Finite Group Databasebeta

← #30650686

Subgroups — #30650686

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#22550526C211yesmaximal, characteristic, complemented
2#22550523C211yesmaximal, characteristic, complemented
3#22550234C211yesmaximal, characteristic, complemented
4#22550208C211yesmaximal, characteristic, complemented
5#22550192C211yesmaximal, characteristic, complemented
6#22550165C211yesmaximal, characteristic, complemented
7#22550104C211yesmaximal, characteristic, complemented
8#22550089C211yesmaximal, characteristic, complemented
9#22550019C211yesmaximal, characteristic, complemented
10#22550008C211yesmaximal, characteristic, complemented
11#22549985C211yesmaximal, characteristic, complemented
12#22549945C211yesmaximal, characteristic, complemented
13#22549878C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#469×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523247—21no—
18C2×D4×D127C2211yescharacteristic
19C2×#14523236C2211yescharacteristic, complemented
20C2×#14523233C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22#16861208C2211yescharacteristic, complemented
23#16861196C2211yescharacteristic
24#16861191—21no—
25#16861191—21no—
26#16861190—21no—
27#16861189C2211yescharacteristic
28#16861186—21no—
29#16861180—21no—
30#16861180C2211yescharacteristic
31#16861177—21no—
32D4×Dic127C2211yescharacteristic, complemented
33#16861171C2211yescharacteristic
34C4×C22⋊D127C2211yescharacteristic
35C2×C4⋊Dic127C2211yescharacteristic, complemented
36C4⋊C4×D127C2211yescharacteristic
37C22⋊C4×D127—21no—
38#16861140C2211yescharacteristic, complemented
39#16861140C2211yescharacteristic, complemented
40Dic127⋊C8C2211yescharacteristic, complemented
41#16861134C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic
51#16861119C2211yescharacteristic, complemented
52C2×D8×C127C2211yescharacteristic
53C2×M16×C127C2211yescharacteristic, complemented
54#121×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic, complemented
56#106×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—22no—
61#14523248—41no—
62#14523247—44no—
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127D411yescharacteristic
70D4×D127—22no—
71D4×D127—41no—
72C22×Dic127—21no—
73C22×Dic127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127C2311yescharacteristic
78#14523236—41no—
79#14523233—22no—
80C2×C127⋊C8C2311yescharacteristic
81C2×C127⋊C8C2311yescharacteristic
82C8×D127—41no—
83C4⋊Dic127—21no—
84C4⋊Dic127—21no—
85C4⋊Dic127C2311yescharacteristic
86C4⋊Dic127C2311yescharacteristic
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127D411yescharacteristic
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127C2311yescharacteristic
99C2×D4×C127—21no—
100C2×D4×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127C2311yescharacteristic
104D8×C127—41no—
105M16×C127—41no—
106C2×C8×C127C2311yescharacteristic
107C2×C8×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C22⋊C4×C127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117Dic254—21no—
118Dic254—21no—
119Dic254—42no—
120D508—21no—
121D508#4112yescomplemented
122C22⋊D127—22no—
123C22⋊D127—42no—
124C22⋊D127—42no—
125C22⋊D127—42no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—42no—
130C2×Dic127—42no—
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C2×Dic127C2×D411yescharacteristic
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—42no—
137C4×D127—42no—
138C4×D127—81no—
139C127⋊C8—21no—
140C127⋊C8—21no—
141C23×C127—21no—
142C23×C127—21no—
143C23×C127C2×D411yescharacteristic
144D4×C127—21no—
145D4×C127—21no—
146D4×C127—21no—
147D4×C127—21no—
148D4×C127—42no—
149D4×C127—81no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127C22×D411yescharacteristic
168C2×D127—21no—
169C2×D127—21no—
170C2×D127—22no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—84no—
174C2×D127AGL(1,ℤ/8)12yes—
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—41no—
187C2×C127—81no—
188C2×C127—81no—
189C2×C127#54611yescharacteristic
190C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C127#57711yescharacteristic
192D127—22no—
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#469D12711yescharacteristic, complemented
203#465—1271no—
204#463—1271no—
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×SD16—1271no—
215C2×D8—1271no—
216C2×D8C2×D12711yescharacteristic
217C2×M16C2×D12711yescharacteristic, complemented
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—2541no—
228#121C2×D12711yescharacteristic
229#121C2×D12711yescharacteristic, complemented
230#120—2541no—
231C4×D4—1271no—
232C4×D4—1271no—
233C4×D4—1271no—
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#97—1271no—
254C24—2541no—
255#41—5081no—
256#41—5084no—
257#41—2542no—
258C2×Q8—1271no—
259C2×Q8—2541no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—1271no—
263C2×D4—1271no—
264C2×D4—5081no—
265C2×D4—2541no—
266C2×D4—2542no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4C22×D12711yescharacteristic
270C2×D4C22×D12711yescharacteristic
271C22×C4—1271no—
272C22×C4—2541no—
273C22×C4—2541no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4—2541no—
277C22×C4C22×D12711yescharacteristic
278SD16—5081no—
279D8—41no—
280D8—2542no—
281M16—41no—
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—
306C42—1271no—
307C23—21no—
308C23—21no—
309C23—1271no—
310C23—10161no—
311C23—5082no—
312C23—5082no—
313C23—2541no—
314C23D4×D12711yescharacteristic
315Q8—5082no—
316Q8—2541no—
317Q8—2541no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—81no—
324D4—1272no—
325D4—5082no—
326D4—5082no—
327D4—5082no—
328D4—2542no—
329D4—2541no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—1271no—
335C2×C4—1271no—
336C2×C4—1271no—
337C2×C4—10161no—
338C2×C4—5082no—
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—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—1272no—
363C22—10164no—
364C22—5081no—
365C22—5081no—
366C22—2542no—
367C22—2541no—
368C22—2541no—
369C22C2×D4×D12711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—5081no—
374C4—5081no—
375C4—5081no—
376C4—2541no—
377C4—2541no—
378C4—2541no—
379C4#41×D12711yescharacteristic
380C4#1686125411yescharacteristic
381C2—41no—
382C2—81no—
383C2—81no—
384C2—5082no—
385C2—2542no—
386C2#2255051511yescharacteristic
387C2AGL(1,ℤ/8)×D12711yescharacteristic
388C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #22550526C2semidirect
2. #22550523C2semidirect
3. #22550234C2semidirect
4. #22550208C2semidirect
5. #22550192C2semidirect
6. #22550165C2semidirect
7. #22550104C2semidirect
8. #22550089C2semidirect
9. #22550019C2semidirect
10. #22550008C2semidirect
11. #22549985C2semidirect
12. #22549945C2semidirect
13. #22549878C2semidirect
14. #102×D127C2semidirect
15. #469×C127C2semidirect
19. C2×#14523236C22semidirect
22. #16861208C22semidirect
32. D4×Dic127C22semidirect
35. C2×C4⋊Dic127C22semidirect
38. #16861140C22semidirect
39. #16861140C22semidirect
40. Dic127⋊C8C22semidirect
41. #16861134C22semidirect
42. #16861134C22semidirect
43. #16861132C22semidirect
49. C22⋊(C127⋊C8)C22semidirect
51. #16861119C22semidirect
53. C2×M16×C127C22semidirect
55. #121×C127C22semidirect
56. #106×C127C22semidirect
58. #102×C127C22semidirect
121. D508#41semidirect
195. C127#3306semidirect
202. #469D127semidirect
217. C2×M16C2×D127semidirect
229. #121C2×D127semidirect
239. #106C2×D127semidirect
250. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505262
#225505232
#225502342
#225502082
#225501922
#225501652
#225501042
#225500892
#225500192
#225500082
#225499852
#225499452
#225498782
#102×D1272
#469×C1272
#3306127