Finite Group Databasebeta

← #30650698

Subgroups — #30650698

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#22550531C211yesmaximal, characteristic, complemented
2#22550519C211yesmaximal, characteristic, complemented
3#22550235C211yesmaximal, characteristic, complemented
4#22550205C211yesmaximal, characteristic, complemented
5#22550193C211yesmaximal, characteristic, complemented
6#22550167C211yesmaximal, characteristic, complemented
7#22550126C211yesmaximal, characteristic, complemented
8#22550102C211yesmaximal, characteristic, complemented
9#22550020C211yesmaximal, characteristic, complemented
10#22550008C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549946C211yesmaximal, characteristic, complemented
13#22549876C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248—21no—
18C2×D4×D127C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523231C2211yescharacteristic, complemented
21C2×C8×D127C2211yescharacteristic
22#16861209C2211yescharacteristic, complemented
23#16861191—21no—
24#16861191—21no—
25#16861191—21no—
26#16861188—21no—
27#16861185C2211yescharacteristic
28#16861182—21no—
29#16861180C2211yescharacteristic
30#16861180C2211yescharacteristic
31#16861180C2211yescharacteristic, complemented
32#16861179—21no—
33C4×D508C2211yescharacteristic
34C4×D508C2211yescharacteristic
35C4⋊C4×D127C2211yescharacteristic
36C4⋊C4×D127C2211yescharacteristic, complemented
37C22⋊C4×D127—21no—
38#16861141C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41#16861133C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51#16861120C2211yescharacteristic, complemented
52C2×SD16×C127C2211yescharacteristic, complemented
53C2×D8×C127C2211yescharacteristic
54#124×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic, complemented
56C4⋊C8×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic
59C23×D127—21no—
60#14523249—22no—
61#14523249—41no—
62#14523248—44no—
63C2×Dic254—21no—
64C2×D508—21no—
65C2×D508D411yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71D4×D127—22no—
72D4×D127—41no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127C2311yescharacteristic
78C2×C4×D127C2311yescharacteristic
79#14523233—22no—
80#14523231—41no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C8×D127—41no—
84C4⋊Dic127—21no—
85C4⋊Dic127D411yescharacteristic
86C4⋊Dic127C2311yescharacteristic
87C4⋊Dic127C2311yescharacteristic
88C4⋊Dic127C2311yescharacteristic
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92C22⋊Dic127—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4D411yescharacteristic
98(C2×D127)⋊C4D411yescharacteristic
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127C2311yescharacteristic
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127—21no—
103SD16×C127—41no—
104D8×C127—41no—
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C22⋊C4×C127—21no—
109C22⋊C4×C127—21no—
110C22⋊C4×C127—21no—
111C42×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117C22×D127C2×D411yescharacteristic
118Dic254—42no—
119D508—22no—
120D508—21no—
121D508—42no—
122D508#4112yescomplemented
123C22⋊D127—42no—
124C22⋊D127—42no—
125C22⋊D127—42no—
126C22⋊D127—81no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—42no—
137C4×D127—42no—
138C4×D127—42no—
139C4×D127—81no—
140C127⋊C8—21no—
141C127⋊C8—21no—
142C23×C127—21no—
143C23×C127—21no—
144Q8×C127—21no—
145Q8×C127—21no—
146D4×C127—21no—
147D4×C127—21no—
148D4×C127—21no—
149D4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—42no—
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127C22×D411yescharacteristic
165C2×D127—21no—
166C2×D127—21no—
167C2×D127—21no—
168C2×D127—22no—
169C2×D127—21no—
170C2×D127—21no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—84no—
174C2×D127AGL(1,ℤ/8)12yes—
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127C2×#4111yescharacteristic
180C4×C127H(5,𝔽2)11yescharacteristic
181Dic127—21no—
182Dic127—21no—
183Dic127—41no—
184Dic127—41no—
185Dic127—41no—
186C2×C127—81no—
187C2×C127—81no—
188C2×C127#54611yescharacteristic
189C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C127#57711yescharacteristic
191D127—22no—
192D127—41no—
193D127—42no—
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#463D12711yescharacteristic, complemented
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×SD16C2×D12711yescharacteristic, complemented
215C2×D8—1271no—
216C2×D8C2×D12711yescharacteristic
217C2×M16—1271no—
218C22×C8—1271no—
219C4⋊Q8—1271no—
220#124C2×D12711yescharacteristic
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×D4C2×D12711yescharacteristic, complemented
234C2×C4⋊C4—1271no—
235C2×C4⋊C4—1271no—
236C2×C22⋊C4—2541no—
237#106—1271no—
238#106—1271no—
239#106—1271no—
240C4⋊C8C2×D12711yescharacteristic, complemented
241#103—1271no—
242#103—1271no—
243#103C2×D12711yescharacteristic, complemented
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102C2×D12711yescharacteristic
251C22⋊C8—1271no—
252C22⋊C8—1271no—
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—5081no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4—2542no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4C22×D12711yescharacteristic
270C2×D4C22×D12711yescharacteristic
271C22×C4—21no—
272C22×C4—1271no—
273C22×C4—1271no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4—2541no—
277C22×C4—2541no—
278SD16—41no—
279D8—41no—
280D8—2542no—
281M16—5081no—
282C2×C8—1271no—
283C2×C8—1271no—
284C2×C8—5081no—
285C2×C8C22×D12711yescharacteristic
286C2×C8C22×D12711yescharacteristic
287C4⋊C4—1271no—
288C4⋊C4—1271no—
289C4⋊C4—1271no—
290C4⋊C4—1271no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4—2541no—
294C4⋊C4—2541no—
295C4⋊C4C22×D12711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—1271no—
300C22⋊C4—1271no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C42C22×D12711yescharacteristic
307C23—21no—
308C23—21no—
309C23—1271no—
310C23—1271no—
311C23—10161no—
312C23—5082no—
313C23—5082no—
314C23—2541no—
315Q8—21no—
316Q8—21no—
317Q8—5082no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—1272no—
323D4—10161no—
324D4—5082no—
325D4—5082no—
326D4—5082no—
327D4—5082no—
328D4—2542no—
329D4—2541no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—1271no—
334C2×C4—1271no—
335C2×C4—10161no—
336C2×C4—5082no—
337C2×C4—5082no—
338C2×C4—5082no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4D4×D12711yescharacteristic
347C2×C4D4×D12711yescharacteristic
348C2×C4C23×D12711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2541no—
352C8—2541no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—1272no—
360C22—10164no—
361C22—5081no—
362C22—5081no—
363C22—2541no—
364C22—2541no—
365C22—2542no—
366C22—2541no—
367C22—2541no—
368C22—2541no—
369C22C2×D4×D12711yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—5081no—
375C4—5081no—
376C4—5081no—
377C4—2541no—
378C4—2541no—
379C4#41×D12711yescharacteristic
380C4#1686125411yescharacteristic
381C2—81no—
382C2—81no—
383C2—5082no—
384C2—5081no—
385C2—2542no—
386C2#2255052311yescharacteristic
387C2AGL(1,ℤ/8)×D12711yescharacteristic
388C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #22550531C2semidirect
2. #22550519C2semidirect
3. #22550235C2semidirect
4. #22550205C2semidirect
5. #22550193C2semidirect
6. #22550167C2semidirect
7. #22550126C2semidirect
8. #22550102C2semidirect
9. #22550020C2semidirect
10. #22550008C2semidirect
11. #22549986C2semidirect
12. #22549946C2semidirect
13. #22549876C2semidirect
14. #102×D127C2semidirect
15. #463×C127C2semidirect
20. C2×#14523231C22semidirect
22. #16861209C22semidirect
31. #16861180C22semidirect
36. C4⋊C4×D127C22semidirect
38. #16861141C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. #16861133C22semidirect
42. #16861133C22semidirect
47. #16861130C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. #16861120C22semidirect
52. C2×SD16×C127C22semidirect
55. C4×D4×C127C22semidirect
56. C4⋊C8×C127C22semidirect
57. #103×C127C22semidirect
122. D508#41semidirect
195. C127#3306semidirect
204. #463D127semidirect
214. C2×SD16C2×D127semidirect
233. C4×D4C2×D127semidirect
240. C4⋊C8C2×D127semidirect
243. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505312
#225505192
#225502352
#225502052
#225501932
#225501672
#225501262
#225501022
#225500202
#225500082
#225499862
#225499462
#225498762
#102×D1272
#463×C1272
#3306127