265C2×D4—1271no—
266C2×D4—5084no—
267C2×D4—5081no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4—2541no—
271C2×D4—2541no—
272C2×D4—2541no—
273C2×D4—2542no—
274C2×D4C22×D12711yescharacteristic
275C2×D4C22×D12711yescharacteristic
276C22×C4—1271no—
277C22×C4—2541no—
278C22×C4—2541no—
279C22×C4—2541no—
280C22×C4C22×D12711yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—5081no—
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—1271no—
294C4⋊C4—1271no—
295C4⋊C4—2541no—
296C4⋊C4—2541no—
297C4⋊C4—2541no—
298C4⋊C4—2541no—
299C4⋊C4—2541no—
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—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—1271no—
316C23—10161no—
317C23—5082no—
318C23—5082no—
319C23—2541no—
320C23D4×D12711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—10161no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—1274no—
329D4—10161no—
330D4—5082no—
331D4—5082no—
332D4—5082no—
333D4—5082no—
334D4—2541no—
335D4—2541no—
336D4#1452324712yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—1271no—
342C2×C4—1271no—
343C2×C4—1271no—
344C2×C4—10161no—
345C2×C4—10161no—
346C2×C4—5082no—
347C2×C4—2541no—
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—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—1272no—
368C22—10164no—
369C22—5081no—
370C22—5081no—
371C22—5081no—
372C22—2541no—
373C22—2541no—
374C22SD16×D12712yes—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—1272no—
380C4—5081no—
381C4—5081no—
382C4—5081no—
383C4—2541no—
384C4—2541no—
385C4C2×#1452324711yescharacteristic
386C4#1686125411yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—10161no—
391C2—2542no—
392C2#2255051311yescharacteristic
393C2C2×SD16×D12711yescharacteristic
394C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radicalD4×C22⋊D127C2
hypercenter#450D127
hypocenterC127#3295
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22550584C2semidirect
3. #22550221C2semidirect
4. #22550195C2semidirect
5. #22550193C2semidirect
6. #22550164C2semidirect
7. #22550107C2semidirect
8. #22550102C2semidirect
9. #22550074C2semidirect
10. #22550025C2semidirect
11. SD16×Dic127C2semidirect
12. (C22⋊D127)⋊C8C2semidirect
13. C2×#16861130C2semidirect
14. #102×D127C2semidirect
15. #450×C127C2semidirect
18. C2×#14523235C22semidirect
19. C22×C127⋊C8C22semidirect
20. C2×C8×D127C22semidirect
21. #16861208C22semidirect
26. #16861185C22semidirect
32. Q8×Dic127C22semidirect
36. C2×C4⋊Dic127C22semidirect
37. C4⋊C4×D127C22semidirect
38. #16861141C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861133C22semidirect
43. #16861131C22semidirect
44. #16861131C22semidirect
47. #16861130C22semidirect
48. #16861130C22semidirect
49. #16861129C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. C8×Dic127C22semidirect
54. C2×SD16×C127C22semidirect
55. C22⋊Q8×C127C22semidirect
57. #102×C127C22semidirect
58. C22⋊C8×C127C22semidirect
125. C22⋊D127SD16semidirect
198. C127#3295semidirect
209. #450D127semidirect
219. C2×SD16C2×D127semidirect
228. C22⋊Q8C2×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
#225505842
D4×C22⋊D1272
#225502212
#225501952
#225501932
#225501642
#225501072
#225501022
#225500742
#225500252
SD16×Dic1272
(C22⋊D127)⋊C82
C2×#168611302
#102×D1272
#450×C1272
#3295127