Finite Group Databasebeta

← #30650687

Subgroups — #30650687

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#22550520C211yesmaximal, characteristic, complemented
3#22550234C211yesmaximal, characteristic, complemented
4#22550206C211yesmaximal, characteristic, complemented
5#22550193C211yesmaximal, characteristic, complemented
6#22550164C211yesmaximal, characteristic, complemented
7#22550128C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550090C211yesmaximal, characteristic, complemented
10#22550022C211yesmaximal, characteristic, complemented
11#22550010C211yesmaximal, characteristic, complemented
12#22549985C211yesmaximal, characteristic, complemented
13#22549947C211yesmaximal, characteristic, complemented
14#22549874C211yesmaximal, characteristic, complemented
15C2×#102×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C2×#14523247C2211yescharacteristic
18C2×#14523237C2211yescharacteristic, complemented
19C2×#14523233C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C2×#14523231C2211yescharacteristic, complemented
22#16861208C2211yescharacteristic, complemented
23#16861196C2211yescharacteristic
24#16861194—21no—
25#16861192—21no—
26#16861190—21no—
27#16861188C2211yescharacteristic
28#16861187—21no—
29#16861184C2211yescharacteristic, complemented
30#16861183C2211yescharacteristic
31#16861181—21no—
32#16861180C2211yescharacteristic
33#16861178—21no—
34D4×Dic127C2211yescharacteristic
35#16861171C2211yescharacteristic, complemented
36C4×C22⋊D127C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic, complemented
38C2×C22⋊Dic127—21no—
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861132C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51#16861119C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C2×C4⋊C4×C127C2211yescharacteristic
55#102×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523248—44no—
60#14523247—22no—
61#14523247—41no—
62C2×Dic254—21no—
63C2×Dic254C2311yescharacteristic
64C2×D508—21no—
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127D411yescharacteristic
70C22×Dic127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2311yescharacteristic
75#14523237—41no—
76#14523233—22no—
77#14523233—41no—
78#14523231—41no—
79C2×C127⋊C8C2311yescharacteristic
80C2×C127⋊C8C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127D411yescharacteristic
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4D411yescharacteristic
97C4×Dic127C2311yescharacteristic
98C24×C127—21no—
99C2×D4×C127—22no—
100C2×D4×C127—21no—
101C2×D4×C127—41no—
102C2×D4×C127C2311yescharacteristic
103C2×D4×C127C2311yescharacteristic
104C22×C4×C127—21no—
105C22×C4×C127C2311yescharacteristic
106C2×C8×C127—41no—
107C2×C8×C127C2311yescharacteristic
108C2×C8×C127C2311yescharacteristic
109C4⋊C4×C127—21no—
110C4⋊C4×C127C2311yescharacteristic
111C4⋊C4×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116Dic254—42no—
117D508—21no—
118D508—21no—
119D508—42no—
120C22⋊D127—22no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C22⋊D127—81no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—42no—
128C2×Dic127—81no—
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—42no—
135C4×D127—42no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127—42no—
141C23×C127—42no—
142C23×C127—81no—
143C23×C127C2×D411yescharacteristic
144D4×C127—21no—
145D4×C127—21no—
146D4×C127—21no—
147D4×C127—42no—
148D4×C127#4112yescomplemented
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×C127—22no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—84no—
167C22×C127AGL(1,ℤ/8)12yes—
168C22×C127C22×D411yescharacteristic
169C2×D127—21no—
170C2×D127—21no—
171C2×D127—21no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C4×C127—21no—
176C4×C127—41no—
177C4×C127—41no—
178C4×C127C2×#4111yescharacteristic
179C4×C127H(5,𝔽2)11yescharacteristic
180Dic127—21no—
181Dic127—21no—
182Dic127—21no—
183Dic127—41no—
184Dic127—41no—
185Dic127—41no—
186C2×C127—22no—
187C2×C127—42no—
188C2×C127—81no—
189C2×C127#54611yescharacteristic
190C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C127#57711yescharacteristic
192D127—41no—
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#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×#102D12711yescharacteristic, complemented
211C2×#41—1271no—
212C2×#41—2541no—
213C22×D4C2×D12711yescharacteristic
214C2×SD16—1271no—
215C2×D8—1271no—
216C2×D8—1271no—
217C2×M16—1271no—
218C22×C8C2×D12711yescharacteristic
219C4⋊Q8—1271no—
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×D4—1271no—
234C2×C4⋊C4—1271no—
235C2×C4⋊C4C2×D12711yescharacteristic
236C2×C22⋊C4—2541no—
237#106—1271no—
238#106—1271no—
239#106—1271no—
240C4⋊C8—1271no—
241#103—1271no—
242#103—1271no—
243#103—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102C2×D12711yescharacteristic
248#102C2×D12711yescharacteristic
249#102C2×D12711yescharacteristic
250#102C2×D12711yescharacteristic
251C22⋊C8—1271no—
252C22⋊C8—1271no—
253#97—1271no—
254C24—21no—
255#41—5084no—
256#41—5081no—
257#41—2542no—
258C2×Q8—1271no—
259C2×Q8—2541no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—41no—
263C2×D4—1271no—
264C2×D4—1271no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4C22×D12711yescharacteristic
270C2×D4C22×D12711yescharacteristic
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—41no—
283C2×C8—1271no—
284C2×C8—1271no—
285C2×C8C22×D12711yescharacteristic
286C2×C8C22×D12711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—1271no—
289C4⋊C4—1271no—
290C4⋊C4—1271no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4—2541no—
294C4⋊C4C22×D12711yescharacteristic
295C4⋊C4C22×D12711yescharacteristic
296C22⋊C4—1271no—
297C22⋊C4—1271no—
298C22⋊C4—2541no—
299C22⋊C4—2541no—
300C22⋊C4—2541no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C42—1271no—
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—1271no—
313C23—2541no—
314C23D4×D12711yescharacteristic
315Q8—5082no—
316Q8—2541no—
317Q8—2541no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—10161no—
323D4—5082no—
324D4—5082no—
325D4—5082no—
326D4—2542no—
327D4—2541no—
328D4—2541no—
329D4#1452324812yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—1271no—
336C2×C4—1271no—
337C2×C4—1271no—
338C2×C4—10161no—
339C2×C4—5082no—
340C2×C4—5082no—
341C2×C4—5082no—
342C2×C4—2541no—
343C2×C4—2541no—
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—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—84no—
362C22—5081no—
363C22—5081no—
364C22—5081no—
365C22—2541no—
366C22—2541no—
367C22—2541no—
368C22#1686123312yes—
369C22C2×D4×D12711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—5081no—
374C4—5081no—
375C4—5081no—
376C4—2541no—
377C4—2541no—
378C4—2541no—
379C4C2×#1452324811yescharacteristic
380C4#1686125411yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—10161no—
385C2—5081no—
386C2#2255051611yescharacteristic
387C2C2×#1686123311yescharacteristic
388C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC22×C127C22×D4
hypercenterC2×#102D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #22550527C2semidirect
2. #22550520C2semidirect
3. #22550234C2semidirect
4. #22550206C2semidirect
5. #22550193C2semidirect
6. #22550164C2semidirect
7. #22550128C2semidirect
8. #22550105C2semidirect
9. #22550090C2semidirect
10. #22550022C2semidirect
11. #22550010C2semidirect
12. #22549985C2semidirect
13. #22549947C2semidirect
14. #22549874C2semidirect
15. C2×#102×C127C2semidirect
18. C2×#14523237C22semidirect
21. C2×#14523231C22semidirect
22. #16861208C22semidirect
29. #16861184C22semidirect
35. #16861171C22semidirect
37. C4⋊C4×D127C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. #16861140C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861132C22semidirect
45. #16861132C22semidirect
46. #16861131C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. #16861119C22semidirect
148. D4×C127#41semidirect
195. C127#3306semidirect
210. C2×#102D127semidirect
329. D4#14523248semidirect

Maximal subgroups

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

subgroupindex
#225505272
#225505202
#225502342
#225502062
#225501932
#225501642
#225501282
#225501052
#225500902
#225500222
#225500102
#225499852
#225499472
#225498742
C2×#102×C1272
#3306127