Finite Group Databasebeta

← #30650696

Subgroups — #30650696

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#22550521C211yesmaximal, characteristic, complemented
3#22550236C211yesmaximal, characteristic, complemented
4#22550208C211yesmaximal, characteristic, complemented
5#22550196C211yesmaximal, characteristic, complemented
6#22550166C211yesmaximal, characteristic, complemented
7#22550128C211yesmaximal, characteristic, complemented
8#22550104C211yesmaximal, characteristic, complemented
9#22550091C211yesmaximal, characteristic, complemented
10#22550018C211yesmaximal, characteristic, complemented
11#22550008C211yesmaximal, characteristic, complemented
12#22549949C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#442×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C2×#14523247C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19C2×#14523236C2211yescharacteristic, complemented
20C2×#14523233C2211yescharacteristic
21C2×#14523231C2211yescharacteristic, complemented
22C2×C8×D127C2211yescharacteristic
23#16861209C2211yescharacteristic, complemented
24#16861197C2211yescharacteristic
25#16861193—21no—
26#16861191—21no—
27#16861191—21no—
28#16861189C2211yescharacteristic
29#16861188—21no—
30#16861184C2211yescharacteristic
31#16861183—21no—
32#16861180C2211yescharacteristic
33#16861180C2211yescharacteristic, complemented
34#16861179—21no—
35C4×D508C2211yescharacteristic
36C4⋊C4×D127C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic, complemented
38C2×(C2×D127)⋊C4—21no—
39#16861140C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41C4⋊(C127⋊C8)C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861129C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52C2×D8×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic, complemented
58#97×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—44no—
61#14523247—22no—
62#14523247—41no—
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65C2×D508—21no—
66C2×D508D411yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72D4×D127—22no—
73D4×D127—41no—
74C22×Dic127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127C2311yescharacteristic
78C2×C4×D127C2311yescharacteristic
79#14523236—41no—
80#14523233—41no—
81#14523231—41no—
82C2×C127⋊C8C2311yescharacteristic
83C2×C127⋊C8C2311yescharacteristic
84C8×D127—41no—
85C4⋊Dic127—21no—
86C4⋊Dic127—21no—
87C4⋊Dic127D411yescharacteristic
88C4⋊Dic127C2311yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92C22⋊Dic127—21no—
93C22⋊Dic127—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4D411yescharacteristic
99(C2×D127)⋊C4D411yescharacteristic
100C2×D4×C127C2311yescharacteristic
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127—21no—
103C22×C4×C127—21no—
104D8×C127—22no—
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
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—21no—
119Dic254—21no—
120Dic254—42no—
121D508—22no—
122D508—21no—
123D508—21no—
124D508—81no—
125C22⋊D127—42no—
126C22⋊D127—42no—
127C22⋊D127—42no—
128C22⋊D127—42no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127—21no—
132C2×Dic127—42no—
133C2×Dic127C2×D411yescharacteristic
134C2×Dic127C2×D411yescharacteristic
135C4×D127—21no—
136C4×D127—21no—
137C4×D127—21no—
138C4×D127—21no—
139C4×D127—42no—
140C4×D127—42no—
141C127⋊C8—21no—
142C127⋊C8—21no—
143C23×C127—21no—
144C23×C127—21no—
145D4×C127—21no—
146D4×C127—21no—
147D4×C127—21no—
148D4×C127#4112yescomplemented
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—42no—
152C2×C4×C127—81no—
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—22no—
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—21no—
169C2×D127—21no—
170C2×D127—41no—
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—42no—
187C2×C127—81no—
188C2×C127#54611yescharacteristic
189C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C127#57711yescharacteristic
191D127—22no—
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#442D12711yescharacteristic, complemented
208#435—1271no—
209#417—1271no—
210C2×#102—1271no—
211C2×#41—1271no—
212C2×#41—2541no—
213C22×D4—1271no—
214C2×SD16—1271no—
215C2×D8—1271no—
216C2×D8C2×D12711yescharacteristic
217C2×M16—1271no—
218C22×C8—1271no—
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×D4C2×D12711yescharacteristic
233C4×D4C2×D12711yescharacteristic, complemented
234C2×C4⋊C4—1271no—
235C2×C4⋊C4—1271no—
236C2×C22⋊C4—2541no—
237#106—1271no—
238#106—1271no—
239#106C2×D12711yescharacteristic, complemented
240C4⋊C8—1271no—
241#103—1271no—
242#103—1271no—
243#103—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102C2×D12711yescharacteristic
250#102C2×D12711yescharacteristic, complemented
251C22⋊C8—1271no—
252C22⋊C8—1271no—
253#97C2×D12711yescharacteristic, complemented
254C24—2541no—
255#41—5084no—
256#41—5081no—
257#41—2542no—
258C2×Q8—1271no—
259C2×Q8—2541no—
260C2×D4—1271no—
261C2×D4—1271no—
262C2×D4—5081no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2542no—
269C2×D4C22×D12711yescharacteristic
270C2×D4C22×D12711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—1271no—
274C22×C4—1271no—
275C22×C4—2541no—
276C22×C4—2541no—
277C22×C4—2541no—
278SD16—5081no—
279D8—22no—
280D8—5081no—
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—2541no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4—2541no—
294C4⋊C4C22×D12711yescharacteristic
295C4⋊C4C22×D12711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—1271no—
299C22⋊C4—1271no—
300C22⋊C4—2541no—
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—5082no—
316Q8—2541no—
317Q8—2541no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—10161no—
322D4—5082no—
323D4—5082no—
324D4—5082no—
325D4—5082no—
326D4—2542no—
327D4—2541no—
328D4—2541no—
329D4#1452324812yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—1271no—
335C2×C4—1271no—
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—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—1272no—
360C22—10164no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—2541no—
365C22—2541no—
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—
379C4C2×#1452324811yescharacteristic
380C4#1686125511yescharacteristic
381C2—42no—
382C2—81no—
383C2—10161no—
384C2—5081no—
385C2—2542no—
386C2#2255052211yescharacteristic
387C2AGL(1,ℤ/8)×D12711yescharacteristic
388C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#442×C127C2
socleC22×C127C22×D4
hypercenter#442D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #22550531C2semidirect
2. #22550521C2semidirect
3. #22550236C2semidirect
4. #22550208C2semidirect
5. #22550196C2semidirect
6. #22550166C2semidirect
7. #22550128C2semidirect
8. #22550104C2semidirect
9. #22550091C2semidirect
10. #22550018C2semidirect
11. #22550008C2semidirect
12. #22549949C2semidirect
13. #22549874C2semidirect
14. #102×D127C2semidirect
15. #442×C127C2semidirect
19. C2×#14523236C22semidirect
21. C2×#14523231C22semidirect
23. #16861209C22semidirect
33. #16861180C22semidirect
37. C4⋊C4×D127C22semidirect
39. #16861140C22semidirect
40. #16861140C22semidirect
41. C4⋊(C127⋊C8)C22semidirect
42. #16861134C22semidirect
43. #16861134C22semidirect
44. #16861132C22semidirect
51. (C2×D127)⋊C8C22semidirect
54. C4×D4×C127C22semidirect
55. #106×C127C22semidirect
57. #102×C127C22semidirect
58. #97×C127C22semidirect
148. D4×C127#41semidirect
195. C127#3306semidirect
207. #442D127semidirect
233. C4×D4C2×D127semidirect
239. #106C2×D127semidirect
250. #102C2×D127semidirect
253. #97C2×D127semidirect
329. D4#14523248semidirect

Maximal subgroups

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

subgroupindex
#225505312
#225505212
#225502362
#225502082
#225501962
#225501662
#225501282
#225501042
#225500912
#225500182
#225500082
#225499492
#225498742
#102×D1272
#442×C1272
#3306127