Finite Group Databasebeta

← #30650693

Subgroups — #30650693

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#22550530C211yesmaximal, characteristic, complemented
2#22550522C211yesmaximal, characteristic, complemented
3#22550236C211yesmaximal, characteristic, complemented
4#22550207C211yesmaximal, characteristic, complemented
5#22550195C211yesmaximal, characteristic, complemented
6#22550167C211yesmaximal, characteristic, complemented
7#22550129C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550092C211yesmaximal, characteristic, complemented
10#22550023C211yesmaximal, characteristic, complemented
11#22550012C211yesmaximal, characteristic, complemented
12#22549989C211yesmaximal, characteristic, complemented
13#22549878C211yesmaximal, characteristic, complemented
14C2×#16861131C211yesmaximal, characteristic, complemented
15#435×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523247—21no—
18C22×D508C2211yescharacteristic
19C2×#14523237C2211yescharacteristic, complemented
20C2×D1016C2211yescharacteristic
21C2×D1016C2211yescharacteristic
22#16861209C2211yescharacteristic, complemented
23#16861197C2211yescharacteristic
24#16861194—21no—
25#16861193—21no—
26#16861193—21no—
27#16861189—21no—
28#16861188C2211yescharacteristic
29#16861183C2211yescharacteristic
30#16861183C2211yescharacteristic
31#16861183C2211yescharacteristic, complemented
32#16861182—21no—
33#16861177—21no—
34C4×D508C2211yescharacteristic
35C4×D508C2211yescharacteristic, complemented
36C2×C4⋊Dic127C2211yescharacteristic
37C2×C4⋊Dic127C2211yescharacteristic, complemented
38C22⋊C4×D127—21no—
39#16861141C2211yescharacteristic, complemented
40#16861141C2211yescharacteristic, complemented
41#16861141C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861131C2211yescharacteristic, complemented
52C2×M16×C127C2211yescharacteristic, complemented
53C22×C8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55C4⋊C8×C127C2211yescharacteristic, complemented
56C22⋊C8×C127C2211yescharacteristic
57C22⋊C8×C127C2211yescharacteristic, complemented
58#97×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—22no—
61#14523248—41no—
62#14523247—44no—
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65C2×D508—22no—
66C2×D508—21no—
67C2×D508—41no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C22×Dic127—21no—
76C22×Dic127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80#14523237—41no—
81D1016—22no—
82D1016—41no—
83C4⋊Dic127—21no—
84C4⋊Dic127—21no—
85C4⋊Dic127—21no—
86C4⋊Dic127C2311yescharacteristic
87C4⋊Dic127C2311yescharacteristic
88C4⋊Dic127C2311yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99C2×D4×C127D411yescharacteristic
100C22×C4×C127C2311yescharacteristic
101C22×C4×C127C2311yescharacteristic
102M16×C127—41no—
103C2×C8×C127—41no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C2×C8×C127C2311yescharacteristic
108C4⋊C4×C127D411yescharacteristic
109C22⋊C4×C127D411yescharacteristic
110C22⋊C4×C127D411yescharacteristic
111C42×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—21no—
115C22×D127—42no—
116C22×D127—42no—
117C22×D127—81no—
118Dic254—21no—
119Dic254—21no—
120Dic254—42no—
121D508—21no—
122D508—21no—
123D508—21no—
124D508—21no—
125D508—21no—
126D508—42no—
127D508#4112yescomplemented
128C22⋊D127—42no—
129C22⋊D127—42no—
130C22⋊D127—42no—
131C22⋊D127—81no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—42no—
138C2×Dic127—42no—
139C4×D127—42no—
140C4×D127—42no—
141C4×D127—81no—
142C23×C127C2×D411yescharacteristic
143C23×C127C2×D411yescharacteristic
144D4×C127—22no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2×D411yescharacteristic
153C2×C4×C127C2411yescharacteristic
154C8×C127—21no—
155C8×C127—21no—
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—21no—
163C22×C127AGL(1,ℤ/8)12yes—
164C22×C127C22×D411yescharacteristic
165C2×D127—22no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—84no—
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—21no—
178C4×C127—21no—
179C4×C127C2×#4111yescharacteristic
180C4×C127H(5,𝔽2)11yescharacteristic
181Dic127—41no—
182Dic127—41no—
183Dic127—41no—
184Dic127—41no—
185Dic127—41no—
186C2×C127—22no—
187C2×C127—41no—
188C2×C127#54611yescharacteristic
189C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C127#57711yescharacteristic
191D127—42no—
192D127—81no—
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#435D12711yescharacteristic, complemented
209#417—1271no—
210C2×#102—1271no—
211C2×#41—1271no—
212C2×#41—2541no—
213C22×D4—1271no—
214C2×SD16—1271no—
215C2×D8—1271no—
216C2×D8—1271no—
217C2×M16C2×D12711yescharacteristic, complemented
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×D4C2×D12711yescharacteristic
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#103—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102—1271no—
251C22⋊C8C2×D12711yescharacteristic
252C22⋊C8C2×D12711yescharacteristic, complemented
253#97C2×D12711yescharacteristic, complemented
254C24—2541no—
255#41—5081no—
256#41—5084no—
257#41—2542no—
258C2×Q8—1271no—
259C2×Q8—2541no—
260C2×D4—1271no—
261C2×D4—1271no—
262C2×D4—1271no—
263C2×D4—5081no—
264C2×D4—2542no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4D50811yescharacteristic
271C22×C4—2541no—
272C22×C4—2541no—
273C22×C4—2541no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4C22×D12711yescharacteristic
277C22×C4C22×D12711yescharacteristic
278SD16—5081no—
279D8—5081no—
280D8—2542no—
281M16—41no—
282C2×C8—41no—
283C2×C8C22×D12711yescharacteristic
284C2×C8C22×D12711yescharacteristic
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⋊C4D50811yescharacteristic
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C22⋊C4—2541no—
300C22⋊C4—2541no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4D50811yescharacteristic
305C22⋊C4D50811yescharacteristic
306C42C22×D12711yescharacteristic
307C23—10161no—
308C23—5082no—
309C23—5082no—
310C23—2541no—
311C23—2541no—
312C23—2541no—
313C23C2×D50811yescharacteristic
314C23C2×D50811yescharacteristic
315Q8—5082no—
316Q8—2541no—
317Q8—2541no—
318D4—22no—
319D4—1272no—
320D4—10161no—
321D4—5082no—
322D4—5082no—
323D4—5082no—
324D4—5082no—
325D4—2541no—
326D4—2541no—
327D4—2541no—
328D4—2541no—
329D4—2541no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—10161no—
335C2×C4—5082no—
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×C4C2×D50811yescharacteristic
345C2×C4C2×D50811yescharacteristic
346C2×C4C2×D50811yescharacteristic
347C2×C4C2×D50811yescharacteristic
348C2×C4C23×D12711yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—10164no—
359C22—5081no—
360C22—5081no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—5081no—
365C22—5081no—
366C22—5081no—
367C22—2542no—
368C22#1686123712yes—
369C22C22×D50811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—5081no—
375C4—5081no—
376C4—5081no—
377C4—5081no—
378C4—5081no—
379C4#41×D12711yescharacteristic
380C4#1686125411yescharacteristic
381C2—22no—
382C2—41no—
383C2—10161no—
384C2—10161no—
385C2—5082no—
386C2#2255052111yescharacteristic
387C2C2×#1686123711yescharacteristic
388C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#435×C127C2
socleC22×C127C22×D4
hypercenter#435D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #22550530C2semidirect
2. #22550522C2semidirect
3. #22550236C2semidirect
4. #22550207C2semidirect
5. #22550195C2semidirect
6. #22550167C2semidirect
7. #22550129C2semidirect
8. #22550105C2semidirect
9. #22550092C2semidirect
10. #22550023C2semidirect
11. #22550012C2semidirect
12. #22549989C2semidirect
13. #22549878C2semidirect
14. C2×#16861131C2semidirect
15. #435×C127C2semidirect
19. C2×#14523237C22semidirect
22. #16861209C22semidirect
31. #16861183C22semidirect
35. C4×D508C22semidirect
37. C2×C4⋊Dic127C22semidirect
39. #16861141C22semidirect
40. #16861141C22semidirect
41. #16861141C22semidirect
42. #16861134C22semidirect
43. #16861134C22semidirect
44. #16861134C22semidirect
51. #16861131C22semidirect
52. C2×M16×C127C22semidirect
55. C4⋊C8×C127C22semidirect
57. C22⋊C8×C127C22semidirect
58. #97×C127C22semidirect
127. D508#41semidirect
195. C127#3306semidirect
208. #435D127semidirect
217. C2×M16C2×D127semidirect
240. C4⋊C8C2×D127semidirect
252. C22⋊C8C2×D127semidirect
253. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505302
#225505222
#225502362
#225502072
#225501952
#225501672
#225501292
#225501052
#225500922
#225500232
#225500122
#225499892
#225498782
C2×#168611312
#435×C1272
#3306127